# Active Calculus

## AppendixCAnswers to Selected Exercises

This appendix contains answers to all non-WeBWorK exercises in the text. For WeBWorK exercises, please use the HTML version of the text for access to answers and solutions.

### 1Understanding the Derivative1.1How do we measure velocity?1.1.4Exercises

#### 1.1.4.1.Average velocity from position.

$$2.5\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}$$
$$3.5\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s}}$$

#### 1.1.4.2.Rate of calorie consumption.

$$162\ {\rm cal}$$
$$\text{The 170 ... one hour}$$
$$\text{Increase}$$

#### 1.1.4.3.Average rate of change - quadratic function.

$$-1$$
$$-4$$
$$-3$$
$$\text{D}$$
$$\text{C}$$
$$\text{B}$$

#### 1.1.4.4.Comparing average rate of change of two functions.

$$\text{g}$$
$$\text{g}$$
$$\text{f}$$
$$\text{f}$$
$$\text{g}$$

#### 1.1.4.5.Matching a distance graph to velocity.

$$3$$

#### 1.1.4.6.

1. $$s(15)-s(0) \approx -98.75\text{.}$$
2. \begin{align*} AV_{[0,15]} &= \frac{s(15)-s(0)}{15-0} \approx -6.58\\ AV_{[0,2]} &= \frac{s(2)-s(0)}{2-0} \approx -47.63\\ AV_{[1,6]} &= \frac{s(6)-s(1)}{6-1} \approx -13.25\\ AV_{[8,10]} &= \frac{s(10)-s(8)}{10-8} \approx -7.35 \end{align*}
3. Most negative average velocity on $$[0,4]\text{;}$$ most positive average velocity on $$[4,8]\text{.}$$
4. $$\frac{21.31+22.25}{2} = 21.78$$ feet per second.
5. The average velocities are negative; the instantaneous velocity was positive. Downward motion corresponds to negative average velocity; upward motion to positive average velocity.

#### 1.1.4.7.

1. Sketch a plot where the diver’s height at time $$t$$ is on the vertical axis. For instance, $$h(2.45) = 0\text{.}$$
2. $$AV_{[2.45,7]} \approx \frac{-3.5-0}{7-2.45}=\frac{-3.5}{4.55}=-0.7692$$ m/sec. The average velocity is not the same on every time interval within $$[2.45,7]\text{.}$$
3. When the diver is going upward, her velocity is positive. When she is going downward, her velocity is negative. At the peak of her dive and when her feet touch the bottom of the pool.
4. It looks like when the position function is steep, the velocity function’s value is farther away from zero, and that whenever the height/position function is rising/increasing, the velocity function has a positive value. Similarly, whenever the position function is decreasing, the velocity is negative.

#### 1.1.4.8.

1. $$15 957$$ people.
2. In an average year the population grew by about $$798$$ people/year.
3. The slope of a secant line through the points $$(a,f(a))$$ and $$(b,f(b))\text{.}$$
4. $$AV_{[0,20]} \approx 798$$ people per year.
5. \begin{align*} AV_{[5,10]} & \approx 734.50\\ AV_{[5,9]} & \approx 733.06\\ AV_{[5,8]} & \approx 731.62\\ AV_{[5,7]} & \approx 730.19\\ AV_{[5,6]} & \approx 728.7535 \end{align*}

### 1.2The notion of limit1.2.4Exercises

#### 1.2.4.1.Limits on a piecewise graph.

$$6$$
$$7$$
$$\text{none}$$
$$-2$$

#### 1.2.4.2.Estimating a limit numerically.

$$6$$

#### 1.2.4.3.Limits for a piecewise formula.

$$3\cdot 4+0$$
$$4\cdot 4-4$$
$$4\cdot 4-4$$

#### 1.2.4.4.Evaluating a limit algebraically.

$$-14$$

#### 1.2.4.5.

1. All real numbers except $$x = \pm 2\text{.}$$
2.  $$x$$ $$f(x)$$ $$2.1$$ $$-8.41$$ $$2.01$$ $$-8.0401$$ $$2.001$$ $$-8.004001$$ $$1.999$$ $$-7.996001$$ $$1.99$$ $$-7.9601$$ $$1.9$$ $$-7.61$$
$$\lim_{x \to 2} f(x) = -8\text{.}$$
3. $$\lim_{x \to 2} \frac{16-x^4}{x^2-4} = -8\text{.}$$
4. False.
5. False.

#### 1.2.4.6.

1. All real numbers except $$x = -3\text{.}$$
2.  $$x$$ $$g(x)$$ $$-2.9$$ $$-1$$ $$-2.99$$ $$-1$$ $$-2.999$$ $$-1$$ $$-3.001$$ $$1$$ $$-3.01$$ $$-1$$ $$-3.1$$ $$-1$$
The limit does not exist.
3. If $$x \gt -3\text{,}$$
\begin{equation*} -\frac{|x+3|}{x+3} = -\frac{x+3}{x+3} = -1; \end{equation*}
if $$x \lt -3\text{,}$$ it follows that
\begin{equation*} -\frac{|x+3|}{x+3} = -\frac{-(x+3)}{x+3} = +1\text{.} \end{equation*}
Hence the limit does not exist.
4. False.
5. False.

#### 1.2.4.8.

1. \begin{equation*} AV_{[1,1+h]} = \frac{100\cos(0.75(1+h)) \cdot e^{-0.2(1+h)} - 100\cos(0.75) \cdot e^{-0.2}}{h} \end{equation*}
2. \begin{equation*} \lim_{h \to 0} AV_{1, 1+h} \approx -53.837\text{.} \end{equation*}
3. The instantaneous velocity of the bungee jumper at the moment $$t = 1$$ is approximately $$-53.837$$ ft/sec.

### 1.3The derivative of a function at a point1.3.3Exercises

#### 1.3.3.1.Estimating derivative values graphically.

$$0.905829$$
$$-0.5$$
$$3.22474$$
$$0.355567$$
$$1.75$$

#### 1.3.3.2.Tangent line to a curve.

$$6.8$$
$$2$$
$$6.8$$
$$\frac{-1\cdot 0.05}{0.06}$$

#### 1.3.3.3.Interpreting values and slopes from a graph.

$${\verb!<!}$$
$${\verb!>!}$$
$${\verb!<!}$$
$${\verb!<!}$$

#### 1.3.3.4.Finding an exact derivative value algebraically.

$$30$$

#### 1.3.3.5.Estimating a derivative from the limit definition.

$$387$$

#### 1.3.3.6.

1. $$AV_{[-3,-1]} \approx 1.15\text{;}$$ $$AV_{[0,2]} \approx -0.4\text{.}$$
2. $$f'(-3) \approx 3\text{;}$$ $$f'(0) \approx -\frac{1}{2}\text{.}$$

#### 1.3.3.7.

1. For instance, you could let $$f(-3) = 3$$ and have $$f$$ pass through the points $$(-3,3)\text{,}$$ $$(-1,-2)\text{,}$$ $$(0,-3)\text{,}$$ $$(1,-2)\text{,}$$ and $$(3,-1)$$ and draw the desired tangent lines accordingly.
2. For instance, you could draw a function $$g$$ that passes through the points $$(-2,3)\text{,}$$ $$(-1,2)\text{,}$$ $$(1,0)\text{,}$$ $$(2,0)\text{,}$$ and $$(3,3)$$ in such a way that the tangent line at $$(-1,2)$$ is horizontal and the tangent line at $$(2,0)$$ has slope $$1\text{.}$$

#### 1.3.3.8.

1. $$AV_{[0,7]}=\frac{0.1175}{7} \approx 0.01679$$ billion people per year; $$P'(7) \approx 0.1762$$ billion people per year; $$P'(7) \gt AV_{[0,7]}\text{.}$$
2. $$AV_{[19,29]} \approx 0.02234$$ billion people/year.
3. We will say that today’s date is July 1, 2015, which means that $$t = 22.5\text{;}$$
\begin{equation*} P'(22.5) = \lim_{h \to 0} \frac{115(1.014)^{22.5+h}-115(1.014)^{22.5}}{h}; \end{equation*}
$$P'(22.5) \approx 0.02186$$ billions of people per year.
4. $$y - 1.57236 = 0.02186(t-22.5)\text{.}$$

#### 1.3.3.9.

1. All three approaches show that $$f'(2) = 1\text{.}$$
2. All three approaches show that $$f'(1) = -1\text{.}$$
3. All three approaches show that $$f'(1) = \frac{1}{2}\text{.}$$
4. All three approaches show that $$f'(1)$$ does not exist.
5. The first two approaches show that $$f'(\frac{\pi}{2}) = 0\text{.}$$

### 1.4The derivative function1.4.3Exercises

#### 1.4.3.1.The derivative function graphically.

$$-2$$
$$1$$
$$0$$
$$2$$

#### 1.4.3.2.Applying the limit definition of the derivative.

$$4\!\left(x+h\right)^{2}-8-\left(4x^{2}-8\right)$$
$$4\cdot 2x$$

#### 1.4.3.3.Sketching the derivative.

$$\text{3}$$

#### 1.4.3.4.Comparing function and derivative values.

$$\text{x4}$$
$$\text{x6}$$
$$\text{x3}$$
$$\text{x5}$$

#### 1.4.3.5.Limit definition of the derivative for a rational function.

$$-1$$
$$-1$$
$$\frac{-1}{4}$$
$$\frac{-1}{16}$$

#### 1.4.3.6.

1. See the figure below.
2. See the figure below.
3. One example of a formula for $$f$$ is $$f(x) = \frac{1}{2}x^2 - 1\text{.}$$

#### 1.4.3.7.

1. $$g'(x) = 2x - 1\text{.}$$
2. $$p'(x) = 10x - 4\text{.}$$
3. The constants $$3$$ and $$12$$ don’t seem to affect the results at all. The coefficient $$-4$$ on the linear term in $$p(x)$$ appears to make the $$-4$$’’ appear in $$p'(x)= 10x - 4\text{.}$$ The leading coefficient $$5$$ in $$(x) = 5x^2 - 4x + 12$$ leads to the coefficient of $$10$$’’ in $$p'(x) = 10x -4\text{.}$$

#### 1.4.3.8.

1. $$g$$ is linear.
2. On $$-3.5 \lt x \lt -2\text{,}$$ $$-2 \lt x \lt 0$$ and $$2 \lt x \lt 3.5\text{.}$$
3. At $$x = -2, 0, 2\text{;}$$ $$g$$ must have sharp corners at these points.

### 1.5Interpreting, estimating, and using the derivative1.5.4Exercises

#### 1.5.4.1.A cooling cup of coffee.

$$\text{negative}$$
$${\text{degC/min}}$$
$$35$$
$$\text{temperature}$$
$$68\ {\rm degC}$$
$$\text{decrease}$$
$$0.75\ {\rm degC}$$

#### 1.5.4.2.A cost function.

$$\text{gallons}$$
$$\text{dollars}$$
$$\text{gallons}$$
$${\text{dollars/gallon}}$$

#### 1.5.4.3.Weight as a function of calories.

$$\text{cal}$$
$$\text{lb}$$
$$\text{cal}$$
$${\text{lb/cal}}$$
$$\text{lb}$$
$$\text{cal}$$
$${\text{lb/cal}}$$
$$2800$$
$$0.02$$

#### 1.5.4.4.Displacement and velocity.

$$2$$
$$2.5$$
$$4$$
$$3.5$$
$$3$$

#### 1.5.4.5.

1. $$F'(10) \approx -3.33592\text{.}$$
2. The coffee’s temperature is decreasing at about $$3.33592$$ degrees per minute.
3. $$F'(20)\text{.}$$
4. We expect $$F'$$ to get closer and closer to $$0$$ as time goes on.

#### 1.5.4.6.

1. If a patient takes a dose of $$50$$ ml of a drug, the patient will experience a body temperature change of $$0.75$$ degrees F.
2. degrees Fahrenheit per milliliter.’’
3. For a patient taking a $$50$$ ml dose, adding one more ml to the dose leads us to expect a temperature change that is about $$0.02$$ degrees less than the temperature change induced by a $$50$$ ml dose.

#### 1.5.4.7.

1. $$t=0\text{.}$$
2. $$v'(1) = -32\text{.}$$
3. feet per second per second’’; $$v'(1) = -32$$ tells us that the ball’s velocity is decreasing at a rate of 32 feet per second per second.
4. The acceleration of the ball.

#### 1.5.4.8.

1. $$AV_{[40000,55000]} \approx -0.153$$ dollars per mile.
2. $$h'(55000) \approx -0.147$$ dollars per mile. During $$55 0001$$st mile, we expect the car’s value to drop by $$0.147$$ dollars.
3. $$h'(30000) \lt h'(80000)\text{.}$$
4. The graph of $$h$$ might have the general shape of the graph of $$y = e^{-x}$$ for positive values of $$x\text{:}$$ always positive, always decreasing, and bending upwards while tending to $$0$$ as $$x$$ increases.

### 1.6The second derivative1.6.5Exercises

#### 1.6.5.1.Comparing $$f, f', f''$$ values.

$$\text{positive}$$
$$\text{positive}$$
$$\text{negative}$$

#### 1.6.5.2.Signs of $$f, f', f''$$ values.

$$\text{zero}$$
$$\text{negative}$$
$$\text{zero}$$
$$\text{positive}$$
$$\text{positive}$$
$$\text{negative}$$
$$\text{zero}$$
$$\text{positive}$$
$$\text{zero}$$
$$\text{negative}$$
$$\text{positive}$$
$$\text{positive}$$
$$\text{positive}$$
$$\text{negative}$$
$$\text{negative}$$

#### 1.6.5.3.Acceleration from velocity.

$$32.05\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s^{2}}}$$
$$23.5\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s^{2}}}$$

#### 1.6.5.4.Rates of change of stock values.

$$\text{negative}$$
$$\text{positive}$$
$$\text{negative}$$
$$\text{positive}$$

#### 1.6.5.5.Interpreting a graph of $$f'$$.

$$\text{x6}$$
$$\text{x1}$$
$$\text{x3}$$
$$\text{x5}$$
$$\text{x6}$$
$$\text{x4}$$

#### 1.6.5.6.

1. $$f$$ is increasing and concave down near $$x=2\text{.}$$
2. Greater.
3. Less.

#### 1.6.5.7.

1. $$g'(2) \approx 1.4\text{.}$$
2. At most one.
3. $$9\text{.}$$
4. $$g''(2) \approx 5.5 \text{.}$$

#### 1.6.5.8.

1. $$h'(4.5) \approx 14.3\text{;}$$ $$h'(5) \approx 21.2\text{;}$$ $$h'(5.5) \approx = 23.9\text{;}$$ rising most rapidly at $$t = 5.5\text{.}$$
2. $$h'(5) \approx 9.6 \text{.}$$
3. Acceleration of the bungee jumper in feet per second per second.
4. $$0 \lt t \lt 2\text{,}$$ $$6 \lt t \lt 10\text{.}$$

### 1.7Limits, Continuity, and Differentiability1.7.5Exercises

#### 1.7.5.1.Limit values of a piecewise graph.

$$-9$$
$$-10$$
$$\text{none}$$
$$17$$

#### 1.7.5.2.Limit values of a piecewise formula.

$$3\cdot 1-2$$
$$3\cdot 1-2$$
$$3\cdot 1-2$$

#### 1.7.5.3.Continuity and differentiability of a graph.

$$\text{none}$$
$$2, 4, 6$$

#### 1.7.5.4.Continuity of a piecewise formula.

$$5\cdot 2$$

#### 1.7.5.5.

1. $$a = 0\text{.}$$
2. $$a = 0, 3\text{.}$$
3. $$a = -2, 0, 1, 2, 3\text{.}$$

#### 1.7.5.6.

1. $$f(x) = |x-2|\text{.}$$
2. Impossible.
3. Let $$f$$ be the function defined to be $$f(x) = 1$$ for every value of $$x \ne -2\text{,}$$ and such that $$f(-2) = 4\text{.}$$

#### 1.7.5.7.

1. $$h$$ must be piecewise linear with slope of $$1$$ or $$-1\text{,}$$ depending on the interval.
2. $$h'(x)$$ is not defined for $$x = -2, 0, 2\text{.}$$
3. It is possible that $$h$$ is not continuous at $$x = -2, 0, 2\text{.}$$
4. Two of the many possible graphs for $$h$$ are shown in the following figure.

#### 1.7.5.8.

1. At $$x = 0\text{.}$$
\begin{align*} g'(0) & = \lim_{h \to 0} \frac{g(0+h) - g(0)}{h}\\ & = \lim_{h \to 0} \frac{\sqrt{|h|} - \sqrt{|0|}}{h}\\ & = \lim_{h \to 0} \frac{\sqrt{|h|}}{h} \end{align*}
2.  $$h$$ $$0.1$$ $$0.01$$ $$0.001$$ $$0.0001$$ $$-0.1$$ $$-0.01$$ $$-0.001$$ $$-0.0001$$ $$\sqrt{|h|}/h$$ $$3.162$$ $$10$$ $$31.62$$ $$100$$ $$-3.162$$ $$-10$$ $$-31.62$$ $$-100$$
$$g'(0)$$ does not exist.

### 1.8The Tangent Line Approximation1.8.4Exercises

#### 1.8.4.1.Approximating $$\sqrt{x}$$.

$$0.0833333333333333$$
$$3$$
$$6.00833333333333$$

#### 1.8.4.2.Local linearization of a graph.

$$3$$
$$8$$
$$8.8$$
$$\text{under}$$

#### 1.8.4.3.Estimating with the local linearization.

$$41.5$$

#### 1.8.4.4.Predicting behavior from the local linearization.

$$\text{negative}$$
$${\text{degC/min}}$$
$$30$$
$$\text{temperature}$$
$$51\ {\rm degC}$$
$$\text{decrease}$$
$$1.125\ {\rm degC}$$

#### 1.8.4.5.

1. $$p(3) = -1$$ and $$p'(3) = -2\text{.}$$
2. $$p(2.79) \approx -0.58\text{.}$$
3. Too large.

#### 1.8.4.6.

1. $$F'(60)\approx 1.56$$ degrees per minute.
2. $$L(t) \approx 1.56(t-60)+324.5\text{.}$$
3. $$F(63)\approx L(63)\approx = 329.18$$ degrees F.
4. Overestimate.

#### 1.8.4.7.

1. $$s(9.34) \approx L(9.34) = 3.592\text{.}$$
2. underestimate.
3. The object is slowing down as it moves toward toward its starting position at $$t=4\text{.}$$

#### 1.8.4.8.

1. $$x=1\text{.}$$
2. On $$-0.37 \lt x \lt 1.37\text{;}$$ $$f$$ is concave up.
3. $$f(1.88) \approx -3.0022\text{,}$$ and this estimate is larger than the true value of $$f(1.88)\text{.}$$

### 2Computing Derivatives2.1Elementary derivative rules2.1.5Exercises

#### 2.1.5.1.Derivative of a power function.

$$\frac{15}{16}x^{\frac{15}{16}-1}$$

#### 2.1.5.2.Derivative of a rational function.

$$-1\cdot 19x^{-1\cdot \left(19+1\right)}$$

#### 2.1.5.3.Derivative of a root function.

$$\frac{1}{2}x^{\frac{1}{2}-1}$$

#### 2.1.5.4.Derivative of a quadratic.

$$2\cdot 3t-7$$

#### 2.1.5.5.Derivative of a sum of power functions.

$$6\cdot 6t^{6-1}-\frac{9}{2}t^{\frac{-1}{2}}-\frac{7}{t^{2}}$$

#### 2.1.5.6.Simplifying a product before differentiating.

$$\frac{2\cdot 3+1}{2}x^{\frac{2\cdot 3-1}{2}}+\frac{9}{2}x^{\frac{-1}{2}}$$

#### 2.1.5.7.Simplifying a quotient before differentiating.

$$\frac{6x^{5}x-\left(x^{6}+9\right)}{x^{2}}$$

#### 2.1.5.8.Finding a tangent line equation.

$$84\!\left(x-3\right)+76$$

#### 2.1.5.9.Determining where $$f'(x) = 0$$.

$$8, -12$$

#### 2.1.5.10.

1. $$h(2) = 27\text{;}$$ $$h'(2) = -19/2\text{.}$$
2. $$L(x) = 27 - \frac{19}{2}(x-2)\text{.}$$
3. $$p$$ is increasing at $$x=2\text{.}$$
4. $$p(2.03) \approx -11.44\text{.}$$

#### 2.1.5.11.

1. $$p$$ is not differentiable at $$x=-1$$ and $$x=1\text{;}$$ $$q$$ is not differentiable at $$x=-1$$ and $$x=1\text{.}$$
2. $$r$$ is not differentiable at $$x=-1$$ and $$x=1\text{.}$$
3. $$r'(-2) = 4\text{;}$$ $$r'(0) = \frac{1}{2}\text{.}$$
4. $$y = 4\text{.}$$

#### 2.1.5.12.

1. $$w'(t) = 3t^t(\ln(t) + 1) + 2\frac{1}{\sqrt{1-t^2}}\text{.}$$
2. $$L(t) = (\frac{3}{\sqrt{2}} - \frac{2\pi}{3}) + (\frac{3}{\sqrt{2}}(\ln(\frac{1}{2}) + 1) + \frac{4}{\sqrt{3}})(t-\frac{1}{2})\text{.}$$
3. $$v$$ is decreasing at $$t = \frac{1}{2}\text{.}$$

#### 2.1.5.13.

1. \begin{align*} f'(x) &= \lim_{h \to 0} \frac{a^{x+h}-a^x}{h}\\ &= \lim_{h \to 0} \frac{a^{x} \cdot a^{h} - a^x}{h}\\ &= \lim_{h \to 0} \frac{a^{x}(a^{h}-1)}{h}\text{,} \end{align*}
2. Since $$a^x$$ does not depend at all on $$h\text{,}$$ we may treat $$a^x$$ as constant in the noted limit and thus write the value $$a^x$$ in front of the limit being taken.
3. When $$a = 2\text{,}$$ $$L \approx 0.6931\text{;}$$ when $$a = 3\text{,}$$ $$L \approx 1.0986\text{.}$$
4. $$a \approx 2.71828$$ (for which $$L \approx 1.000$$)
5. $$\frac{d}{dx}[2^x] = 2^x \cdot \ln(2)$$ and $$\frac{d}{dx}[3^x] = 3^x \cdot \ln(3)$$
6. \begin{equation*} \frac{d}{dx}[e^x] = e^x\text{.} \end{equation*}

### 2.2The sine and cosine functions2.2.3Exercises

#### 2.2.3.1.

1. $$V'(2) = 24 \cdot 1.07^2 \cdot \ln(1.07) + 6 \cos(2) \approx -0.63778$$ thousands of dollars per year.
2. $$V''(2) = 24 \cdot 1.07^2 \cdot \ln(1.07)^2 - 6 \sin(2) \approx -5.33$$ thousands of dollars per year per year. At this moment, $$V'$$ is decreasing and we expect the derivative’s value to decrease by about $$5.33$$ thousand dollars per year over the course of the next year.
3. See the figure below. Adding the term $$6\sin(t)$$ to $$A$$ to create the function $$V$$ adds volatility to the value of the portfolio.

#### 2.2.3.2.

1. $$f'\left(\frac{\pi}{4}\right) = -5\left(\frac{\sqrt{2}}{2}\right)\text{.}$$
2. $$L(x) = 3+2(x-\pi)\text{.}$$
3. Decreasing.
4. The tangent line to $$f$$ lies above the curve at this point.

#### 2.2.3.3.

1. Hint: in the numerator of the difference quotient, combine the first and last terms and remove a factor of $$\sin(x)\text{.}$$
2. Hint: divide each part of the numerator by $$h$$ and consider the sum of two separate limits.
3. $$\lim_{h \to 0} \left( \frac{\cos(h)-1}{h} \right) = 0$$ and $$\lim_{h \to 0} \left( \frac{\sin(h)}{h} \right) = 1 \text{.}$$
4. $$f'(x) = \sin(x) \cdot 0 + \cos(x) \cdot 1\text{.}$$
5. Hint: $$\cos(\alpha + \beta)$$ is $$\cos(\alpha + \beta) = \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta)\text{.}$$

### 2.3The product and quotient rules2.3.5Exercises

#### 2.3.5.1.Derivative of a basic product.

$$13^{x}+x\ln\!\left(13\right)\cdot 13^{x}$$

#### 2.3.5.2.Derivative of a product.

$$\left(9x^{9-1}-\frac{1}{3}x^{-1+\frac{1}{3}}\right)\cdot 2^{x}+\left(x^{9}-x^{\frac{1}{3}}\right)\ln\!\left(2\right)\cdot 2^{x}$$

#### 2.3.5.3.Derivative of a quotient of linear functions.

$$\frac{2\!\left(8t+7\right)-8\!\left(2t+7\right)}{\left(8t+7\right)^{2}}$$

#### 2.3.5.4.Derivative of a rational function.

$$\frac{3r^{3-1}\!\left(9r+13\right)-9r^{3}}{\left(9r+13\right)^{2}}$$

#### 2.3.5.5.Derivative of a product of trigonometric functions.

$$6\!\left(-1\sin\!\left(q\right)\sin\!\left(q\right)+\cos\!\left(q\right)\cos\!\left(q\right)\right)$$

#### 2.3.5.6.Derivative of a product of power and trigonmetric functions.

$$5x^{5-1}\cos\!\left(x\right)-x^{5}\sin\!\left(x\right)$$

#### 2.3.5.7.Derivative of a sum that involves a product.

$$\sin\!\left(t\right)+t\cos\!\left(t\right)+\frac{1}{\cos^{2}\!\left(t\right)}$$

#### 2.3.5.8.Product and quotient rules with graphs.

$$0.666667\cdot 1.33333+0\cdot 1.33333$$
$$\frac{-4\cdot \left(-1\right)--1\cdot \left(-2\right)}{\left(-1\right)^{2}}$$

#### 2.3.5.9.Product and quotient rules with given function values.

$$4\cdot 5+5\cdot 4$$
$$\frac{5\cdot 4-4\cdot 5}{4\cdot 4}$$

#### 2.3.5.10.

1. $$h(2) = -15\text{;}$$ $$h'(2) = 23/2\text{.}$$
2. $$L(x) = -15 + 23/2(x-2)\text{.}$$
3. Increasing.
4. $$r(2.06) \approx -0.5796\text{.}$$

#### 2.3.5.11.

1. $$w'(t) = t^t\left(\ln t+1\right)\cdot\left(\arccos t\right)+t^t\cdot\frac{-1}{\sqrt{1-t^2}} \text{.}$$
2. $$L(t) \approx 0.740-0.589(t-0.5)\text{.}$$
3. Increasing.

#### 2.3.5.12.

1. $$r'(-2) = 5$$ and $$r'(0) = 1\text{.}$$
2. At $$x = -1$$ and $$x = 1\text{.}$$
3. $$L(x) = 2\text{.}$$
4. $$z'(0) = -\frac{1}{4}$$ and $$z'(2) = -1\text{.}$$
5. At $$x = -1\text{,}$$ $$x = 1\text{,}$$ $$x = -1.5\text{,}$$ and $$x = 1\text{.}$$

#### 2.3.5.13.

1. $$C(t) = A(t)Y(t)$$ bushels in year $$t\text{.}$$
2. $$1 190 000$$ bushels of corn.
3. $$C'(t) = A(t)Y'(t) + A'(t)Y(t)\text{.}$$
4. $$C'(0) = 158 000$$ bushels per year.
5. $$C(1) \approx 1 348 000$$bushels.

#### 2.3.5.14.

1. $$g(80) = 20$$ kilometers per liter, and $$g'(80) = -0.16\text{.}$$ kilometers per liter per kilometer per hour.
2. $$h(80) = 4$$ liters per hour and $$h'(80) = 0.082$$ liters per hour per kilometer per hour.
3. Think carefully about units and how each of the three pairs of values expresses fundamentally the same facts.

### 2.4Derivatives of other trigonometric functions2.4.3Exercises

#### 2.4.3.1.A sum and product involving $$\tan(x)$$.

$$\tan\!\left(t\right)+t\frac{1}{\cos^{2}\!\left(t\right)}+-1\sin\!\left(t\right)$$

#### 2.4.3.2.A quotient involving $$\tan(t)$$.

$$\frac{5\sec^{2}\!\left(x\right)x-5\tan\!\left(x\right)}{x^{2}}$$
$$2.56387$$

#### 2.4.3.3.A quotient of trigonometric functions.

$$\frac{\sec^{2}\!\left(x\right)\sec\!\left(x\right)-\left(\tan\!\left(x\right)-2\right)\sec\!\left(x\right)\tan\!\left(x\right)}{\sec^{2}\!\left(x\right)}$$
$$2.22324$$

#### 2.4.3.4.A quotient that involves a product.

$$\frac{\left(2\cdot 2x\tan\!\left(x\right)+2x^{2}\sec^{2}\!\left(x\right)\right)\sec\!\left(x\right)-2x^{2}\tan\!\left(x\right)\sec\!\left(x\right)\tan\!\left(x\right)}{\sec^{2}\!\left(x\right)}$$
$$-33.0254$$

#### 2.4.3.5.Finding a tangent line equation.

$$3\cdot 2$$
$$3-6\cdot 0.785398$$

#### 2.4.3.6.

1. $$h'(2) = \frac{-2\sin(2) - 2\cos(2) \ln(1.2)}{1.2^2} \approx -1.1575$$ feet per second.
2. $$h''(2) = \frac{\cos(2)(-2 + 2ln^2(1.2)) + 4\ln(1.2)\sin(2))}{1.2^2} \approx 1.0193$$ feet per second per second.
3. The object is falling and slowing down.

#### 2.4.3.7.

1. $$f'(x) = \sin(x) \cdot (-\csc^2(x)) + \cot(x) \cdot \cos(x)\text{.}$$
2. False.
3. $$f'(x) = \frac{-\sin^2(x)}{\sin(x)} = -\sin(x)$$ for $$x \ne \frac{\pi}{2} + k\pi$$ for some integer value of $$k\text{.}$$

#### 2.4.3.8.

1. $$\displaystyle p'(z) = \frac{\left(z^2\sec(z) +1 \right)\left(z\sec^2(z)+\tan(z)\right) - z\tan(z) \left(z^2\sec(z)\tan(z)+2z\sec(z)\right)}{\left(z^2\sec(z) + 1\right)^2} +3e^z$$
2. $$y - 4 = 3(x-0)\text{.}$$
3. Increasing.

### 2.5The chain rule2.5.5Exercises

#### 2.5.5.1.Mixing rules: chain, product, sum.

$$5e^{5x}\!\left(x^{2}+7^{x}\right)+e^{5x}\!\left(2x+\ln\!\left(7\right)\cdot 7^{x}\right)$$

#### 2.5.5.2.Mixing rules: chain and product.

$$6t^{6-1}e^{-1ct}-ct^{6}e^{-1ct}$$

#### 2.5.5.3.Using the chain rule repeatedly.

$$\frac{-1\cdot 5te^{-1\cdot 5t^{2}}}{\sqrt{e^{-1\cdot 5t^{2}}+9}}$$

#### 2.5.5.4.Derivative involving arbitrary constants $$a$$ and $$b$$.

$$ae^{-1bx+12}-abxe^{-1bx+12}$$

#### 2.5.5.5.Chain rule with graphs.

$$-0.5$$
$$-0.5$$
$$-0.5$$

#### 2.5.5.6.Chain rule with function values.

$$4$$
$$6\cdot 6$$
$$3$$
$$4\cdot 5$$
$$\frac{5\cdot 5-1\cdot 6}{5^{2}}$$

#### 2.5.5.7.A product involving a composite function.

$$2\sin\!\left(6x\right)+2\cdot 6x\cos\!\left(6x\right)$$

#### 2.5.5.8.

1. $$h'\left( \frac{\pi}{4} \right) = \frac{3}{2\sqrt{2}}\text{.}$$
2. $$r'(0.25) = \cos(0.25^3) \cdot 3(0.25)^2 \approx 0.1875 \gt h'(0.25) = 3\sin^2(0.25) \cdot \cos(0.25) \approx 0.1779\text{;}$$ $$r$$ is changing more rapidly.
3. $$h'(x)$$ is periodic; $$r'(x)$$ is not.

#### 2.5.5.9.

1. $$p'(x) = e^{u(x)} \cdot u'(x)\text{.}$$
2. $$q'(x) = u'(e^x) \cdot e^x\text{.}$$
3. $$r'(x) = -\csc^2(u(x) \cdot u'(x)\text{.}$$
4. $$s'(x) = u'(\cot(x)) \cdot (-\csc^2(x))\text{.}$$
5. $$a'(x) = u'(x^4) \cdot 4x^3\text{.}$$
6. $$b'(x) = 4(u(x))^3 \cdot u'(x)\text{.}$$

#### 2.5.5.10.

1. $$C'(0) = 0$$ and $$C'(3) = -\frac{1}{2}\text{.}$$
2. Consider $$C'(1)\text{.}$$ By the chain rule, we’d expect that $$C'(1) = p'(q(1)) \cdot q'(1)\text{,}$$ but we know that $$q'(1)$$ does not exist since $$q$$ has a corner point at $$x = 1\text{.}$$ This means that $$C'(1)$$ does not exist either.
3. Since $$Y(x) = q(q(x))\text{,}$$ the chain rule implies that $$Y'(x) = q'(q(x)) \cdot q'(x)\text{,}$$ and thus $$Y'(-2) = q'(q(-2)) \cdot q'(-2) = q'(-1) \cdot q'(-2)\text{.}$$ But $$q'(-1)$$ does not exist, so $$Y'(-2)$$ also fails to exist. Using $$Z(x) = q(p(x))$$ and the chain rule, we have $$Z'(x) = q'(p(x)) \cdot p'(x)\text{.}$$ Therefore $$Z'(0) = q'(p(0)) \cdot p'(0) = q'(-0.5) \cdot p'(0) = 0 \cdot 0.5 = 0\text{.}$$

#### 2.5.5.11.

1. $$\left. \frac{dV}{dh} \right|_{h=1} = 7 \pi$$ cubic feet per foot.
2. $$h'(2) = \pi \cos(2\pi) = \pi$$ feet per hour.
3. $$\left. \frac{dV}{dt} \right|_{t=2} = 7 \pi^2$$ cubic feet per hour.
4. In (a) we are determining the instantaneous rate at which the volume changes as we increase the height of the water in the tank, while in (c) we are finding the instantaneous rate at which volume changes as we increase time.

### 2.6Derivatives of Inverse Functions2.6.6Exercises

#### 2.6.6.1.Composite function involving logarithms and polynomials.

$$\frac{3t^{2}}{t^{3}+3}$$

#### 2.6.6.2.Composite function involving trigonometric functions and logarithms.

$$\frac{-1\sin\!\left(\ln\!\left(t\right)\right)}{t}$$

#### 2.6.6.3.Product involving $$\arcsin(w)$$.

$$5\sin^{-1}\!\left(w\right)+\frac{5w}{\sqrt{1-w^{2}}}$$

#### 2.6.6.4.Derivative involving $$\arctan(x)$$.

$$0$$

#### 2.6.6.5.Composite function from a graph.

$$\frac{5\cdot 6^{5-1}\!\left(6.2-6\right)}{2.1-2}$$
$$\frac{2.1-2}{6.2-6}$$

#### 2.6.6.6.Composite function involving an inverse trigonometric function.

$$7\frac{1}{\sqrt{1-\left(x^{3}\right)^{2}}}\cdot 3x^{2}$$

#### 2.6.6.7.Mixing rules: product, chain, and inverse trig.

$$8\cdot 4x^{4-1}\tan^{-1}\!\left(3x^{3}\right)+\frac{8x^{4}\cdot 3\cdot 3x^{3-1}}{1+3\cdot 3x^{2\cdot 3}}$$

#### 2.6.6.8.Mixing rules: product and inverse trig.

$$8\cos\!\left(x\right)\frac{1}{\sqrt{1-x^{2}}}-8\sin\!\left(x\right)\sin^{-1}\!\left(x\right)$$

#### 2.6.6.9.

1. $$f'(x) = \frac{1}{2\arctan(x) + 3\arcsin(x) + 5} \cdot \left(\frac{2}{1+x^2} + \frac{3}{\sqrt{1-x^2}}\right)\text{.}$$
2. $$r'(z) = \frac{1}{1+\left(\ln(\arcsin(z))\right)^2} \cdot \left( \frac{1}{\arcsin(z)} \right) \cdot \frac{1}{\sqrt{1-z^2}}\text{.}$$
3. $$q'(t) = \arctan^2(3t) \cdot \left[4\arcsin^3(7t) \left( \frac{7}{\sqrt{1-(7t)^2}} \right)\right] + \arcsin^4(7t) \cdot \left[2\arctan(3t) \left(\frac{3}{1+(3t)^2}\right) \right]\text{.}$$
4. $$\displaystyle g'(v) = \frac{1}{\frac{\arctan(v)}{\arcsin(v) + v^2}} \cdot \frac{(\arcsin(v) + v^2) \cdot \frac{1}{1+v^2} - \arctan(v) \cdot \left(\frac{1}{\sqrt{1-v^2}} + 2v \right)}{(\arcsin(v) + v^2)^2}$$

#### 2.6.6.10.

1. $$f'(1) \approx 2\text{.}$$
2. $$(f^{-1})'(-1) \approx 1/2\text{.}$$

#### 2.6.6.11.

1. $$f$$ passes the horizontal line test.
2. $$f^{-1}(x) = g(x) = \sqrt[3]{4x-16}\text{.}$$
3. $$f'(x) = \frac{3}{4}x^2\text{;}$$ $$f'(2) = 3\text{.}$$ $$g'(x) = \frac{1}{3}(4x-16)^{-2/3} \cdot 4\text{;}$$ $$g'(6) = \frac{1}{3}\text{.}$$ These two derivative values are reciprocals.

#### 2.6.6.12.

1. $$h$$ passes the horizontal line test.
2. The equation $$y = x + \sin(x)$$ can’t be solved for $$x$$ in terms of $$y\text{.}$$
3. $$(h^{-1})'(\frac{\pi}{2} + 1) = 1\text{.}$$

### 2.7Derivatives of Functions Given Implicitly2.7.3Exercises

#### 2.7.3.1.Implicit differentiaion in a polynomial equation.

$$\frac{1-2xy}{x^{2}-5}$$

#### 2.7.3.2.Implicit differentiation in an equation with logarithms.

$$\frac{y\!\left(3-x\ln\!\left(y\right)\right)}{x\!\left(x+3y^{3}\right)}$$

#### 2.7.3.3.Implicit differentiation in an equation with inverse trigonometric functions.

$$\frac{3x^{3-1}y-y^{3}-x^{2\cdot 3}y^{3+2}}{3xy^{3-1}+3x^{2\cdot 3+1}y^{3+1}-x^{3}}$$

#### 2.7.3.4.Slope of the tangent line to an implicit curve.

$$-{\frac{8}{11}}$$

#### 2.7.3.5.Equation of the tangent line to an implicit curve.

$$x+10y = 14$$

#### 2.7.3.6.

Horizontal tangent lines: $$(0,-1)\text{,}$$ $$(0,-0.618)\text{,}$$ $$(0,1.618)\text{,}$$ $$(1,-1)\text{,}$$ $$(1,-0.618)\text{,}$$ $$(1,1.618)\text{,}$$ $$(0.5,-1.0493)\text{,}$$ $$(0.5,0.2104)\text{,}$$ $$(0.5, 1.6139)\text{.}$$ Vertical tangent lines: $$(-0.1756,-0.379)\text{,}$$ $$(0.2912,-0.379)\text{,}$$ $$(0.7088,-0.379)\text{,}$$ $$(1.1756,-0.379)\text{,}$$ $$(-0.8437, 1.235)\text{,}$$ and $$(1.8437, 1.235)\text{.}$$

#### 2.7.3.7.

$$y = \frac{\pi}{2} - \left(x-\frac{\pi}{2}\right)\text{.}$$

#### 2.7.3.8.

1. $$x = \frac{\ln(y)}{\ln(a)}\text{.}$$
2. $$1 = \frac{1}{\ln(a)} \cdot \frac{1}{y}\frac{dy}{dx}\text{.}$$
3. $$\frac{d}{dx}[a^x] = a^x \ln(a)\text{.}$$

### 2.8Using Derivatives to Evaluate Limits2.8.4Exercises

#### 2.8.4.1.L’Hôpital’s Rule with graphs.

$$\text{zero}$$
$$\text{zero}$$

#### 2.8.4.2.L’Hôpital’s Rule to evaluate a limit.

$$\frac{1}{32}$$

#### 2.8.4.3.Determining if L’Hôpital’s Rule applies.

$$5.44444444444445$$
$$1.12467028923762$$

#### 2.8.4.4.Using L’Hôpital’s Rule multiple times.

$$0$$

#### 2.8.4.5.

$$\lim_{x \to 3} h(x) = -2\text{.}$$

#### 2.8.4.6.

Horizontal asymptote: $$y = \frac{3}{5}\text{;}$$ vertical asymptote: $$x = c\text{;}$$ hole: $$(a, \frac{3(a-b)}{5(a-c)})\text{.}$$ $$R$$ is not continuous at $$x = a$$ and $$x = c\text{.}$$

#### 2.8.4.7.

1. $$\ln(x^{2x}) = 2x \cdot \ln(x)\text{.}$$
2. $$x = \frac{1}{\frac{1}{x}}\text{.}$$
3. $$\lim_{x \to 0^+} h(x) = 0\text{.}$$
4. $$\lim_{x \to 0^+} g(x) = \lim_{x \to 0^+} x^{2x} = 1\text{.}$$

#### 2.8.4.8.

1. Show that $$\lim_{x \to \infty}\frac{\ln(x)}{\sqrt{x}} = 0\text{.}$$
2. Show that $$\lim_{x \to \infty}\frac{\ln(x)}{\sqrt[n]{x}} = 0\text{.}$$
3. Consider $$\lim_{x \to \infty} \frac{p(x)}{e^x}$$ By repeated application of LHR, the numerator will eventually be simply a constant (after $$n$$ applications of LHR), and thus with $$e^x$$ still in the denominator, the overall limit will be $$0\text{.}$$
4. Show that $$\lim_{x \to \infty} \frac{\ln(x)}{x^n} = 0$$
5. For example, $$f(x) = 3x^2 + 1$$ and $$g(x) = -0.5x^2 + 5x - 2\text{.}$$

### 3Using Derivatives3.1Using derivatives to identify extreme values3.1.4Exercises

#### 3.1.4.1.Finding critical points and inflection points.

$$0$$
$$-0.707106781186547, 0.707106781186547$$
$$0$$
$$-0.707106781186547, 0.707106781186547$$

#### 3.1.4.2.Finding inflection points.

$$-7, 0.125$$

A
B
C

#### 3.1.4.4.

1. $$f'$$ is positive for $$-1 \lt x lt 1$$ and for $$x \gt 1\text{;}$$ $$f'$$ is negative for all $$x \lt -1\text{.}$$ $$f$$ has a local minimum at $$x = -1\text{.}$$
2. A possible graph of $$y = f''(x)$$ is shown at right in the figure.
3. $$f''(x)$$ is negative for $$-0.35 \lt x \lt 1\text{;}$$ $$f''(x)$$ is positive everywhere else; $$f$$ has points of inflection at $$x \approx -0.35$$ and $$x = 1\text{.}$$
4. A possible graph of $$y = f(x)$$ is shown at left in the figure.

#### 3.1.4.5.

1. Neither.
2. $$g''(2) = 0\text{;}$$ $$g''$$ is negative for $$1 \lt x \lt 2$$ and positive for $$2 \lt x \lt 3\text{.}$$
3. $$g$$ has a point of inflection at $$x = 2\text{.}$$

#### 3.1.4.6.

1. $$h$$ can have no, one, or two real zeros.
2. One root is negative and the other positive.
3. $$h$$ will look like a line with slope $$3\text{.}$$
4. $$h$$ is concave up everywhere; $$h$$ is almost linear for large values of $$|x|\text{.}$$

#### 3.1.4.7.

1. $$p''(x)$$ is negative for $$-1 \lt x \lt 2$$ and positive for all other values of $$x\text{;}$$ $$p$$ has points of inflection at $$x = -1$$ and $$x = 2\text{.}$$
2. Local maximum.
3. Neither.

### 3.2Using derivatives to describe families of functions3.2.3Exercises

#### 3.2.3.1.Drug dosage with a parameter.

$$C$$
$$\frac{C}{2}$$

#### 3.2.3.2.Using the graph of $$g'$$.

$$0$$
$$2$$
$$-2$$
$$\left(-\infty ,-8\right)$$
<

#### 3.2.3.3.

1. $$x = 0$$ and $$x = \frac{2a}{3}\text{.}$$
2. $$x = \frac{a}{3}\text{;}$$ $$p''(x)$$ changes sign from negative to positive at $$x = \frac{a}{3}\text{.}$$
3. As we increase the value of $$a\text{,}$$ both the location of the critical number and the inflection point move to the right along with $$a\text{.}$$

#### 3.2.3.4.

1. $$x=c$$ is a vertical asymptote because $$\lim_{x \to c^+} \frac{e^{-x}}{x-c} = \infty$$ and $$\lim_{x \to c^-} \frac{e^{-x}}{x-c} = -\infty\text{.}$$
2. $$\lim_{x \to \infty} \frac{e^{-x}}{x-c} = 0\text{;}$$ $$\lim_{x \to -\infty} \frac{e^{-x}}{x-c} = -\infty\text{.}$$
3. The only critical number for $$q$$ is $$x=c-1\text{.}$$
4. When $$x \lt c-1\text{,}$$ $$q'(x) \gt 0\text{;}$$ when $$x \gt c-1\text{,}$$ $$q'(x) \lt 0\text{;}$$ $$q$$ has a local maximum at $$x = c-1\text{.}$$

#### 3.2.3.5.

1. $$x = m\text{.}$$
2. $$E$$ is increasing for $$x \lt m$$ and decreasing for $$x \gt m\text{,}$$ with a local maximum at $$x = m\text{.}$$
3. $$x = m \pm s\text{.}$$
4. $$\lim_{x \to \infty} E(x) = \lim_{x \to -\infty} E(x) = 0\text{.}$$

### 3.3Global Optimization3.3.4Exercises

#### 3.3.4.1.

1. Not enough information is given.
2. Global minimum at $$x = b\text{.}$$
3. Global minimum at $$x = a\text{;}$$ global maximum at $$x = b\text{.}$$
4. Not enough information is provided.

#### 3.3.4.2.

1. Absolute maximum $$p(0) = p(a) = 0\text{;}$$ absolute minimum $$p\left( \frac{a}{\sqrt{3}} \right) = -\frac{2a^3}{3\sqrt{3}}\text{.}$$
2. Absolute max $$r\left( \frac{1}{b} \right) \approx 0.368 \frac{a}{b}\text{;}$$ absolute min $$r\left( \frac{2}{b} \right) \approx 0.270 \frac{a}{b}\text{.}$$
3. Absolute minimum $$g(b) = a(1-e^{-b^2})\text{;}$$ absolute maximum $$g(3b) = a(1-e^{-3b^2})\text{.}$$
4. Absolute max $$s\left( \frac{\pi}{2k} \right) = 1\text{;}$$ absolute min $$s\left( \frac{5\pi}{6k} \right) = \frac{1}{2}\text{.}$$

#### 3.3.4.3.

1. Global maximum at $$x=a\text{;}$$ global minimum at $$x=b\text{.}$$
2. Global maximum at $$x=c\text{;}$$ global minimum at either $$x=a$$ or $$x=b\text{.}$$
3. Global minimum at $$x=a$$ and $$x=b\text{;}$$ global maximum somewhere in $$(a,b)\text{.}$$
4. Global minimum at $$x=c\text{;}$$ global maximum value at $$x = a\text{.}$$

#### 3.3.4.4.

1. Absolute max $$s(\frac{5\pi}{12}) = 8\text{;}$$ absolute min $$s(\frac{11\pi}{12}) = 2\text{.}$$
2. Absolute max $$s(\frac{5\pi}{12}) = 8\text{;}$$ absolute min $$s(0) = 5 - \frac{3\sqrt{3}}{2} \approx 2.402\text{.}$$
3. Absolute max $$s(\frac{5\pi}{12}) = 8\text{;}$$ absolute min $$s(\frac{11\pi}{12}) = 2\text{.}$$ (There are other points at which the function achieves these values on the given interval.)
4. Absolute max $$s(\frac{5\pi}{12}) = 8\text{;}$$ absolute min $$s(\frac{5\pi}{6}) \approx 2.402\text{.}$$

### 3.4Applied Optimization3.4.3Exercises

#### 3.4.3.1.Maximizing the volume of a box.

$$13.8735\times5.8735$$
$$2.06325$$

#### 3.4.3.2.Minimizing the cost of a container.

$$\418.28$$

#### 3.4.3.3.Maximizing area contained by a fence.

$$6864.29\ {\rm ft^{2}}$$

#### 3.4.3.4.Minimizing the area of a poster.

$$29.0587\ {\rm cm}$$
$$38.745\ {\rm cm}$$

#### 3.4.3.5.Maximizing the area of a rectangle.

$$1.15$$
$$0.66667$$

#### 3.4.3.6.

The absolute maximum volume is $$V\left( \sqrt{\frac{5}{3}} \right) = \frac{15}{12}\left( \sqrt{\frac{5}{3}} \right) - \frac{1}{4}\left( \sqrt{\frac{5}{3}} \right)^3 \approx 1.07583$$ cubic feet.

#### 3.4.3.7.

The maximum possible area that each of the four pens can enclose is 390625 square feet.

#### 3.4.3.8.

$$172.047$$ feet of cable.

#### 3.4.3.9.

The minimum cost is \$1165.70.

### 3.5Related Rates3.5.3Exercises

#### 3.5.3.1.Height of a conical pile of gravel.

$$\frac{40}{529\pi }$$

#### 3.5.3.2.Movement of a shadow.

$$11.1429$$

#### 3.5.3.3.A leaking conical tank.

$$207953.1076$$

#### 3.5.3.4.

The boat is approaching the dock at a rate of $$\frac{13}{6} \approx 2.167$$ feet per second.

#### 3.5.3.5.

The depth of the water is increasing at
\begin{equation*} \left. \frac{dh}{dt}\right|_{h = 5} = 1.28 \end{equation*}
feet per minute. The depth of the water is increasing at a decreasing rate.

#### 3.5.3.6.

$$\left. \frac{d\theta}{dt} \right|_{x = 30} = -0.24$$ radians per second.

#### 3.5.3.7.

$$\left. \frac{dh}{dt}\right|_{V=1000} = \frac{10}{\pi \left( \sqrt[3]{\frac{3000}{\pi}} \right)^2} \approx 0.0328$$ feet per minute.

### 4The Definite Integral4.1Determining distance traveled from velocity4.1.5Exercises

#### 4.1.5.1.Estimating distance traveled from velocity data.

$$126\ {\rm ft}$$
$$306\ {\rm ft}$$

#### 4.1.5.2.Distance from a linear velocity function.

$$550\ {\rm m}$$

#### 4.1.5.3.Change in position from a linear velocity function.

$$16\ {\rm cm}$$

#### 4.1.5.5.Finding average acceleration from velocity data.

$$34.09\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s^{2}}}$$
$$25\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut s^{2}}}$$

#### 4.1.5.6.Change in position from a quadratic velocity function.

$$26.8333333333334$$

#### 4.1.5.7.

1. At time $$t = 1\text{,}$$ $$12$$ miles north of the lake.
2. $$s(2) - s(0) = 1$$ mile north of the lake.
3. $$40$$ miles.

#### 4.1.5.8.

1. $$t = \frac{500}{32} = \frac{125}{8} = 15.625$$ is when the rocket reaches its maximum height.
2. $$A = 3906.25\text{,}$$ the vertical distance traveled on $$[0, 15.625]\text{.}$$
3. $$s(t) = 500t - 16t^2\text{.}$$
4. $$s(15.625) - s(0) = 3906.25$$ is the change of the rocket’s position on $$[0,15.625]\text{.}$$
5. $$s(5) - s(1) = 1616\text{;}$$ the rocket rose $$1616$$ feet on $$[1,5]\text{.}$$

#### 4.1.5.9.

1. $$\frac{1}{2} + \frac{1}{4} \pi \approx 1.285\text{.}$$
2. $$s(5)-s(2) = -2$$ is the change in position of the object on $$[2,5]\text{.}$$
3. On the time interval $$[5,7]\text{.}$$
4. $$s$$ is increasing on the intervals $$(0,2)$$ and $$(5,7)\text{;}$$ the position function has a relative maximum at $$t=2\text{.}$$

#### 4.1.5.10.

1. Think about the product of the units involved: units of pollution per day’’ times days’’. Connect this to the area of a thin vertical rectangle whose height is given by the curve.
2. An underestimate is $$336$$ units of pollution.

### 4.2Riemann Sums4.2.5Exercises

#### 4.2.5.1.Evaluating Riemann sums for a quadratic function.

$$14.125$$
$$\text{there is ambiguity}$$
$$13.125$$
$$\text{there is ambiguity}$$

#### 4.2.5.2.Estimating distance traveled with a Riemann sum from data.

$$2\ {\rm ft}$$
$$20\ {\rm ft}$$
$$10\ {\rm ft}$$
$$20\ {\rm ft}$$

#### 4.2.5.3.Writing basic Riemann sums.

$$e^{0}\cdot 0.5$$
$$e^{0.5}\cdot 0.5$$
$$e^{0.5}\cdot 0.5$$
$$e^{1}\cdot 0.5$$

#### 4.2.5.4.

1. $$M_4 = 43.5\text{.}$$
2. $$A = \frac{87}{2}\text{.}$$
3. The rectangles with heights that come from the midpoint have the same area as the trapezoids that are formed by the function values at the two endpoints of each subinterval.
$$M_n$$ will give the exact area for any value of $$n\text{.}$$ Neither $$L_n$$ nor $$R_n$$ will be exact for any $$n\text{.}$$
4. For any linear function $$g$$ of the form $$g(x) = mx + b$$ such that $$g(x) \ge 0$$ on the interval of interest.

#### 4.2.5.5.

1. $$f(x) = x^2 + 1$$ on the interval $$[1,3]\text{.}$$
2. If $$S$$ is a left Riemann sum, $$f(x) = x^2 + 1$$ on the interval $$[1.4, 3.4]\text{.}$$ If $$S$$ is a middle Riemann sum, $$f(x) = x^2 + 1$$ on the interval $$[1.2, 3.2]\text{.}$$
3. The area under $$f(x) = x^2 + 1$$ on $$[1,3]\text{.}$$
4. $$R_{10} = \sum_{i=1}^{10} \left( (1+0.2i)^2 + 1 \right) \cdot 0.2\text{.}$$

#### 4.2.5.6.

1. $$M_3 = 99.6$$ feet.
2. $$L_6 = 114 \text{,}$$
\begin{equation*} R_6 = 84\text{,} \end{equation*}
and $$\frac{1}{2}(L_6 + R_6) = 99\text{.}$$
3. $$114$$ feet.

#### 4.2.5.7.

1. $$M_4 \approx 6.4\text{.}$$
2. The total tonnage of pollution escaping the scrubbing process in the time interval $$[0,4]$$ weeks.
3. $$L_5 \approx 5.19620599\text{.}$$
4. $$6.4$$ tons.

### 4.3The Definite Integral4.3.5Exercises

#### 4.3.5.1.Evaluating definite integrals from graphical information.

$$-18$$
$$6$$
$$-12$$
$$24$$

#### 4.3.5.2.Estimating definite integrals from a graph.

$$-3$$
$$-A$$

#### 4.3.5.3.Finding the average value of a linear function.

$$39.5$$

#### 4.3.5.4.Finding the average value of a function given graphically.

$$\frac{3}{8}$$
$$\frac{1}{4}$$
$$0$$

#### 4.3.5.5.Estimating a definite integral and average value from a graph.

$$-5$$

#### 4.3.5.6.Using rules to combine known integral values.

$$10-8-10$$
$$-10\cdot \left(-8\right)+8\cdot \left(-6\right)-8\cdot \left(-7.5\right)$$

#### 4.3.5.7.

1. The total change in position is $$P = \int_0^{4} v(t) dt\text{.}$$
2. $$P = -2.625$$ feet.
3. $$D = 3.375$$ feet.
4. $$AV = -0.65625$$ feet per second.
5. $$s(t) = -t^2+t\text{.}$$

#### 4.3.5.8.

1. The total change in position, $$P\text{,}$$ is $$P = \int_0^1 v(t) \, dt + \int_1^3 v(t) \, dt + \int_3^4 v(t) \, dt = \int_0^4 v(t) \, dt\text{.}$$
2. $$P = \int_0^4 v(t) \, dt \approx 2.665\text{.}$$
3. The total distance traveled, $$D\text{,}$$ is $$D = \int_0^1 v(t) \, dt - \int_1^3 v(t) \, dt + \int_3^4 v(t) \, dt\text{.}$$
4. $$D \approx 8.00016\text{.}$$
5. \begin{equation*} v_{\operatorname{AVG} [0,4]} \approx 0.66625 \end{equation*}
feet per second.

#### 4.3.5.9.

1. $$\int_0^1 [f(x) + g(x)] \,dx = 1-\frac{\pi}{4}\text{.}$$
2. $$\int_1^4 [2f(x) - 3g(x)] \, dx = -\frac{15}{2} - 3\pi\text{.}$$
3. $$h_{\operatorname{AVG} [0,4]} = \frac{5}{8} + \frac{3\pi}{16}\text{.}$$
4. $$c = -\frac{3}{8} + \frac{3\pi}{16}\text{.}$$

#### 4.3.5.10.

1. $$A_1 = \int_{-1}^{1} (3-x^2) \, dx\text{.}$$
2. $$A_2 = \int_{-1}^{1} 2x^2 \, dx\text{.}$$
3. The exact area between the two curves is $$\int_{-1}^{1} (3-x^2) \, dx - \int_{-1}^{1} 2x^2 \, dx\text{.}$$
4. Use the sum rule for definite integrals over the same interval.
5. Think about subtracting the area under $$q$$ from the area under $$p\text{.}$$

### 4.4The Fundamental Theorem of Calculus4.4.5Exercises

#### 4.4.5.1.Finding exact displacement.

$$-12$$

#### 4.4.5.2.Evaluating the definite integral of a rational function.

$$\frac{-1}{4}+\frac{1}{2}$$

#### 4.4.5.3.Evaluating the definite integral of a linear function.

$$224$$

#### 4.4.5.4.Evaluating the definite integral of a quadratic function.

$$288$$

#### 4.4.5.5.Simplifying an integrand before integrating.

$$535.957$$

#### 4.4.5.6.Evaluating the definite integral of a trigonometric function.

$$16$$

#### 4.4.5.7.

1. $$20$$ meters.
2. $$\displaystyle v_{\operatorname{AVG} [12,24]} = 12.5$$ meters per minute.
3. The object’s maximum acceleration is $$3$$ meters per minute per minute at the instant $$t = 2\text{.}$$
4. $$c = 5\text{.}$$

#### 4.4.5.8.

1. $$-\frac{5}{6}\text{.}$$
2. $$\displaystyle f_{\operatorname{AVG} [0,5]} = \frac{1}{2}\text{.}$$
3. $$g(x) = f(x)$$ for $$0 \le x \lt 5$$ and $$g(x) = -\frac{5}{4}(x - 5)$$ on $$5 \le x \le 7\text{.}$$

#### 4.4.5.9.

1.  $$h$$ (feet) $$0$$ $$1000$$ $$2000$$ $$3000$$ $$4000$$ $$5000$$ $$6000$$ $$7000$$ $$8000$$ $$9000$$ $$10{,}000$$ $$c$$ (ft/min) $$925$$ $$875$$ $$830$$ $$780$$ $$730$$ $$685$$ $$635$$ $$585$$ $$535$$ $$490$$ $$440$$ $$m$$ (min/ft) $$\frac{1}{925}$$ $$\frac{1}{875}$$ $$\frac{1}{830}$$ $$\frac{1}{780}$$ $$\frac{1}{730}$$ $$\frac{1}{685}$$ $$\frac{1}{635}$$ $$\frac{1}{585}$$ $$\frac{1}{535}$$ $$\frac{1}{490}$$ $$\frac{1}{440}$$
2. The antiderivative function tells the total number of minutes it takes for the plane to climb to an altitude of $$h$$ feet.
3. $$M = \int_{0}^{10000} m(h) \, dh \text{.}$$
4. It takes the plane aabout $$M_5 \approx 15.27$$ minutes.

Yes.

#### 4.4.5.11.

1. $$G'(x) = \frac{1}{-x} \cdot (-1) = \frac{1}{x}\text{.}$$
2. Since for those values of $$x\text{,}$$ $$G'(x) = \frac{1}{x}\text{.}$$
3. If $$x \lt 0\text{,}$$ then $$H(x) = \ln(|x|) = \ln(-x) = G(x)\text{;}$$ if $$x \gt 0\text{,}$$ then $$H(x) = \ln(|x|) = \ln(x) = F(x)\text{.}$$
4. For $$x \lt 0\text{,}$$ $$H'(x) = G'(x) = \frac{1}{x}\text{;}$$ for $$x \gt 0\text{,}$$ $$H'(x) = F'(x) = \frac{1}{x}$$ for all $$x \ne 0\text{.}$$

### 5Evaluating Integrals5.1Constructing Accurate Graphs of Antiderivatives5.1.5Exercises

#### 5.1.5.1.

$$1$$
$$1.5$$
$$1$$
$$0.166666666666667$$
$$-0.333333333333333$$
$$-0.5$$

#### 5.1.5.2.

$$-2$$
$$4$$
$$6$$

#### 5.1.5.3.

$$1, 2.25$$

F
A
B
D
A
C

#### 5.1.5.5.

1. $$s(1) = \frac{5}{3}\text{,}$$ $$s(3) = -1\text{,}$$ $$s(5) = -\frac{11}{3}\text{,}$$ $$s(6) = -\frac{5}{2}\text{.}$$
2. $$s$$ is increasing on $$0 \lt t \lt 1$$ and $$5 \lt t \lt 6\text{;}$$ decreasing for $$1 \lt t \lt 5\text{.}$$
3. $$s$$ is concave down for $$t \lt 2\text{;}$$ concave up for $$t \gt 2\text{.}$$
4. $$s(t) = -2t + \frac{1}{6}(t-3)^3 + 5\text{.}$$

#### 5.1.5.6.

1. $$C$$ measures the total number of calories burned in the workout since $$t = 0\text{.}$$
2. $$C(5) = 12.5\text{,}$$ $$C(10) = 50\text{,}$$ $$C(15) = 125\text{,}$$ $$C(20) = 187.5\text{,}$$ $$C(25) = 237.5\text{,}$$ $$C(30) = 262.5\text{.}$$
3. $$C(t) = 12.5 + 7.5(t-5)$$ on this interval.

#### 5.1.5.7.

1. $$B(-1) = -1\text{,}$$ $$B(0) = 0\text{,}$$ $$B(1) = \frac{1}{2}\text{,}$$ $$B(2) = 0\text{,}$$ $$B(3) = -1\text{,}$$ $$B(4) = -\frac{3}{2}\text{,}$$ $$B(5) = -1\text{,}$$ $$B(6) = 0\text{.}$$ Also, $$A(x) = 1+B(x)$$ and $$C(x) = B(x) - \frac{1}{2}\text{.}$$
 $$x$$ $$-1$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$A(x)$$ $$0$$ $$1$$ $$1.5$$ $$1$$ $$0$$ $$-0.5$$ $$0$$ $$1$$ $$B(x)$$ $$-1$$ $$0$$ $$0.5$$ $$0$$ $$-1$$ $$-1.5$$ $$-1$$ $$0$$ $$C(x)$$ $$-1.5$$ $$-0.5$$ $$0$$ $$-0.5$$ $$-1.5$$ $$-2$$ $$-1.5$$ $$-0.5$$
2. $$A\text{,}$$ $$B\text{,}$$ and $$C$$ are vertical translations of each other.
3. $$A' = f\text{.}$$

### 5.2The Second Fundamental Theorem of Calculus5.2.5Exercises

#### 5.2.5.1.

$$2$$
$$5$$

#### 5.2.5.2.

$$x^{6}$$
$$15625$$

#### 5.2.5.3.

$$0$$
$$2.00876$$
$$\left(1.65,4.59\right)$$
$$6.22504$$

#### 5.2.5.4.

$$\left(-\infty ,\frac{-1}{2\cdot 3}\right)$$

#### 5.2.5.5.

$$F$$ is increasing on $$x \lt -1\text{,}$$ $$0.5 \lt x \lt 4\text{,}$$ and $$5 \lt x \lt 6.5\text{;}$$ decreasing on $$-1 \lt x \lt 0.5$$ and $$4 \lt x \lt 5\text{;}$$ concave up on approximately $$-0.4 \lt x \lt 2$$ and $$4.5 \lt x \lt 6\text{;}$$ concave down on approximately $$2 \lt x \lt 4.5$$ and $$x \gt 6\text{;}$$ $$F(2) = 0\text{;}$$ $$F(0.5) = -6.06\text{;}$$ $$F(-1) = -1.77\text{;}$$ $$F(4) = 6.69\text{;}$$ $$F(5) = 6.33\text{;}$$ $$F(6.5) = 8.12\text{.}$$

#### 5.2.5.6.

1. The total sand removed on this time interval is
\begin{equation*} \int_0^6 \left[2 + 5\sin \left( \frac{4\pi t}{25} \right) \right] \, dt\text{.} \end{equation*}
2. The total amount of sand on the beach at time $$x$$ is given by
\begin{equation*} Y(x) = \int_0^x \left[ S(t) - R(t) \right] \, dt = \int_0^x \left[ \frac{15t}{1+3t} - \left( 2 + 5\sin \left( \frac{4\pi t}{25} \right) \right) \right] \, dt\text{.} \end{equation*}
3. $$Y'(4) = S(4) - R(4) \approx -1.90875$$ cubic yards per hour.
4. $$Y$$ has an absolute minimum on $$[0,6]$$ of $$Y(5.118) \approx 2492.368\text{.}$$

#### 5.2.5.7.

1. $$m(h) = \frac{1}{c(h)}\text{.}$$
 $$h$$ (feet) $$0$$ $$1000$$ $$2000$$ $$3000$$ $$4000$$ $$5000$$ $$6000$$ $$7000$$ $$8000$$ $$9000$$ $$10{,}000$$ $$c$$ (ft/min) $$925$$ $$875$$ $$830$$ $$780$$ $$730$$ $$685$$ $$635$$ $$585$$ $$535$$ $$490$$ $$440$$ $$m$$ (min/ft) $$\frac{1}{925}$$ $$\frac{1}{875}$$ $$\frac{1}{830}$$ $$\frac{1}{780}$$ $$\frac{1}{730}$$ $$\frac{1}{685}$$ $$\frac{1}{635}$$ $$\frac{1}{585}$$ $$\frac{1}{535}$$ $$\frac{1}{490}$$ $$\frac{1}{440}$$
2. The antiderivative function tells us the total number of minutes that it takes for the plane to climb to an altitude of $$h$$ feet.
3. The number of minutes required for the airplane to ascend to $$10{,}000$$ feet of altitude is given by the definite integral
\begin{equation*} M = \int_{0}^{10000} m(h) \, dh\text{.} \end{equation*}
4. The number of minutes required for the airplane to ascend to $$h$$ feet of altitude is given by the definite integral
\begin{equation*} M(h) = \int_{0}^{h} m(t) \, dt\text{.} \end{equation*}
5. Estimating the desired integral using $$3$$ subintervals and midpoints,
\begin{equation*} \int_{0}^{6000} m(h) \, dh \approx 7.77\text{.} \end{equation*}
Using $$5$$ subintervals and midpoints,
\begin{equation*} \int_{0}^{10000} m(h) \, dh \approx 15.27\text{.} \end{equation*}

### 5.3Integration by Substitution5.3.5Exercises

#### 5.3.5.1.

$$\sec\!\left(2x\right)$$
$$2\sec\!\left(2x\right)\tan\!\left(2x\right)dx$$
$$-\frac{1.5}{u^{2}}du$$
$$1.5u^{-1}+C$$
$$1.5\!\left(\sec\!\left(2x\right)\right)^{-1}+C$$

#### 5.3.5.2.

$$\cot\!\left(-3x\right)$$
$$3\csc^{2}\!\left(-3x\right)dx$$
$$2.33333e^{u}du$$
$$2.33333e^{u}+C$$
$$2.33333e^{\cot\!\left(-3x\right)}+C$$

#### 5.3.5.3.

$$\frac{\left(t^{3}-1\right)^{3}}{9}+C$$

#### 5.3.5.4.

$$1.5\sin\!\left(x^{4}\right)+C$$

#### 5.3.5.5.

$$0.166667\ln^{6}\!\left(z\right)+C$$

#### 5.3.5.6.

$$0.25\ln\!\left(e^{4x}+9\right)+C$$

#### 5.3.5.7.

$$-1.71828182845905$$

#### 5.3.5.8.

$$7x^{3+2}+7x^{2}$$
$$1.16667x^{6}+2.33333x^{3}+C$$
$$2.33333w$$
$$1.16667\!\left(x^{3}+1\right)^{2}+C$$
$$1.16666666666667$$

#### 5.3.5.9.

$$\frac{-1.66667\!\left(8\sin\!\left(t\right)\cos^{5}\!\left(t\right)+12\cos^{4}\!\left(t\right)-8\sin\!\left(t\right)\cos^{3}\!\left(t\right)-12\cos^{2}\!\left(t\right)-14\cos^{6}\!\left(t\right)-6\sin\!\left(t\right)\cos\!\left(t\right)-1\right)}{\cos^{6}\!\left(t\right)}+C+c$$

#### 5.3.5.10.

$$0.5\ln^{2}\!\left(\sin\!\left(x\right)\right)+C+c$$

#### 5.3.5.11.

1. $$\int \tan(x) \, dx = \ln\left(|\sec(x)|\right) + C\text{.}$$
2. $$\int \cot(x) \, dx = -\ln\left(|\csc(x)|\right) + C\text{.}$$
3. $$\int \frac{\sec^2(x) + \sec(x) \tan(x)}{\sec(x) + \tan(x)} \, dx = \ln\left(|\sec(x) + \tan(x)|\right) + C\text{.}$$
4. $$\frac{\sec^2(x) + \sec(x) \tan(x)}{\sec(x) + \tan(x)} = \sec(x)\text{.}$$
5. $$\int \sec(x) \, dx = \ln\left(|\sec(x) + \tan(x)|\right) + C\text{.}$$
6. $$\int \csc(x) \, dx = -\ln\left(|\csc(x) + \cot(x)|\right) + C\text{.}$$

#### 5.3.5.12.

1. $$\int x \sqrt{x-1} \, dx = \int (u+1) \sqrt{u} \, du\text{.}$$
2. $$\int x \sqrt{x-1} \, dx = \frac{2}{5} (x-1)^{5/2} + \frac{2}{3} (x-1)^{3/2} + C\text{.}$$
3. $$\int x^2 \sqrt{x-1} \, dx = \frac{2}{7} (x-1)^{7/2} + \frac{4}{5} (x-1)^{5/2} + \frac{2}{3} (x-1)^{3/2} + C\text{.}$$
$$\int x \sqrt{x^2 - 1} \, dx = \frac{1}{3} (x^2-1)^{3/2} + C\text{.}$$

#### 5.3.5.13.

1. We don’t have a function-derivative pair.
2. $$\sin^3(x) = \sin(x) (1-\cos^2(x))\text{.}$$
3. $$u = \cos(x)$$ and $$du = -\sin(x) \, dx\text{.}$$
4. $$\int \sin^3(x) \, dx = \frac{1}{3}\cos^3(x) - \cos(x) + C\text{.}$$
5. $$\int \cos^3(x) \, dx = \sin(x) - \frac{1}{3}\sin^3(x) + C\text{.}$$

#### 5.3.5.14.

1. The model is reasonable because it appears to be periodic and the rate of consumption seems to peak at the times of day where people are most active in their homes.
2. The total energy consumed in $$24$$ hours, measured in megawatt-hours.
3. $$\int_0^{24} r(t) \, dt \approx 95.7809$$ megawatt-hours of power used in $$24$$ hours.
4. $$\displaystyle r_{\operatorname{AVG} [0,24]} \approx 3.99087$$ megawatt-hours.

### 5.4Integration by Parts5.4.7Exercises

#### 5.4.7.1.

$$x$$
$$\cos\!\left(4x\right)dx$$
$$dx$$
$$0.25\sin\!\left(4x\right)$$
$$x\cdot 0.25\sin\!\left(4x\right)$$
$$0.25\sin\!\left(4x\right)dx$$
$$0.25x\sin\!\left(4x\right)+0.0625\cos\!\left(4x\right)+C$$

#### 5.4.7.2.

$$x$$
$$\sec^{2}\!\left(5x\right)dx$$
$$dx$$
$$0.2\tan\!\left(5x\right)$$
$$x\cdot 0.2\tan\!\left(5x\right)$$
$$0.2\tan\!\left(5x\right)dx$$
$$x\cdot 0.2\tan\!\left(5x\right)+0.04\ln\!\left(\cos\!\left(5x\right)\right)+C$$

#### 5.4.7.3.

$$\ln\!\left(-6x\right)$$
$$x^{8}dx$$
$$\frac{1}{x}dx$$
$$0.111111x^{9}$$
$$\ln\!\left(-6x\right)\cdot 0.111111x^{9}$$
$$0.111111x^{9}\frac{1}{x}dx$$
$$\ln\!\left(-6x\right)\cdot 0.111111x^{9}-0.0123457x^{9}+C$$

#### 5.4.7.4.

$$0.959572318005487$$

#### 5.4.7.5.

$$0.8y\!\left(y+5\right)^{1.25}-0.355556\!\left(y+5\right)^{2.25}+C$$

#### 5.4.7.6.

$$\left(\frac{1+z}{7}-0.0204082\right)e^{7z}+C$$

#### 5.4.7.7.

$$\frac{x^{3}}{3}\tan^{-1}\!\left(7x\right)-\frac{x^{2}}{6\cdot 7}+\frac{1}{6\cdot 343}\ln\!\left(1+49x^{2}\right)$$

#### 5.4.7.8.

$$5\!\left(-t^{2}\cos\!\left(t\right)+2t\sin\!\left(t\right)+2\cos\!\left(t\right)\right)+C$$

#### 5.4.7.9.

$$x\ln^{2}\!\left(2x\right)-2x\ln\!\left(2x\right)+2x$$

#### 5.4.7.10.

$$5\cdot 5--3--7$$

#### 5.4.7.11.

$$\frac{1}{4}x^{4}\sin\!\left(x^{4}\right)+\frac{1}{4}\cos\!\left(x^{4}\right)$$

#### 5.4.7.13.

1. $$F'(x) = xe^{-2x}\text{.}$$
2. $$\displaystyle F(x) = -\frac{1}{2}xe^{-2x} - \frac{1}{4}e^{-2x} + \frac{1}{4}$$
3. Increasing.

#### 5.4.7.14.

1. $$\int e^{2x} \cos(e^x) \, dx = \int z \cos(z) \, dz\text{.}$$
2. $$\int e^{2x} \cos(e^x) \, dx = e^x \sin(e^x) + \cos(e^x) + C\text{.}$$
3. $$\displaystyle \int e^{2x} \cos(e^{2x}) \, dx = \frac{1}{2} \sin(e^{2x}) + C$$
• $$\int e^{2x} \sin(e^x) \, dx = \sin(e^x) - z\cos(e^x) + C\text{.}$$
• $$\int e^{3x} \sin(e^{3x}) \, dx = -\frac{1}{3} \cos(e^{3x}) + C\text{.}$$
• $$\int xe^{x^2} \cos(e^{x^2}) \sin(e^{x^2}) \, dx = \frac{1}{4} \sin^2(e^{x^2}) + C\text{.}$$

#### 5.4.7.15.

1. $$u$$-substitution; $$\int x^2 \cos(x^3) \ dx = \frac{1}{3} \sin(x^3) + C\text{.}$$
2. Both are needed; $$\int x^5 \cos(x^3) \ dx = \frac{1}{3} \left(x^3\sin(x^3) + \cos(x^3) \right) + C\text{.}$$
3. Integration by parts; $$\int x ln(x^2) \ dx = \frac{x^2}{2} \ln(x^2) - \frac{x^2}{2} + C\text{.}$$
4. Neither.
5. $$u$$-substitution; $$\int x^3 \sin(x^4) \ dx = -\frac{1}{4} \cos(x^4) + C\text{.}$$
6. Both are needed; $$\int x^7 \sin(x^4) \ dx = -\frac{1}{4} x^4 \cos(x^4) + \frac{1}{4} \sin(x^4) + C\text{.}$$

### 5.5Other Options for Finding Algebraic Antiderivatives5.5.5Exercises

#### 5.5.5.1.Partial fractions: linear over difference of squares.

$$\frac{1}{x-5}$$
$$\frac{1}{x+5}$$
$$\ln\!\left(\left|x-5\right|\right)$$
$$\ln\!\left(\left|x+5\right|\right)$$
$$\frac{1}{w}$$
$$\ln\!\left(\left|w\right|\right)$$
$$\ln\!\left(\left|x^{2}-25\right|\right)$$

#### 5.5.5.2.Partial fractions: constant over product.

$$0.5\ln\!\left(\left|x+6\right|\right)-0.5\ln\!\left(\left|x+8\right|\right)+C$$

#### 5.5.5.3.Partial fractions: linear over quadratic.

$$15\ln\!\left(\left|x-2\right|\right)-8\ln\!\left(\left|x-1\right|\right)+C$$

#### 5.5.5.4.Partial fractions: cubic over 4th degree.

$$6$$
$$4$$
$$1$$
$$4$$
$$\frac{-6}{x}+4\ln\!\left(x\right)+0.5\ln\!\left(x^{2}+16\right)+1\tan^{-1}\!\left(\frac{x}{4}\right)$$

#### 5.5.5.5.Partial fractions: quadratic over factored cubic.

$$1$$
$$2$$
$$0$$
$$\ln\!\left(\left|x-5\right|\right)+\frac{2\ln\!\left(x^{2}+4\right)}{2}+C$$

#### 5.5.5.6.

1. \begin{equation*} \int \frac{x^3 + x + 1}{x^4 - 1} \, dx = -\frac{1}{2} \arctan(x) + \frac{1}{4} \ln|x+1| + \frac{3}{4} \ln|x-1| + C \end{equation*}
2. \begin{equation*} \int \frac{x^5 + x^2 + 3}{x^3 - 6x^2 + 11x - 6} \, dx = \frac{x^3}{3} + 3x^2 + 25x + \frac{255}{2} \ln|x-3| - 39 \ln|x-2| + \frac{5}{2} \ln|x-1| + C\text{.} \end{equation*}
3. \begin{equation*} \int \frac{x^2 - x - 1}{(x-3)^3} \, dx = \ln|x-3| - \frac{5}{x-3} - \frac{5}{2(x-3)^2} + C\text{.} \end{equation*}

#### 5.5.5.7.

1. $$\int \frac{1}{x \sqrt{9x^2 + 25}} \, dx = -\frac{1}{5} \ln \left| \frac{5+\sqrt{9x^2 + 5^2}}{3x} \right| + C\text{.}$$
2. $$\displaystyle \int x \sqrt{1 + x^4} \, dx = \frac{1}{2} \left( \frac{x^2}{2}\sqrt{x^4 + 1} + \frac{1}{2}\ln|x^2 + \sqrt{x^4 + 1}| \right) + C$$
3. $$\displaystyle \int e^x \sqrt{4 + e^{2x}} \, dx = \frac{e^x}{2}\sqrt{e^{2x} + 4} + 2\ln|e^x + \sqrt{e^{2x} + 4}| + C$$
4. $$\int \frac{\tan(x)}{\sqrt{9 - \cos^2(x)}} \, dx = \frac{1}{3} \ln \left| \frac{3 + \sqrt{9 - \cos^2(x)}}{\cos(x)} \right| + C\text{.}$$

#### 5.5.5.8.

1. Try $$u = 1+x^2$$ or $$u = x + \sqrt{1+x^2}\text{.}$$
2. Try $$u = \sqrt{x+\sqrt{1+x^2}}$$ and $$dv = \frac{1}{x} \, dx\text{.}$$
3. No.
4. It appears that the function $$\frac{\sqrt{x+\sqrt{1+x^2}}}{x}$$ does not have an elementary antiderivative.

### 5.6Numerical Integration5.6.6Exercises

#### 5.6.6.1.

$$729.811$$
$$659.193$$
$$682.732$$

#### 5.6.6.2.

$$\frac{2\cdot 67.5+81}{3}$$

#### 5.6.6.3.

$$e^{4}-1$$
$$2\!\left(1+e^{2}\right)$$
$$2\!\left(e^{2}+e^{4}\right)$$
$$1\!\left(1+2e^{2}+e^{4}\right)$$
$$2\!\left(e^{1}+e^{3\cdot 1}\right)$$
$$\frac{2\cdot 45.6076+70.3763}{3}$$
$$53.5982-16.7781$$
$$53.5982-123.974$$
$$53.5982-70.3763$$
$$53.5982-45.6076$$
$$53.5982-53.8638$$
$$1\!\left(1+e^{1}+e^{2}+e^{3\cdot 1}\right)$$
$$1\!\left(e^{1}+e^{2}+e^{3\cdot 1}+e^{4}\right)$$
$$0.5\!\left(1+2e^{1}+2e^{2}+2e^{3\cdot 1}+e^{4}\right)$$
$$1\!\left(e^{0.5}+e^{3\cdot 0.5}+e^{5\cdot 0.5}+e^{7\cdot 0.5}\right)$$
$$\frac{2\cdot 51.4284+57.9919}{3}$$
$$53.5982-31.1929$$
$$53.5982-84.791$$
$$53.5982-57.9919$$
$$53.5982-51.4284$$
$$53.5982-53.6162$$
$$\frac{36.8201}{22.4053}$$
$$\frac{-70.3758}{-31.1928}$$
$$\frac{-16.7781}{-4.3937}$$
$$\frac{7.9906}{2.1698}$$
$$\frac{-0.2656}{-0.018}$$

RIGHT(n)
MID(n)
Exact
TRAP(n)
LEFT(n)

#### 5.6.6.5.

1. $$u$$-substitution fails since there’s not a composite function present; try showing that each of the choices of $$u = x$$ and $$dv = \tan(x) \, dx\text{,}$$ or $$u = \tan(x)$$ and $$dv = x \, dx\text{,}$$ fail to produce an integral that can be evaluated by parts.
• $$\displaystyle L_4 = 0.25892$$
• $$\displaystyle R_4 = 0.64827$$
• $$\displaystyle M_4 = 0.41550$$
• $$\displaystyle T_4 = \frac{L_4 + R_4}{2} = 0.45360$$
• $$\displaystyle S_8 = \frac{2M_4 + T_4}{3} = 0.42820$$
2. $$L_4$$ and $$M_4$$ are underestimates; $$R_4$$ and $$M_4$$ are overestimates.

#### 5.6.6.6.

1. Decreasing.
2. Concave down.
3. $$\int_3^6 f(x) \approx 7.03\text{.}$$

#### 5.6.6.7.

1. $$\int_0^{60} r(t) \, dt \text{.}$$
2. $$\int_0^{60} r(t) \, dt \gt M_3 = 204000\text{.}$$
3. $$\int_0^{60} r(t) \, dt \approx S_6 = \frac{619000}{3} \approx 206333.33\text{.}$$
4. $$\frac{1}{60} S_6 \approx 3438.89 \text{;}$$ $$\frac{2000+2100+2400+3000+3900+5100+6500}{7} = \frac{25000}{7} \approx 3571.43 \text{.}$$ each estimates the average rate at which water flows through the dam on $$[0,60]\text{,}$$ and the first is more accurate.

### 6Using Definite Integrals6.1Using Definite Integrals to Find Area and Length6.1.5Exercises

#### 6.1.5.1.

$$257.6$$

#### 6.1.5.2.

$$0.166667$$

#### 6.1.5.3.

$$9$$

#### 6.1.5.4.

$$18.9736659610103$$

#### 6.1.5.5.

$$0$$
$$1$$
$$1-x^{\frac{1}{4}}$$
$$0$$
$$1$$
$$y^{4}$$
$$\frac{1}{5}$$

#### 6.1.5.6.

$$-6$$
$$2$$
$$-6$$
$$2$$
$$\frac{12-y^{2}}{4}-y$$
$$21.3333$$

#### 6.1.5.7.

$$0$$
$$2$$
$$\sqrt{1+\left(20x^{4}+14\right)^{2}}$$

#### 6.1.5.8.

$$31.0944537700081$$

#### 6.1.5.9.

$$3.93517760459196$$

#### 6.1.5.10.

1. $$A = \int_{\frac{3 - \sqrt{3}}{2}}^{\frac{3 + \sqrt{3}}{2}} -2y^2 + 6y - 3 \ dy = \sqrt{3} \text{.}$$
2. $$A = \int_{\pi/4}^{3\pi/4} \sin(x) - \cos(x) \ dx = \sqrt{2} \text{.}$$
3. $$A = \int_{-1}{5/2} \frac{y+1}{2} - (y^2-y-2) \ dy = \frac{343}{48} \text{.}$$
4. $$A = \int_{\frac{m - \sqrt{m^2+4}}{2}}^{\frac{m + \sqrt{m^2+4}}{2}} mx - \left(x^2-1\right) \ dx = - \frac{1}{3}\left(\frac{m + \sqrt{m^2+4}}{2}\right)^3 - \frac{1}{3}\left(\frac{m - \sqrt{m^2+4}}{2}\right)^3 + \frac{m}{2}\left(\frac{m + \sqrt{m^2+4}}{2}\right)^2 - \frac{m}{2}\left(\frac{m - \sqrt{m^2+4}}{2}\right)^2 + \left(\frac{m + \sqrt{m^2+4}}{2} - \frac{m - \sqrt{m^2+4}}{2} \right) \text{.}$$

#### 6.1.5.11.

$$a = \frac{1}{2}\text{.}$$

#### 6.1.5.12.

1. $$r = \frac{4}{3} \text{.}$$
2. $$A_1 = A_2 = \frac{4 \sqrt{6}}{27}\text{.}$$
3. Yes.

### 6.2Using Definite Integrals to Find Volume6.2.5Exercises

#### 6.2.5.1.

$$0.771013094252786$$

#### 6.2.5.2.

$$2.35619449019234$$

#### 6.2.5.3.

$$2.89993168023673$$

#### 6.2.5.4.

$$29.8278687110064$$

#### 6.2.5.5.

$$1.5707963267949$$

#### 6.2.5.7.

$$\frac{9}{2}\pi$$

#### 6.2.5.8.

$$\pi \!\left(\frac{4}{3}-\frac{2}{15}\right)$$

#### 6.2.5.9.

1. $$L = \int_0^{1.84257} \sqrt{1+\left( -3 \sin \left(\frac{x^3}{4}\right) \cdot \frac{3}{4}x^2 \right)^2} \, dx \approx 4.10521\text{.}$$
2. $$A = \int_0^{1.84527} 3 \cos\left( \frac{x^3}{4} \right) \, dx \approx 4.6623 \text{.}$$
3. $$V = \int_0^{1.84527} \pi \cdot 9 \cos^2 \left( \frac{x^3}{4} \right) \, dx \approx 40.31965 \text{.}$$
4. $$V = \int_0^3 \pi \left( 4\arccos \left(\frac{y}{3} \right) \right)^{2/3} \, dy \approx 23.29194 \text{.}$$

#### 6.2.5.10.

1. $$A = \int_0^{\frac{\pi}{4}} ( \cos(x) - \sin(x) ) \, dx \text{.}$$
2. $$V = \int_0^{\frac{\pi}{4}} \pi (\cos^2(x) - \sin^2(x)) \, dx \text{.}$$
3. $$\displaystyle V = \int_0^{\frac{\sqrt{2}}{2}} \pi \arcsin^2(y) \, dy + \int_{\frac{\sqrt{2}}{2}}^1 \pi \arccos^2(y) \, dy$$
4. $$V = \int_0^{\frac{\pi}{4}} \pi [(2 - \sin(x))^2 - (2 - \cos(x))^2] \, dx \text{.}$$
5. $$\displaystyle V = \int_0^{\frac{\sqrt{2}}{2}} \pi [ (1+\arcsin(y))^2 - 1^2 ] \, dy + \int_{\frac{\sqrt{2}}{2}}^1 \pi [ (1+\arccos(y))^2 - 1^2 ] \, dy$$

#### 6.2.5.11.

1. $$A = \int_0^{1.5} 1+\frac{1}{2}(x-2)^2 - \frac{1}{2}x^2 \ dx = 2.25\text{.}$$
2. $$\displaystyle V = \int_0^{1.5} \pi\left[\left(2+\frac{1}{2}(x-2)^2\right)^2 - \left(1+\frac{1}{2}x^2\right)^2 \right] \ dx = \frac{315}{32} \pi$$
3. $$V = \int_{0}^{1.125} \pi \left(\sqrt{2y}\right)^2 \ dy + \int_{1.125}^3 \pi \left(2 - \sqrt{2(y-1)}\right)^2 \ dy \approx 7.06858347 \text{.}$$
4. $$P = 3 + \int_0^{1.5} \sqrt{1+(x-2)^2} + \sqrt{1+x^2} \ dx \approx 7.387234642 \text{.}$$

### 6.3Density, Mass, and Center of Mass6.3.5Exercises

#### 6.3.5.1.Center of mass for a linear density function.

$$\left(5+3x\right)Dx$$
$$28.5\ {\rm g}$$

#### 6.3.5.2.Center of mass for a nonlinear density function.

$$2+2\pi$$
$$\frac{\pi }{2}$$

#### 6.3.5.3.Interpreting the density of cars on a road.

$$250\!\left(2+\sin\!\left(4\sqrt{x+0.15}\right)\right)Dx$$
$$\frac{1}{8}\cdot 250\!\left(320+4\sqrt{0.15}\cos\!\left(4\sqrt{0.15}\right)-4\sqrt{20+0.15}\cos\!\left(4\sqrt{20+0.15}\right)-\sin\!\left(4\sqrt{0.15}\right)+\sin\!\left(4\sqrt{20+0.15}\right)\right)$$

#### 6.3.5.4.Center of mass in a point-mass system.

$$2.2\ {\rm cm}$$
$$\text{to the right of the origin}$$

#### 6.3.5.5.

1. $$a = -10 \ln(0.7) \approx 3.567 \text{ cm}\text{.}$$
2. Left of the midpoint.
3. $$\overline{x} \approx \frac{50.3338}{30} \approx 1.687\text{.}$$
4. $$q = -10\ln(0.85) \approx 1.625$$ cm.

#### 6.3.5.6.

1. $$M_1 = \arctan(10) \approx 1.47113\text{;}$$ $$M_2 = 10 - 10e^{-1} \approx 6.32121\text{.}$$
2. $$\overline{x_1} \approx 1.56857 \text{;}$$ $$\overline{x_2} \approx 4.18023 \text{.}$$
1. \begin{equation*} M = \int_0^{10} \rho(x) \, dx + \int_0^{10} p(x) \, dx \approx 1.47113 + 6.32121 = 7.79234\text{.} \end{equation*}
2. \begin{equation*} \int_0^{10} x(\rho(x) + p(x))) \, dx = 28.73167\text{.} \end{equation*}
3. False.

#### 6.3.5.7.

1. $$V = \int_0^{30} \pi (2xe^{-1.25x} + (30-x) e^{-0.25(30-x)})^2 \, dx \approx 52.0666$$ cubic inches.
2. $$W \approx 0.6 \cdot 52.0666 = 31.23996$$ ounces.
3. At a given $$x$$-location, the amount of weight concentrated there is approximately the weight density ($$0.6$$ ounces per cubic inch) times the volume of the slice, which is $$V_{text{slice}} \approx \pi f(x)^2\text{.}$$
4. $$\displaystyle \overline{x} \approx 23.21415$$

### 6.4Physics Applications: Work, Force, and Pressure6.4.5Exercises

#### 6.4.5.1.Work to empty a conical tank.

$$520187.424855521\ {\rm J}$$

#### 6.4.5.2.Work to empty a cylindrical tank.

$$511451\ {\rm ft\,lb}$$

#### 6.4.5.3.Work to empty a rectangular pool.

$$5.39136\times 10^{6}\ {\rm ft\,lb}$$

#### 6.4.5.4.Work to empty a cylindrical tank to differing heights.

$$225833\ {\rm lbf}$$
$$414027\ {\rm lbf}$$
$$169375\ {\rm lbf}$$

#### 6.4.5.5.Force due to hydrostatic pressure.

$$2940\ {\rm N}$$
$$1176\ {\rm N}$$
$$392\ {\rm N}$$

#### 6.4.5.6.

1. $$W \approx 1353.55$$ foot-pounds.
2. $$W = \int_0^h 3744 x \cos \left( \frac{x^3}{4} \right) \, dx \text{.}$$
3. $$F \approx 462.637$$ pounds.

#### 6.4.5.7.

1. $$W = 1404(19\pi - 8) \approx 305179.3$$ foot-pounds.
2. \begin{equation*} F \approx 1123.2 \end{equation*}
pounds.

### 6.5Improper Integrals6.5.5Exercises

#### 6.5.5.1.

$$\frac{3e^{-0\cdot 0\cdot 0}}{3}$$

#### 6.5.5.2.

$$4.11340632411132\times 10^{-5}$$

#### 6.5.5.3.

$$\text{diverges}$$

#### 6.5.5.4.

$$\text{diverges}$$

#### 6.5.5.5.

$$6$$

#### 6.5.5.6.

$$\text{Diverges}$$

#### 6.5.5.7.

$$1406250$$

#### 6.5.5.8.

$$8264.46281$$

#### 6.5.5.9.

$$\text{Diverge}$$
$$\text{Diverge}$$
$$\text{Not sufficient information}$$

#### 6.5.5.10.

$$\frac{A}{r}\!\left(1-\frac{a}{\sqrt{r^{2}+a^{2}}}\right)$$

#### 6.5.5.11.

1. Diverges.
2. Diverges.
3. Converges to $$1\text{.}$$
4. $$\int_e^{\infty} \frac{1}{x(\ln(x))^p} \, dx$$ diverges if $$p \leq 1$$ and converges to $$\frac{1}{p-1}$$ if $$p \gt 1\text{.}$$
5. Diverges.
6. Converges to $$-1\text{.}$$

1. converges
2. diverges
3. diverges

### 7Differential Equations7.1An Introduction to Differential Equations7.1.5Exercises

#### 7.1.5.3.

$$71.1722791608475$$

#### 7.1.5.4.

$$-2107$$

#### 7.1.5.6.

$$18$$
$$32.4$$
$$58.32$$
$$104.976$$

#### 7.1.5.7.

1. $$\frac{dT}{dt}\vert_{T=105} = -2\text{;}$$ when $$T = 105\text{,}$$ the coffee’s temperature is decreasing at an instantaneous rate of $$-2$$ degrees F per minute.
2. $$T$$ decreasing at $$t=0\text{.}$$
3. $$T(1) \approx 103$$ degrees F.
4. For $$T \lt 75\text{,}$$ $$T$$ increases. For $$T \gt 75\text{,}$$ $$T$$ decreases.
5. Room temperature is $$75$$ degrees F.
6. Substitute $$T(t) = 75 + 30e^{-t/15}$$ in for $$T$$ in the differential equation $$\frac{dT}{dt}= -\frac1{15}T+5$$ and verify the equality holds; $$T(0) = 75 + 30e^0 = 75 + 30 = 105\text{;}$$ $$T(t) = 75 + 30e^{-t/15} \to 75$$ as $$t \to \infty\text{.}$$

#### 7.1.5.8.

1. $$1 \lt P \lt 3\text{.}$$
2. $$P \lt 1$$ and $$3 \lt P \lt 4\text{.}$$
3. $$P$$ will not change at all.
4. The population will decrease toward $$P = 0$$ with $$P$$ always being positive.
5. The population will increase toward $$P = 3$$ with $$P$$ always being between $$1$$ and $$3\text{.}$$
6. The population will decrease toward $$P = 3$$ with $$P$$ always being above $$3\text{.}$$
7. There’s a maximum threshold of $$P = 3\text{.}$$

#### 7.1.5.9.

1. $$y(t) = t + 1 + 2e^t$$ is a solution to the DE.
2. $$y(t) = t + 1$$ is a solution to the DE.
3. $$y(t) = t + 2$$ is a not solution to the DE.
1. $$k = 9\text{.}$$

### 7.2Qualitative behavior of solutions to DEs7.2.4Exercises

#### 7.2.4.1.

$$\text{negative}$$
$$\text{negative}$$
$$\text{positive}$$
$$\text{negative}$$

#### 7.2.4.3.

$$-4$$
$$-2$$
$$-1$$
$$2$$

#### 7.2.4.4.

$$-1, 2$$

#### 7.2.4.5.

$$x+1$$

#### 7.2.4.6.

1. Sketch curves through appropriate points in the slope field above.
2. $$y(t) = t-1\text{.}$$
3. $$t$$ and $$y$$ are equal.

#### 7.2.4.7.

1. Any solution curve that starts with $$P(0) \gt 3$$ will decrease to $$P(t) = 3$$ as $$t \to \infty\text{;}$$ any curve that starts with $$1 \lt P(0) \lt 3$$ will increase to $$P(t) = 3\text{;}$$ any curve that starts with $$0 \lt P(0) \lt 1$$ will decrease to $$P(t) = 0\text{.}$$
2. $$P = 0\text{,}$$ $$P = 1\text{,}$$ and $$P = 3\text{.}$$ $$P = 1$$ is unstable; $$P = 0$$ and $$P = 3$$ are stable.
3. The population will stabilize either at the value $$P = 3$$ or at $$P = 0\text{.}$$
4. $$P(t) = 1$$ is the threshold.

#### 7.2.4.8.

1. A graph of $$f$$ against $$P$$ is given in blue in the figure below. The equilibrium solutions are $$P=0$$ (unstable) and $$P=6$$ (stable).
2. $$\frac{dP}{dt} = g(P) = P(6-P)-1$$ ; the equilibrium at $$P\approx 0.172$$ is unstable; the equilibrium at $$P \approx 5.83$$ is stable.
3. If $$P \lt \frac{6-\sqrt{32}}{2}\text{,}$$ then the fish population will die out. If $$\frac{6-\sqrt{32}}{2} \lt \text{,}$$ then the fish population will approach $$\frac{6+\sqrt{32}}{2}$$ thousand fish.
4. $$\frac{dP}{dt} = g(P) = P(6-P)-h \text{;}$$ equilibrium solutions $$P = \frac{6+\sqrt{36-4h}}{2}, \ \frac{6-\sqrt{36-4h}}{2} \text{.}$$
5. $$9000$$ fish; harvesting at that rate will maintain the number of fish we start with, provided it’s at least $$3000\text{.}$$

#### 7.2.4.9.

1. $$\frac{dy}{dt} = 20y\text{.}$$
2. $$\displaystyle \frac{dy}{dt} = 20y - C\frac{y}{2+y}$$
3. For positive $$y$$ near $$0\text{,}$$ $$M(y) = \frac{y}{2+y} \approx 0\text{;}$$ for large values of $$y\text{,}$$ $$M(y) = \frac{y}{2+y} \approx 1\text{.}$$
4. The only equilibrium solution is $$y = 0\text{,}$$ which is unstable.
5. The equilibrium solutions are $$y = 0$$ (stable) and $$y = 1$$ (unstable).
6. At least $$41$$ cats.

### 7.3Euler’s method7.3.4Exercises

#### 7.3.4.1.

$$0.8$$
$$1.28$$
$$2.048$$
$$3.2768$$
$$5.24288$$

#### 7.3.4.2.

$$1512$$
$$1514.24$$
$$1515.405024$$

#### 7.3.4.3.

$$3.5$$
$$3.75$$
$$3+1x^{2}$$
$$0.5$$
$$0.25$$
$$2$$

#### 7.3.4.4.

$$0.2561$$
$$-1.4$$

#### 7.3.4.5.

$$2.31836502375202$$

#### 7.3.4.6.

1. Alice’s coffee: $$\frac{dT_A}{dt} \vert_{T = 100} = -0.5(30) = -15$$ degrees per minute; Bob’s coffee: $$\frac{dT_B}{dt} \vert_{T = 100} = -0.1(30) = -3$$ degrees per minute.
2. Consider the insulation of the containers.
3. Alice’s coffee:
\begin{equation*} \frac{dT_A}{dt} = -0.5(T_A-(70+10\sin t))\text{,} \end{equation*}
with the inital condition $$T_A(0) = 100\text{.}$$
 $$t$$ $$T_A(t)$$ $$0.0$$ $$100$$ $$0.1$$ $$98.5$$ $$0.2$$ $$97.12492$$ $$0.3$$ $$95.86801$$ $$0.4$$ $$94.72237$$ $$\vdots$$ $$\vdots$$ $$49.6$$ $$65.56715$$ $$49.7$$ $$65.48008$$ $$49.8$$ $$65.43816$$ $$49.9$$ $$65.44183$$ $$50$$ $$65.49103$$
4.  $$t$$ $$T_A(t)$$ $$0.0$$ $$100$$ $$0.1$$ $$99.7$$ $$0.2$$ $$99.41298$$ $$0.3$$ $$99.13872$$ $$0.4$$ $$98.87689$$ $$\vdots$$ $$\vdots$$ $$49.6$$ $$69.39515$$ $$49.7$$ $$69.33946$$ $$49.8$$ $$69.29248$$ $$49.9$$ $$69.25467$$ $$50$$ $$69.22638$$
5. Compare the rate of initial decrease and amplitude of oscillation.

#### 7.3.4.7.

1. $$K = 1.054\text{;}$$ $$y(1) = 2.6991\text{.}$$
2. $$K = 1.272\text{;}$$ $$y(1) = 2.7169\text{.}$$
3. $$K = 0.122$$ and $$y(0.3) = 0.0412\text{.}$$

#### 7.3.4.8.

1. $$y(1) \approx y_5 = 2.7027\text{.}$$
2. $$y(1) \approx y_{10} = 2.7141\text{.}$$
3. The square of $$\Delta t\text{.}$$

### 7.4Separable differential equations7.4.3Exercises

#### 7.4.3.1.

$$y = \frac{3}{e^{\frac{1}{3}}}e^{\frac{x^{3}}{3}}$$

#### 7.4.3.2.

$$y^{2}-\left(\frac{x}{5}\right)^{2}$$
$$5y-x = 0$$
$$y = -\sqrt{\left(\frac{x}{5}\right)^{2}+9}$$

#### 7.4.3.3.

$$-115e^{0.7t}+150$$

#### 7.4.3.4.

$$\frac{16}{2+\left(1-t\right)\!\left(15+t\right)\cdot 8}$$

#### 7.4.3.5.

$$\frac{-\ln\!\left(e^{-3\cdot 5}+\frac{5}{10}-\frac{5}{10}e^{10t}\right)}{5}$$

#### 7.4.3.6.

$$3\exp\!\left(2x^{10}\right)$$

#### 7.4.3.7.

$$0.2$$
$$9$$
$$0.4$$
$$9.0000071$$
$$0.6$$
$$9.0002347$$
$$0.8$$
$$9.0019626$$
$$1$$
$$9.0092428$$
$$\sqrt{\frac{2x^{6}}{6}+2\cdot 40.5}$$
$$9.0184995$$

#### 7.4.3.8.

1. $$\frac{dM}{dt} = kM \text{.}$$
2. $$M(t) = M_0e^{kt} \text{.}$$
3. $$\displaystyle M(t) = M_0e^{-\frac{\ln(2)}{5730}t} \approx M_0e^{-0.000121t}$$
4. $$t = \frac{5730\ln(4)}{\ln(2)} \approx 11460$$ years.
5. $$t = -\frac{5730\ln(0.3)}{\ln(2)} \approx 9952.8$$ years.

#### 7.4.3.9.

1. $$y = \sqrt{64 - t^2} \text{.}$$
2. $$-8 \le t \le 8\text{.}$$
3. $$y(8) = 0\text{.}$$
4. $$\frac{dy}{dt} = -\frac ty$$ is not defined when $$y = 0\text{.}$$

#### 7.4.3.10.

1. $$\frac{dh}{dt} = k \sqrt{h} \text{.}$$
2. The tank with $$k = -10$$ has water leaving the tank much more rapidly.
3. $$k = -5\text{.}$$
4. $$h(t) = \left( 5 - 2.5t \right)^2 \text{.}$$
5. $$2.5$$ minutes.
6. No.

#### 7.4.3.11.

1. $$P = 3$$ is stable.
2. $$P(t) = 3e^{\ln \left(\frac{1}{3} \right) e^{-t}} \text{.}$$
3. $$P(t) = 3e^{\ln \left( 2 \right) e^{-t}} \text{.}$$
4. Yes.

### 7.5Modeling with differential equations7.5.3Exercises

#### 7.5.3.1.Mixing problem.

$$0\ {\rm kg}$$
$$0.21-7\frac{S}{1700}$$
$$51\!\left(1-e^{-0.0041176t}\right)$$
$$13.55\ {\rm kg}$$

#### 7.5.3.2.Mixing problem.

$$0.035$$
$$57.3169853067621$$
$$0.0175$$

#### 7.5.3.3.Population growth problem.

$$620\cdot 2.71828^{0.346574t}$$
$$4959.99999999999$$
$$2.2876715451473$$

#### 7.5.3.4.Radioactive decay problem.

$$450.151343578041$$
$$6.52700077955987$$

#### 7.5.3.5.Investment problem.

$$\frac{k}{0.09}\!\left(\exp\!\left(0.09t\right)-1\right)$$
$$3725.07265205081$$

#### 7.5.3.6.

1. $$\displaystyle \frac{dA}{dt} = 1 + 0.05A$$
2. $$A(25) = 49.80686$$ million dollars.
3. $$A(25) = 34.90343$$ million dollars.
4. The first.
5. $$t = 20 \ln(2) \approx 13.86 \ \text{years} \text{.}$$

#### 7.5.3.7.

1. $$\displaystyle \frac{dv}{dt} = 9.8 - kv$$
2. $$v = \frac{9.8}{k}$$ is a stable equilibrium.
3. $$\displaystyle v(t) = \frac{9.8 - 9.8e^{-kt}}{k}$$
4. $$k = 9.8/54 \approx 0.181481\text{.}$$
5. $$t = \frac{\ln(0.5)}{-0.181481} \approx 3.1894$$ seconds.

#### 7.5.3.8.

1. $$\frac{dw}{dt} = \frac{k}{w} \text{.}$$
2. $$w(t) = \sqrt{7t+64} \text{;}$$ $$w(12) = \sqrt{148} \approx 12.17$$ pounds.
3. The model is unrealistic.

#### 7.5.3.9.

1. The inflow and outflow are at the same rate.
2. $$60$$ grams per minute.
3. $$\displaystyle \frac{S(t)}{100} \frac{\text{grams}}{\text{gallon}}$$
4. $$\frac{3S(t)}{100} \frac{\text{grams}}{\text{minute}} \text{.}$$
5. $$\frac{dS}{dt} = 60 - \frac{3}{100} S \text{.}$$
6. $$S = 2000$$ is a stable equilibrium solution.
7. $$S(t) = 2000 - 2000e^{-\frac{3}{100}t} \text{.}$$
8. $$S(t) \to 2000\text{.}$$

### 7.6Population Growth and the Logistic Equation7.6.4Exercises

#### 7.6.4.1.Analyzing a logistic equation.

$$0.25$$
$$0.5$$
$$1.5$$
$$0.5$$
$$\left(0,0.5\right)$$
$$\left(0.5,\infty \right)$$
$$0.5$$
$$0.25$$
$$\left(0,0.5\right)$$
$$\left(0.5,\infty \right)$$
$$0, 0.5$$
$$\frac{0.5}{2}$$

#### 7.6.4.2.Analyzing a logistic model.

$$0.65$$
$$12.7$$
$$3.65$$
$$1985$$
$$20.8$$
$$72$$
$$75$$

#### 7.6.4.3.Finding a logistic function for an infection model.

$$\frac{4250}{1+\frac{\left(4250-10\right)e^{-1.7t}}{10}}$$
$$\frac{4250}{2}$$

#### 7.6.4.4.Analyzing a population growth model.

$$\left(0,1.74796\right)$$
$$\left(1.74796,\infty \right)$$
$$\infty$$
$$\frac{b}{a}$$
$$0$$

#### 7.6.4.5.

1. $$p(t) \to 1$$ as $$t \to \infty$$ provided $$p(0) \gt 0\text{.}$$
2. $$p(t) = \frac{1}{9e^{-0.2t} + 1} \text{.}$$
3. $$t = -5 \ln(1/9) \approx 10.986$$ days.
4. $$t = -5 \ln(0.25/9) \approx 17.19$$ days.

#### 7.6.4.6.

1. $$\displaystyle \frac{db}{dt} = \frac{1}{3000} b(15000 - b)$$
2. $$b = 15000\text{.}$$
3. When $$b = 7500\text{.}$$
4. $$t = -\frac{1}{5} \ln(1/70) \approx 0.8497$$ days.

#### 7.6.4.7.

1. 10000 fish.
2. $$\frac{dP}{dt} = 0.1P(10 - P) - 0.2P \text{.}$$
3. $$8000$$ fish.
4. $$P(1) \approx 8.7899$$ thousand fish.
5. $$t = -1.25 \ln(5/11) \approx 0.986$$ years.

### 8Taylor Polynomials and Taylor Series8.1Approximating $$f(x) = e^x$$8.1.5Exercises

#### 8.1.5.1.

$$\text{-}$$
$$\text{+}$$
$$\text{+}$$

#### 8.1.5.2.

$$3$$
$$-7$$
$$8$$

### 8.2Taylor Polynomials8.2.4Exercises

#### 8.2.4.1.

$$4x$$
$$4x-10.6667x^{3}$$
$$4x-10.6667x^{3}+8.53333x^{5}$$

#### 8.2.4.2.

$$0$$
$$-1$$
$$0$$
$$0$$
$$-1$$
$$0$$
$$0.166666666666667$$

#### 8.2.4.3.

$$0.693147+0.5\!\left(x-1\right)-0.125\!\left(x-1\right)^{2}$$
$$0.0416667\!\left(x-1\right)^{3}$$

#### 8.2.4.4.

$$e^{0.2}+e^{0.2}\!\left(x-0.2\right)+\frac{e^{0.2}}{2}\!\left(x-0.2\right)^{2}$$
$$0.0591338$$

#### 8.2.4.5.

$$3x^{2}-9x+2$$

#### 8.2.4.6.

$$1+2\!\left(x-4\right)+\frac{1}{2!}\!\left(-1\right)\!\left(x-4\right)^{2}$$
$$1+2\!\left(x-4\right)+\frac{1}{2!}\!\left(-1\right)\!\left(x-4\right)^{2}+\frac{1}{3!}\!\left(-4\right)\!\left(x-4\right)^{3}$$
$$0.795$$
$$0.795667$$

### 8.3Geometric Sums8.3.5Exercises

#### 8.3.5.1.

B
$$na$$
$$na$$
A
$$5x^{5}$$
$$x$$

#### 8.3.5.2.

$$16$$
$$6.66666666511446$$
$$7$$
$$0.00162750482559204$$

#### 8.3.5.3.

$$-6.66656494140625$$
$$0.0185185185185185$$

#### 8.3.5.4.

$$0.666666666666667$$

#### 8.3.5.5.

$$-10.6666666666667$$

#### 8.3.5.6.

$$\frac{7\cdot 4^{2+1}}{7^{0+1}\!\left(7-4\right)}$$

#### 8.3.5.8.

1. $$30 \cdot 500 = 1500$$ dollars.
2.  Day Pay on this day Total amount paid to date $$1$$ $$\dollar0.01$$ $$\dollar0.01$$ $$2$$ $$\dollar0.02$$ $$\dollar0.03$$ $$3$$ $$\dollar0.04$$ $$\dollar0.07$$ $$4$$ $$\dollar0.08$$ $$\dollar0.15$$ $$5$$ $$\dollar0.16$$ $$\dollar0.31$$ $$6$$ $$\dollar0.32$$ $$\dollar0.63$$ $$7$$ $$\dollar0.64$$ $$\dollar1.27$$ $$8$$ $$\dollar1.28$$ $$\dollar2.55$$ $$9$$ $$\dollar2.56$$ $$\dollar5.11$$ $$10$$ $$\dollar5.12$$ $$\dollar10.23$$
3. $$\dollar0.01\left(2^{30}-1\right) = \dollar10,737,418.23 \text{.}$$

#### 8.3.5.9.

1. $$h_1 = \left(\frac{3}{4}\right)h \text{.}$$
2. $$h_2 = \left(\frac{3}{4}\right)h_1 = \left(\frac{3}{4}\right)^2h \text{.}$$
3. $$h_3 = \left(\frac{3}{4}\right)h_2 = \left(\frac{3}{4}\right)^3h \text{.}$$
4. $$h_n = \left(\frac{3}{4}\right)h_{n-1} = \left(\frac{3}{4}\right)^nh \text{.}$$
5. The distance traveled by the ball is $$7h\text{,}$$ which is finite.

### 8.4Taylor Series8.4.4Exercises

#### 8.4.4.1.

$$\frac{15x^{2}}{2}-\left(2+3x\right)-\frac{6x^{3}}{6}$$

#### 8.4.4.2.

$$2.71828$$
$$2.71828$$
$$1.35914$$
$$0.453047$$
$$0.113262$$

#### 8.4.4.3.

$$\frac{4}{3}$$
$$\frac{-4\!\left(x-3\right)}{3^{2}}$$
$$\frac{4\!\left(x-3\right)^{2}}{3^{3}}$$
$$\frac{-4\!\left(x-3\right)^{3}}{3^{4}}$$

#### 8.4.4.4.

$$\ln\!\left(9\right)$$
$$\frac{1}{9}\!\left(x-9\right)$$
$$\frac{-\left(\frac{1}{9}\right)^{2}}{2}\!\left(x-9\right)^{2}$$
$$\frac{\left(\frac{1}{9}\right)^{3}}{3}\!\left(x-9\right)^{3}$$
$$\frac{-\left(\frac{1}{9}\right)^{4}}{4}\!\left(x-9\right)^{4}$$

#### 8.4.4.5.

$$\frac{-5}{\sqrt{2}}$$
$$\frac{5\!\left(x+\frac{\pi }{4}\right)}{\sqrt{2}}$$
$$\frac{5\!\left(x+\frac{\pi }{4}\right)^{2}}{2\sqrt{2}}$$
$$\frac{-5\!\left(x+\frac{\pi }{4}\right)^{3}}{6\sqrt{2}}$$

#### 8.4.4.6.

$$\sin^{-1}\!\left(0.3\right)$$
$$\cos^{-1}\!\left(0.1\right)$$

#### 8.4.4.7.

$$-0.756802495307928$$
$$7.38905609893065$$

#### 8.4.4.9.

1. $$P_4(x) = x^2 \text{.}$$
1. $$g(x) = T(x^2) = x^2 - \frac{1}{3!}x^6 + \frac{1}{5!}x^10 - \cdots \text{.}$$
2. All real numbers.

### 8.5Finding and Using Taylor Series8.5.5Exercises

#### 8.5.5.1.

$$\left(k+1\right)\!\left(k+3\right)x^{k+3-1}$$

#### 8.5.5.2.

8.5.5.2.a Part 1.
$$\frac{\left(-1\right)^{n}\!\left(n+1\right)x^{n}}{5^{n+2}}$$
8.5.5.2.b Part 2.
$$\frac{\left(-1\right)^{n}\!\left(n+1\right)\!\left(n+2\right)x^{n}}{2\cdot 5^{n+3}}$$
8.5.5.2.c Part 3.
$$\frac{\left(-1\right)^{n}\!\left(n+1\right)\!\left(n+2\right)x^{n+2}}{2\cdot 5^{n+3}}$$

#### 8.5.5.3.

$$\frac{\left(-1\right)^{n}t^{6n+2}}{6n+2}$$
$$\frac{t^{6\cdot 0+2}}{6\cdot 0+2}$$
$$\frac{-1t^{6\cdot 1+2}}{6\cdot 1+2}$$
$$\frac{t^{6\cdot 2+2}}{6\cdot 2+2}$$
$$\frac{-1t^{6\cdot 3+2}}{6\cdot 3+2}$$
$$\frac{t^{6\cdot 4+2}}{6\cdot 4+2}$$
$$\left(-1,1\right)$$

#### 8.5.5.4.

$$\frac{\left(-1\right)^{n+1}x^{3+n+1}}{n\!\left(3+n+1\right)}$$
$$\frac{x^{3+2}}{1\cdot \left(3+2\right)}$$
$$\frac{-1x^{3+3}}{2\cdot \left(3+3\right)}$$
$$\frac{x^{3+4}}{3\cdot \left(3+4\right)}$$
$$\frac{-1x^{3+5}}{4\cdot \left(3+5\right)}$$
$$\frac{x^{3+6}}{5\cdot \left(3+6\right)}$$
$$\left(-1,1\right)$$

#### 8.5.5.5.

$$4$$

#### 8.5.5.6.

$$9$$

### 9Supplementary material9.1Review of Prerequsites

#### Exercises

##### 9.1.1.
$$0$$
$$I$$
##### 9.1.2.
$$0$$
##### 9.1.3.
$$60$$
$$60$$
$$60x-\left(-360\right)+\left(-172\right)$$
##### 9.1.4.
$$0.333333\!\left(t^{-0.666667}+t^{-1.33333}\right)$$
##### 9.1.5.
$$\ln\!\left(8\right)\cdot 8^{x}$$
##### 9.1.6.
$$\sec^{2}\!\left(x\right)\csc\!\left(x\right)-\csc\!\left(x\right)\cot\!\left(x\right)\tan\!\left(x\right)$$
##### 9.1.7.
$$\frac{\cos\!\left(x\right)+x\sin\!\left(x\right)}{\cos^{2}\!\left(x\right)}$$
##### 9.1.8.
$$\frac{-5\sin\!\left(3\ln\!\left(x\right)\right)\cdot 3}{x}$$
##### 9.1.9.
$$9\tan\!\left(t\right)-2.25t^{4}$$
##### 9.1.10.
$$593.23432548299$$

### 9.2Integrating Rational Functions9.2.2Exercises

#### 9.2.2.5.

$$6$$
$$3$$
$$4$$
$$2$$
$$\frac{6x^{2}}{2}+3x+4\ln\!\left(x-5\right)+2\ln\!\left(x+5\right)$$

#### 9.2.2.6.

$$-2\frac{-1}{x+a}+C+c$$
$$\frac{-2}{b-a}\ln\!\left(\left|\frac{x+a}{x+b}\right|\right)+C+c$$

#### 9.2.2.8.

$$\frac{1}{x-3}$$
$$\frac{1}{x+3}$$
$$\ln\!\left(\left|x-3\right|\right)$$
$$\ln\!\left(\left|x+3\right|\right)$$
$$\frac{1}{w}$$
$$\ln\!\left(\left|w\right|\right)$$
$$\ln\!\left(\left|x^{2}-9\right|\right)$$

### 9.3Integration with Trigonometric Functions9.3.2Exercises

#### 9.3.2.1.

$$-0.466667\!\left(\frac{\sec^{5}\!\left(-15x\right)}{5}-\frac{\sec^{3}\!\left(-15x\right)}{3}\right)+C$$

#### 9.3.2.2.

$$4\!\left(\frac{\tan^{7}\!\left(3x\right)}{7}+\frac{\tan^{5}\!\left(3x\right)}{5}\right)+C$$

#### 9.3.2.3.

$$-1.25\!\left(\frac{\csc^{5}\!\left(4x\right)}{5}-\frac{\csc^{3}\!\left(4x\right)}{3}\right)+C$$

#### 9.3.2.4.

$$\frac{192x}{8}+8\!\left(\frac{\sin\!\left(16x\right)}{4}+\frac{\sin\!\left(32x\right)}{32}\right)+C$$

#### 9.3.2.5.

$$\frac{1}{8}\tan^{8}\!\left(x\right)+\frac{1}{10}\tan^{10}\!\left(x\right)+C$$

#### 9.3.2.6.

$$6\!\left(\frac{0.166667\sin^{4}\!\left(x\right)}{\cos^{6}\!\left(x\right)}+\frac{0.0833333\sin^{4}\!\left(x\right)}{\cos^{4}\!\left(x\right)}\right)+C$$

#### 9.3.2.7.

$$-7\ln\!\left(\left|\sin\!\left(x\right)\right|\right)-\left(-7\right)\sin^{2}\!\left(x\right)+\left(-1.75\right)\sin^{4}\!\left(x\right)+C$$

#### 9.3.2.8.

$$-0.5\!\left(\frac{\cos\!\left(22x\right)}{22}+\frac{\cos\!\left(-8x\right)}{-8}\right)$$

#### 9.3.2.9.

$$\frac{\sec\!\left(x\right)\tan\!\left(x\right)+\ln\!\left(\left|\sec\!\left(x\right)+\tan\!\left(x\right)\right|\right)}{2}$$

### 9.4Parametric Curves

#### Exercises

##### 9.4.3.
$$0$$
$$-3$$
$$3$$
##### 9.4.4.
$$-2x+11$$
$$x-1$$
$$\text{NONE}$$
##### 9.4.5.
$$-2$$
$$\text{to the left}$$
$$\text{up}$$
$$2$$
$$\text{to the right}$$
$$\text{up}$$
##### 9.4.6.
$$7\frac{2t}{3t^{2}-12}$$
$$7\frac{-2\!\left(t^{2}+4\right)}{9\!\left(t^{2}-4\right)^{3}}$$
$$-2$$
$$2$$
##### 9.4.7.
$$8\pi$$
##### 9.4.8.
$$3.65685424949238$$
##### 9.4.9.
$$\pi ab$$
##### 9.4.10.
$$21.1189225314382$$

### 9.5Calculus in Polar Coordinates

#### Exercises

##### 9.5.1.
$$1.4142135623731$$
$$0.785398163397448$$
$$-1.4142135623731$$
$$3.92699081698724$$
$$4$$
$$5.75958653158129$$
$$-4$$
$$2.61799387799149$$
##### 9.5.3.
$$0.197396$$
##### 9.5.4.
$$-3.14159265358979$$
##### 9.5.5.
$$4\sin\!\left(5\cdot 0.785398\right)\sin\!\left(0.785398\right)+1.5\!\left(x-4\sin\!\left(5\cdot 0.785398\right)\cos\!\left(0.785398\right)\right)$$
##### 9.5.6.
$$5806383.56624947$$
##### 9.5.7.
$$202.633$$
$$31.8086$$
$$6.28318530717959$$
$$3.14159265358979$$
$$95614960306955696$$