# Active Calculus

## Section9.1Review of Prerequsites

### ExercisesExercises

#### 1.

Let $$P$$ and $$Q$$ be polynomials.
Find $$\displaystyle \lim_{x \to \infty} \frac{P(x)}{Q(x)}$$
if the degree of $$P$$ is (a) less than the degree of $$Q\text{,}$$ and (b) greater than the degree of $$Q\text{.}$$ If the answer is infinite, enter "I" below.
(a)
(b)

#### 2.

Evaluate the limit using L'Hospital's rule if necessary.
\begin{equation*} \lim_{ x \rightarrow +\infty } \frac{x^{11}}{e^x} \end{equation*}

#### 3.

Suppose that $$f(x) = -5 x^2 + 8\text{.}$$
(A) Find the slope of the line tangent to $$f(x)$$ at $$x=-6\text{.}$$
(B) Find the instantaneous rate of change of $$f(x)$$ at $$x=-6\text{.}$$
(C) Find the equation of the line tangent to $$f(x)$$ at $$x=-6\text{.}$$ $$y=$$

#### 4.

Differentiate the following function:
\begin{equation*} \displaystyle f(t) = \sqrt[3]{t}-\frac{1}{\sqrt[3]{t}} \end{equation*}
$$f'(t)=$$

#### 5.

Find the derivative of $$y = 8^x + 7\text{.}$$
$${dy\over dx} =$$

#### 6.

Use the Product Rule to find the derivative of $$f\text{.}$$
$$f(x) = \csc\!\left(x\right)\tan\!\left(x\right)$$
$$f'(x) =$$

#### 7.

Differentiate $$\displaystyle y =\frac{x}{\cos{x}}\text{.}$$
$$y'=$$

#### 8.

If $$f(x) = 5 \cos(3\ln(x))\text{,}$$ find $$f'( x )\text{.}$$
Consider the function $$f(t) = 9 \sec ^2(t) - 9 t^ { 3 }\text{.}$$ Let $$F(t)$$ be the antiderivative of $$f(t)$$ with $$F(0) = 0\text{.}$$ Find $$F(t)\text{.}$$
$$\displaystyle \int_0^{5} (4 e^x+5 \sin x)\, dx$$ =