Active Calculus

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)

Answer:

Suppose that \(f(x) = -7 x^2 + 4\text{.}\)

(A) Find the slope of the line tangent to \(f(x)\) at \(x=1\text{.}\)

(B) Find the instantaneous rate of change of \(f(x)\) at \(x=1\text{.}\)

(C) Find the equation of the line tangent to \(f(x)\) at \(x=1\text{.}\) \(y=\)

\(f'(t)=\)

Find the derivative of \(y = 6^x + 2\text{.}\)

\({dy\over dx} =\)

Use the Product Rule to find the derivative of \(f\text{.}\)

\(f(x) = \csc\mathopen{}\left(x\right)\tan\mathopen{}\left(x\right)\)

\(f'(x) =\)

Differentiate \(\displaystyle y =\frac{x}{\cos{x}}\text{.}\)

\(y'=\)

If \(f(x) = 4 \cos(2\ln(x))\text{,}\) find \(f'( x )\text{.}\)

Answer:

Consider the function \(f(t) = 8 \sec ^2(t) - 7 t^ { 2 }\text{.}\) Let \(F(t)\) be the antiderivative of \(f(t)\) with \(F(0) = 0\text{.}\) Find \(F(t)\text{.}\)

Answer:

\(\displaystyle \int_0^{5} (3 e^x+4 \sin x)\, dx\) =