Active Calculus

Let $P$ and $Q$ be polynomials.

Find ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{P(x)}{Q(x)}}$

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

(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 $x=1\text{.}$ $y=$

$f^{\prime }(t)=$

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

$\frac{dy}{dx}=$

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

$f^{\prime }(x)=$

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

$y^{\prime }=$

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

Answer:

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

Answer:

${\displaystyle {\int }_0^5(3e^x+4\sin x)dx}$ =