Introduction and Definitions

1. What is the definition of the derivative?

2. If a function represents distance as a function of time, what does its derivative represent?

3. If a function represents the cost to produce $x$ items, what does its derivative cost?

4. If the volume of a sphere is a function of its radius, what is the relationship between the rate of change of the volume and the rate of change of the radius?

Find the derivative of the function $f(x)=5x+8$.

If $f(x)=\displaystyle\frac{1}{x^4}$, find

1. $f'(x)$

2. The equation of the line tangent at $(-1,1)$

If $y=cx$, where $c$ is a constant, find $\displaystyle\frac{dx}{dy}$.

Differentiate.

1. $y=8x$

2. $f(x)=-14x$

If $f(x)=6x^4$, find $\dfrac{dy}{dx}$

AP Calculus Exam: Multiple Choice Question

AP Calculus Exam: Multiple Choice Question

Differentiate each function.

1. $f(x)=x^2+3x-4$

2. $V(r)=\dfrac{4}{3}\pi r^3$

3. $F(x)=(16x)^3$

4. $Y(t)=6t^{-9}$

What information is contained in the following quotients?

1. $\dfrac{R_A(3)}{R_B(3)}\approx 1.003$

2. $\dfrac{R'_A(3)}{R'_B(3)}=\dfrac{4623}{4097}\approx 1.13$

3. $\left[\dfrac{R_A(t)}{R_B(t)}\right]'\approx 1.001$ when $t=3$.

Using the function $$f(x)=\frac{\text{Number of votes}}{\text{Cost of those votes}}=\frac{2.3x}{7.1x^2+210}$$

Find the equation of the tangent line to the graph of $y=\left( 1+\sqrt{x}\right)\left(x-2\right)$ at $x=4$.

Differentiate $y=\dfrac{\left( x^2+1\right)\left(2x^2+1\right)}{3x^2+1}$

Find the derivative of $f(x)=\displaystyle\frac{(2x-1)^4}{(3x+2)}$ at $x=-1$ and interpret this result.

Find the derivative of $f(x)=x^5(4x-1)^{1/4}$.

Differentiate each function

1. $y=\left(5x^3+4x^2\right)^5$

2. $y=x^5$

3. $y=\left(x^2+1\right)^{100}$

4. $\sqrt{4+3x}$

5. $y=\displaystyle\frac{1}{\left(t^4+1\right)^3}$

AP Calculus Exam: Multiple Choice Question

Suppose both $y$ and $x$ are differentiable functions of $t$ and that the relationship between $y$ and $x$ is expressed by the equation $4x^3+3y^5=960$. Find and interpret $\displaystyle\frac{dy}{dt}$ when $\displaystyle\frac{dx}{dt}=4$, $x=6$, and $y=2$.

Find the rate at which the area is increasing at the time that the radius of the spill is $600$ feet.

Use the function $x=\displaystyle\frac{20000}{\sqrt[3]{2p^2-5}}+350$ to find the rate at which the number of instruments sold is changing with respect to time, when the price of an instrument is $\$400$ and is changing at a rate of $\$1$ per month.

1. Find $\displaystyle\frac{dy}{dx}$ for $x^2+y^2=16$.

2. Find $\displaystyle\frac{dy}{dx}$ for $xy-x-3y-3=0$.

3. Find the slope of the line tangent to $y^3=x^3(2-x)$ at $(1,-1)$

AP Calculus Exam: Multiple Choice Question