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