Definition of Differentiation (College Board AP® Calculus AB): Exam Questions

Let $f$ be the function defined by $f\left(x\right)=e^x\cos \left(\frac{x}{2}\right)$.

Find the average rate of change of $f$ on the interval $0\le x\le \pi$.

What is the slope of the line tangent to the graph of $f\left(x\right)=e^x\cos \left(\frac{x}{2}\right)$ at $x=3\pi$ ?

A curve has the equation $y=a{x}^3+b{x}^2+cx$, where $a,b$ and $c$ are constants. The point $\left(p,q\right)$, where $p$ and $q$ are constants, lies on this curve.

Find the equation of the tangent to this graph at the point $\left(p,q\right)$ in terms of $x$ and $y$, and the constants $a,b,c,p$ and $q$.

A new eco-friendly plastic decomposes over time. The mass of the material is modeled by $f\left(t\right)=4.98e^{-0.027t}$ for $0\le t\le 90$, where $f\left(t\right)$ is measured in pounds and $t$ is measured in days.

Find the average rate of decomposition of the material over the interval $0\le t\le 90$. Indicate units of measure.

A function $f$ is defined by $f\left(x\right)=\sin \left(-\frac{x}{2}\right)$, and $f\text{'}\left(\frac{\pi }{2}\right)=-\frac{\sqrt{2}}{4}$.

Find the equation of the line that is tangent to the graph of $g$ at $x=\frac{\pi }{2}$.

The function $f$ is defined by

$f\left(x\right)=\{\begin{array}{ll}{x}^2-3& \mathrm{for}x<1\\ 4x-5& \mathrm{for}x\ge 1\end{array}$

Is $f$ differentiable at $x=1$? Justify your answer.

Let $f$ be the function defined by $f\left(x\right)=e^x \text{cos}x$.

Find the average rate of change of $f$ on the interval $0\le x\le \pi$.

Functions $f$, $g$, and $h$ are twice-differentiable functions with $g\left(2\right)=h\left(2\right)=4$. The line $y=4+\frac{2}{3}\left(x-2\right)$ is tangent to both the graph of $g$ at $x=2$ and the graph of $h$ at $x=2$.

Find $h\text{'}\left(2\right)$.

The table above gives some values of $g\left(x\right)$ for a function $g$.

Find the equation of a line that is tangent to the graph of $g$ at $x=4$.

The function $f$ is defined by

$f\left(x\right)=\{\begin{array}{ll}2x+1& \mathrm{for}x<3\\ 3x-2& \mathrm{for}x\ge 3\end{array}$

Is $f$ differentiable for all values of $x$?

Use the definition of the derivative of the function, $f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(a+h\right)-f\left(a\right)}{h}$, at $x=a$ to justify your answer.

Using the definition of the derivative of the function,

$f\text{'}\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(a+h\right)-f\left(a\right)}{h}$, at $x=a$,

find the equation of the line that is normal to the graph $f\left(x\right)=4x^2$ at $x=-1$

The velocity of a particle is defined by the function

$v\left(t\right)=p{t}^2+qt+4$,

where $v\left(t\right)$ is measured in meters per second and $t$ is measured in seconds.

The velocity of the particle at $t=3$ is $7$ meters per second. The average rate of change of the velocity of the particle over the interval $3\le t\le 6$ is $-11$ meters per second per second. What are the values of $p$ and $q$?

A function $f$ is shown on the graph above. Two points, $p$ and $q$, are indicated on the graph. Point $p$ has coordinates $\left(5,8\right)$ and point $q$ is positioned where $x=7$.

A tangent to the graph at the midpoint between $p$ and $q$ has equation $y=3x-7$. What is $f\left(7\right)$?