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

Let $f$ be the function defined by $f\left(x\right)=\sqrt{\left|x-5\right|}$ for all $x$. Which of the following statements is true?

$\underset{h\to 0}{\lim }\frac{\sin \left(\pi +h\right)-\sin \left(\pi \right)}{h}$ is

The graph of the function $f$ shown in the figure above has a vertical tangent at the point (3, 0) and horizontal tangents at the points (2, 0) and (5, -2).

For what values of $x$, $-1<x<7$ is $f$ not differentiable?

Let $f$ be the function defined by $f\left(x\right)=2x^3+4x-1$. Which of the following is an equation of the line tangent to the graph of $f$ at the point where $x=1$?

Let $f$ be a differentiable function such that $f\left(8\right)=10$ and $f\text{'}\left(x\right)\ge 2$ for all $x$. Of the following, which is not a possible value for $f\left(6\right)$?

Let $g$ be the function given by $g\left(x\right)=|x-2|$. Which of the following statements about $g$ are true?

I. $g$ is continuous at $x=2$.

II. $g$ is differentiable at $x=2$.

III. $g$ has an absolute minimum at $x=2$.

$f\left(x\right)=\{\begin{array}{l}5-x^2,x<-3\\ \frac{6}{x}-1,-3\le x<-1\\ 2x-5,x\ge -1\end{array}$

Let $f$ be the function defined above. At what values of $x$, if any, is $f$ not differentiable?

Which of the following is an equation of the tangent line to the graph of $f\left(x\right)=2x^4+3x^2$ at the point where $f^{\text{'}}\left(x\right)=-1$?

$f\left(x\right)=\{\begin{array}{l}ax-b,x\le 2\\ x^2+\left(10-a\right)x,x>2\end{array}$

Let $f$ be the function defined above, where $a$ and $b$ are constants. If $f$ is differentiable at $x=2$, what is the value of $a-b$?

In the $xy$-plane, the line $8x+y=k$, where $k$ is a constant, is tangent to the graph with derivative $\frac{dy}{dx}=6x-2$ at $\left(x,10\right)$. What is the value of $k$?

A differentiable function $f$ has the property that $f\text{'}\left(x\right)<-1$ for $-2\le x\le 7$ and $f\left(6\right)=-4$. Which of the following could be true?

I. $f\left(3\right)=-5$

II. $f\left(-1\right)=3$

III. $f\left(2\right)=4$