AP-Style Practice Questions
Topics 5.1 – 5.12AB
Show all supporting work on scratch paper. On the AP Exam, Section I is split into a no-calculator and a calculator-allowed part — each question below is labeled accordingly.
Let $f(x)=x^{2}-3x$ on $[0,4]$. The value $c$ guaranteed by the MVT is
The critical points of $f(x)=x^{3}-3x$ are
$f(x)=x^{3}-6x^{2}+9x$ is decreasing on
If $f'$ changes from negative to positive at $x=c$, then $f$ has
$f(x)=x^{4}-6x^{2}$ is concave up on
For $f(x)=x^{3}-3x$, at $x=-1$ the function has
The graph of $y=x^{5}-5x^{4}$ has a point of inflection at
The graph of $f'$ is shown. On which interval is $f$ increasing and concave down?
The absolute maximum of $f(x)=x^{3}-3x$ on $[-2,2]$ is
A rectangle with one side on the $x$-axis has its upper two vertices on the parabola $y=12-x^{2}$. The maximum area of such a rectangle is
The point on $y=\sqrt{x}$ closest to $(3,0)$ has $x$-coordinate
If $f'(x)>0$ and $f''(x)<0$ on $(a,b)$, then the graph of $f$ on $(a,b)$ is
On $[0,2]$, the absolute maximum of $f(x)=x\sqrt{2-x}$ is
The table gives values of a continuous function $f''$. The graph of $f$ must have an inflection point in which interval(s)?
| $x$ | $0$ | $1$ | $2$ | $3$ | $4$ |
|---|---|---|---|---|---|
| $f''$ | $-3$ | $-1$ | $2$ | $1$ | $-1$ |
A page must have a printed area of $96$ in², with $1$ in. margins on each side and $1.5$ in. margins top and bottom. The dimensions (width × height) that minimize total page area are closest to
If the graph of $f'$ crosses the $x$-axis at $x=1$ (from + to −) and has a local min at $x=3$, then $f$ has
Which hypothesis of Rolle's Theorem is violated for $f(x)=|x|$ on $[-1,1]$?
Which of the following must be true if $f''(x)>0$ for all $x$?
Curve-sketching and optimization FRQs require sign charts for $f'$ and $f''$, clear tests (first / second derivative / closed-interval), and interval justifications.
Let $f(x)=x^{3}-6x^{2}+9x+2$.
Let $f(x)=\dfrac{x}{x^{2}+1}$ on $[0,3]$.
A rectangular box with a square base and open top has volume $256$ in³. Material for the base costs $\$6$/in² and for the sides costs $\$2$/in².
The graph of $f'$, the derivative of a function $f$, is shown on $[-2,6]$. The graph consists of two line segments and a parabolic arc.
A farmer has $800$ meters of fencing and wants to enclose a rectangular field along a straight river (no fencing is needed along the river) and then divide it with one fence parallel to the river.