AP Calculus AB Unit 2.7: Derivative Rules

Practice AP Calculus Unit 2.7 questions on differentiating familiar trigonometric, exponential, and logarithmic functions.

Covered syllabus points

FUN-3.A—Calculate derivatives of familiar functions—Topic 2.7 LIM-3.A—Interpret a limit as a definition of a derivative

[Maximum number: 2]

The shaded region R is bounded by the graphs of the functions f and g, where $f(x)=x^{2}-2 x$

and $g(x)=x+\sin (\pi x)$ , as shown in the figure.

Figure for Question 2.7 Derivatives of cos x, sin x, eˣ, and ln x question 1 — AP Calculus AB

(Note: Your calculator should be in radian mode.)

It can be shown that $g^{\prime}(x)=1+\pi \cos (\pi x)$ . Find the value of x, for 0<x<1, at which the line

tangent to the graph of f is parallel to the line tangent to the graph of g.

D It can be shown that $g^{\prime}(x)=1+\pi \cos (\pi x)$ . Find the value of x, for 0<x<1, at which the line tangent

to the graph of f is parallel to the line tangent to the graph of g.

$f^{\prime}(x)=g^{\prime}(x) \Rightarrow 2 x-2=1+\pi \cos (\pi x)$	$f^{\prime}(x)=g^{\prime}(x)$	Point 8 (P8)
$\Rightarrow \quad x=0.675819$ <br>The lines tangent to the graphs of f and g are parallel at x=0.676 (or 0.675).	Answer	Point 9 (P9)
Scoring Notes for Part D

- P8 is earned for a general statement, such as $f^{\prime}(x)=g^{\prime}(x)$ , or for any correct equation formed by

substituting 2 x-2 for $f^{\prime}(x), 1+\pi \cos (\pi x)$ for $g^{\prime}(x)$ , or both.

- P9 is earned for the correct answer, with or without supporting work. A reported answer should be

accurate to three places after the decimal point, rounded or truncated. An inappropriately rounded

answer does not earn the point, unless an earlier point was not earned due to inappropriate rounding.

Part B (AB or BC): Graphing calculator not allowed

[Maximum number: 1]

$\lim _{h \rightarrow 0} \frac{\sin \left(\frac{\pi}{3}+h\right)-\sin \left(\frac{\pi}{3}\right)}{h}$ is

$\frac{1}{2}$

$\frac{\sqrt{3}}{2}$

nonexistent

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Covered syllabus points

2.7 Derivatives of cos x, sin x, eˣ, and ln x question 1

2.7 Derivatives of cos x, sin x, eˣ, and ln x question 2