AP Calculus BC : 2.7 Derivatives of cos x, sin x, ex, and ln x- Exam Style questions with Answer- MCQ

Question

If g(x)=3sinx+2cosx+5, then \(g'(\frac{\pi }{3})\)=
A \(\frac{3}{2}-\sqrt{3}\)
B \(-\frac{3}{2}+\sqrt{3}\)
C \(\frac{3}{2}+\sqrt{3}\)
D \(6+\frac{3}{2}\sqrt{3}\)

Answer/Explanation

Ans:A
This question involves using the basic rules for the differentiation of trigonometric functions, and then evaluating the derivative at π3.
g′(x)=3cosx−2sinx
\(g'(\frac{\pi }{3})=3\cos (\frac{\pi }{3})-2\sin (\frac{\pi }{3})\)

\(=3(\frac{1}{2})-2(\frac{\sqrt{3}}{2})\)

 

Question

Let g be the function given by \(\lim_{h\rightarrow 0}\frac{\cos (x+h)-\cos x}{h}\). What is the instantaneous rate of change of g with respect to x at \(x=\frac{\pi }{3}\)  ?
A \(\frac{\sqrt{3}}{2}\)

B \(\frac{1}{2}\)

C \(-\frac{1}{2}\)

D \(-\frac{3}{\sqrt{2}}\)

Answer/Explanation

Ans:C

Question

\(\lim_{h\rightarrow 0}\frac{7e^{x}-7e^{(x+h)}}{4h}=\)
A \(-7e^{x}\)

B \(7e^{x}\)

C \(-\frac{7}{4}e^{x}\)

D \(\frac{7}{4}e^{x}\)

Answer/Explanation

Ans:C

Question

If \(f(x)=e^{1/x}\), then f'(x)=
(A)\(-\frac{e^{1/x}}{x^{2}}\)         (B)\(-e^{1/x}\)     (C)\(\frac{e^{1/x}}{x}\)                        (D)\(\frac{e^{1/x}}{x^{2}}\)                            (E)\(\frac{1}{x}e^{(1/x)-1}\)

Answer/Explanation

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