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)\(\frac{1}{x}e^{(1/x)-1}\)