Question
The derivative of a function f is given by \(f'(x)=0.2x+e^{0.15x}\). Which of the following procedures can be used to determine the value of x at which the line tangent to the graph of f has slope 2 ?
A Evaluate \(0.2x+e^{0.15x}\) at x=2.
B Evaluate \(\frac{\mathrm{d} }{\mathrm{d} x}(0.2x+e^{0.15x})\) at x=2.
C Solve \(0.2x+e^{0.15x}\) for x.
D Solve \(\frac{\mathrm{d} }{\mathrm{d} x}(0.2x+e^{0.15x})=2\) for x.
Answer/Explanation
Ans:C
Question
The derivative of the function f is given by f′(x)=−2x+4 for all x, and f(−1)=5. Which of the following is an equation for the line tangent to the graph of f at x=−1 ?
A y=−2x+3
B y=−2x+4
C y=6x+5
D y=6x+11
Answer/Explanation
Ans:D
Question
The graph of f′, the derivative of a function f, is shown above. The points (2,7) and (4,18.8) are on the graph of f. Which of the following is an equation for the line tangent to the graph of f at x=2 ?
A y=2x−1
B y=4x−1
C y=4x−8
D y=5.9x−4.8A
Answer/Explanation
ns:B
Question
What is \(\lim_{\Delta x\rightarrow 0}\frac{\sin (\frac{\pi }{4}+\Delta x)\cos (\frac{\pi }{4}+\Delta x)-\sin (\frac{\pi }{4})\cos (\frac{\pi }{4})}{\Delta x}\) ?
(A) -1
(B) 0
(C) 12
(D) 1
Answer/Explanation
Ans:(B)
The definition of the derivative is as follows:
\(f{}'(x)=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}\)
so the limit given may be found by finding
\(f{}'(x)=\frac{\mathrm{d} }{\mathrm{d} x}(\sin x.\cos x)\)
evaluated at \(x=\frac{\pi }{4}\). Using the Chain Rule:
\(f{}'(x)=\cos x.\cos x+(-\sin x).(\sin x)\)
\(f{}'(x)=\cos ^{2}x-\sin ^{2}x\)