91Ó°ÊÓ

Find A\( and \)B\( in \)y=A \sin x+B \cos x\( so that \)y^{\prime \prime}-y=\sin x

The velocity of a falling body is \(v=8 \sqrt{s-t}+1\) feet per second at the instant \(t(\mathrm{sec})\) the body has fallen \(s\) feet from its starting point. Show that the body's acceleration is 32 \(\mathrm{ft} / \mathrm{sec}^{2}\) .

What is the largest value possible for the slope of the curve \(y=\sin (x / 2) ?\)

True or False \(\frac{d}{d x}\left(\pi^{3}\right)=3 \pi^{2} .\) Justify your answer.

Radioactive Decay The amount \(A\) (in grams) of radioactive plutonium remaining in a 20 -gram sample after \(t\) days is given by the formula $$ A=20 \cdot(1 / 2)^{t / 140} $$ At what rate is the plutonium decaying when \(t=2\) days? Answer in appropriate units. rate \(\approx 0.098\) grams/day

For any positive constant \(k,\) the derivative of \(\ln (k x)\) is 1\(/ x\) (a) by using the Chain Rule. (b) by using a property of logarithms and differentiating.

True or False The graph of \(f(x)=1 / x\) has no horizontal tangents. Justify your answer.

Write an equation for the tangent to the curve \(y=\sin m x\) at the origin.

Derivatives of Exponential and Logarithmic Functions 179 Let \(f(x)=2^{x}\)\ (a) Find \(f^{\prime}(0) . \quad\) ln 2 (b) Use the definition of the derivative to write \(f^{\prime}(0)\) as a limit (c) Deduce the exact value of \(\lim _{h \rightarrow 0} \frac{2^{h}-1}{h}\) (d) What is the exact value of \(\lim _{h \rightarrow 0} \frac{7^{h}-1}{h} ?\)

Multiple Choice Let \(y=u v\) be the product of the functions \(u\) and \(v .\) Find \(y^{\prime}(1)\) if \(u(1)=2, u^{\prime}(1)=3, v(1)=-1,\) and \(v^{\prime}(1)=1\) \((\mathbf{A})-4 \quad(\mathbf{B})-1 \quad(\mathbf{C}) 1, \quad(\mathbf{D}) 4 \quad(\mathbf{E})7\)