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Chapter 4: Problem 31
Find the intervals on which \(f\) is increasing and decreasing. $$f(x)=\tan ^{-1} x$$
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Suppose you make a deposit of \(\$ P\) into a savings account that earns interest at a rate of \(100 \mathrm{r} \%\) per year. a. Show that if interest is compounded once per year, then the balance after \(t\) years is \(B(t)=P(1+r)^{t}\). b. If interest is compounded \(m\) times per year, then the balance after \(t\) years is \(B(t)=P(1+r / m)^{m t} .\) For example, \(m=12\) corresponds to monthly compounding, and the interest rate for each month is \(r / 12 .\) In the limit \(m \rightarrow \infty,\) the compounding is said to be continuous. Show that with continuous compounding, the balance after \(t\) years is \(B(t)=\overline{P e^{r t}}\).
Show that the general quartic (fourth-degree) polynomial \(f(x)=x^{4}+a x^{3}+b x^{2}+c x+d\) has either zero or two inflection points, and the latter case occurs provided that \(b<3 a^{2} / 8.\)
Linear approximation a. Write an equation of the line that represents the linear approximation to the following functions at a. b. Graph the function and the linear approximation at a. c. Use the linear approximation to estimate the given quantity. d. Compute the percent error in your approximation. $$f(x)=\tan x ; a=0 ; \tan 3^{\circ}$$
The velocity function and initial position of Runners \(A\) and \(B\) are given. Analyze the race that results by graphing the position functions of the runners and finding the time and positions (if any) at which they first pass each other. $$\text { A: } v(t)=2 e^{-t}, s(0)=0 ; \quad \text { B: } V(t)=4 e^{-4 t}, S(0)=10$$
Use analytical methods to evaluate the following limits. $$\lim _{x \rightarrow 6} \frac{\sqrt[5]{5 x+2}-2}{1 / x-1 / 6}$$
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