91Ó°ÊÓ

$$\begin{gathered} \mathrm{The}\mathrm{table}\mathrm{that}\mathrm{follows}\mathrm{describes}\mathrm{the}\mathrm{activity}\mathrm{in}\mathrm{a}\mathrm{college}\mathrm{tuition}\mathrm{savings}\mathrm{account}\mathrm{over}\mathrm{four}\mathrm{years}. \\ \mathrm{Notice}\mathrm{that}2008\mathrm{was}\mathrm{a}\mathrm{particularly}\mathrm{bad}\mathrm{year}\mathrm{for}\mathrm{investing}! \\ \mathrm{Let}I\left(t\right)\mathrm{be}\mathrm{the}\mathrm{amount}\mathrm{by}\mathrm{which}\mathrm{your}\mathrm{account}\mathrm{increased}\mathrm{or}\mathrm{decreased}\mathrm{in}\mathrm{year}t, \\ \mathrm{and}\mathrm{let}B\left(t\right)\mathrm{be}\mathrm{the}\mathrm{balance}\mathrm{of}\mathrm{your}\mathrm{account}\mathrm{at}\mathrm{the}\mathrm{end}\mathrm{of}\mathrm{year}t. \end{gathered}$$

$$\begin{gathered} \left(\mathrm{a}\right)\mathrm{Describe}\mathrm{in}\mathrm{your}\mathrm{own}\mathrm{words}\mathrm{how}B\left(t\right)\mathrm{is}\mathrm{the}\mathrm{accumulation}\mathrm{function}\mathrm{of}I\left(t\right). \\ \left(\mathrm{b}\right)\mathrm{Plot}a\mathrm{step}-\mathrm{function}\mathrm{graph}\mathrm{of}I\left(t\right),\mathrm{and}\mathrm{describe}\mathrm{how}B\left(t\right)\mathrm{relates}\mathrm{to}\mathrm{the}\mathrm{area}\mathrm{under}\mathrm{graph}. \\ \left(\mathrm{c}\right)\mathrm{What},\mathrm{if}\mathrm{anything},\mathrm{can}\mathrm{you}\mathrm{say}\mathrm{about}B\left(t\right)\mathrm{when}I\left(t\right)\mathrm{is}\mathrm{positive}?\mathrm{Negative}? \\ \mathrm{If}\mathrm{you}\mathrm{had}\mathrm{to}\mathrm{guess}\mathrm{that}\mathrm{one}\mathrm{of}\mathrm{these}\mathrm{functions}\mathrm{was}\mathrm{related}\mathrm{to}\mathrm{the}\mathrm{derivative}\mathrm{of}\mathrm{other},\mathrm{which}\mathrm{one}\mathrm{would}\mathrm{it}\mathrm{be}? \end{gathered}$$

Use the Fundamental Theorem of Calculus to find the exact

values of each of the definite integrals in Exercises 19–64. Use

a graph to check your answer. (Hint: The integrands that involve

absolute values will have to be considered piecewise.)

${∫}_{\raisebox{1ex}{$\mathrm{π}$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$\mathrm{π}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}x\left(cs{c}^2\right(x^2\left)\right)dx$

For each function f and interval [a, b] in Exercises 53–55, approximate the average value of f from x = a to x = b, using a sample size of at least eight.

$f\left(x\right)=\sin x,\left[0,2\mathrm{Ï€}\right]$

Use integration formulas to solve each integral in Exercises 21–62. You may have to use algebra, educated guess- and- check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating. (Hint for Exercise 54: $\tan \left(x\right)=\frac{\sin \left(x\right)}{\cos \left(x\right)}$).

$∫(e^x\sqrt{x}+\frac{e^x}{2\sqrt{x}})dx.$

For the logarithmic value, use Riemann sums with at least four rectangles to find an over approximation and an under-approximation for the value.

$\ln 0.4$

Use integration formulas to solve each integral in Exercises 21–62. You may have to use algebra, educated guess- and- check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating. (Hint for Exercise 54: $\tan \left(x\right)=\frac{\sin \left(x\right)}{\cos \left(x\right)}$).

$∫\frac{1}{2\sqrt{x}}{e}^{\sqrt{x}}dx.$

$$\begin{gathered} \mathrm{Suppose}100\mathrm{mg}\mathrm{of}\mathrm{a}\mathrm{drug}\mathrm{is}\mathrm{administered}\mathrm{to}\mathrm{a}\mathrm{patient}\mathrm{each}\mathrm{morning}\mathrm{in}\mathrm{pill}\mathrm{form} \\ \mathrm{and}\mathrm{it}\mathrm{is}\mathrm{known}\mathrm{that}\mathrm{after}24\mathrm{hours}\mathrm{the}\mathrm{body}\mathrm{processes}80\%\mathrm{of}\mathrm{the}\mathrm{drug}\mathrm{from}\mathrm{such}\mathrm{a}\mathrm{pill}, \\ \mathrm{leaving}20\%\mathrm{of}\mathrm{the}\mathrm{drug}\mathrm{in}\mathrm{the}\mathrm{body}.\mathrm{The}\mathrm{amount}\mathrm{of}\mathrm{the}\mathrm{drug}\mathrm{in}\mathrm{the}\mathrm{body}\mathrm{right}\mathrm{after}\mathrm{the}\mathrm{first}\mathrm{pill}\mathrm{is}\mathrm{taken}\mathrm{is} \\ A\left(1\right)=100mg.24\mathrm{hours}\mathrm{later},\mathrm{after}\mathrm{the}\mathrm{second}\mathrm{pill}\mathrm{has}\mathrm{been}\mathrm{taken},\mathrm{the}\mathrm{amount}\mathrm{in}\mathrm{the}\mathrm{body}\mathrm{is} \\ A\left(2\right)=100(0.2)+100=120mg.48\mathrm{hours}\mathrm{later},\mathrm{the}\mathrm{amount}\mathrm{in}\mathrm{the}\mathrm{body}\mathrm{right}\mathrm{after}\mathrm{taking}\mathrm{the}\mathrm{third}\mathrm{pill}\mathrm{is} \\ A\left(3\right)=100(0.2)(0.2)+100(0.2)+100=124mg. \end{gathered}$$

$$\begin{gathered} \left(\mathrm{a}\right)\mathrm{Explain}\mathrm{the}\mathrm{calculations}\mathrm{for}A\left(1\right),A\left(2\right),\mathrm{and}A\left(3\right)\mathrm{described}\mathrm{in}\mathrm{the}\mathrm{exercise}. \\ \mathrm{Which}\mathrm{term}\mathrm{in}A\left(3\right)\mathrm{corresponds}\mathrm{to}\mathrm{the}\mathrm{drug}\mathrm{left}\mathrm{from}\mathrm{the}\mathrm{first}\mathrm{pill}? \\ \left(\mathrm{b}\right)\mathrm{Interpret}\mathrm{the}\mathrm{given}\mathrm{graph}\mathrm{in}\mathrm{the}\mathrm{context}\mathrm{of}\mathrm{this}\mathrm{problem}.\mathrm{What}\mathrm{do}\mathrm{the}\mathrm{marked}\mathrm{points}\mathrm{represent}? \\ \left(\mathrm{c}\right)\mathrm{Express}A\left(n\right)\mathrm{in}\mathrm{sigma}\mathrm{notation}. \\ \left(\mathrm{d}\right)\mathrm{Calculate}\mathrm{the}\mathrm{amount}\mathrm{of}\mathrm{drug}\mathrm{in}\mathrm{the}\mathrm{body}\mathrm{after}\mathrm{the}{4}^{th}\mathrm{through}{10}^{th}\mathrm{pills}. \\ \mathrm{Do}\mathrm{you}\mathrm{notice}\mathrm{anything}\mathrm{special}\mathrm{about}A\left(n\right)\mathrm{as}\mathrm{n}\mathrm{gets}\mathrm{larger}? \end{gathered}$$

Use the definition of the definite integral as a limit of Riemann sums to prove Theorem 4.12(a): For any function f and real number a,

${∫}_a^{a}f\left(x\right)dx=0$

For each function f and interval[a, b] in Exercises 56–67, use definite integrals and the Fundamental Theorem of Calculus to find the exact average value of f from x = a to x = b. Then use a graph of f to verify that your answer is reasonable.

$f\left(x\right)=x-1,\left[-1,3\right]$

Use the Fundamental Theorem of Calculus to find the exact

values of each of the definite integrals in Exercises 19–64. Use

a graph to check your answer. (Hint: The integrands that involve

absolute values will have to be considered piecewise.)

${∫}_{-\raisebox{1ex}{$\mathrm{π}$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$\mathrm{π}$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{x}^2\left(se{c}^2\right(x^2\left)\right)dx$