4.1E: Exercises
 Page ID
 31097
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left#1\right}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Verbal
1) Explain why the values of an increasing exponential function will eventually overtake the values of an increasing linear function.
 Answer:

Linear functions have a constant rate of change. Exponential functions increase based on a percent of the original.
2) Given a formula for an exponential function, is it possible to determine whether the function grows or decays exponentially just by looking at the formula? Explain.
3)The Oxford Dictionary defines the word nominal as a value that is “stated or expressed but not necessarily corresponding exactly to the real value.” Develop a reasonable argument for why the term nominal rate is used to describe the annual percentage rate of an investment account that compounds interest.
 Answer:

When interest is compounded, the percentage of interest earned to principal ends up being greater than the annual percentage rate for the investment account. Thus, the annual percentage rate does not necessarily correspond to the real interest earned, which is the very definition of nominal.
Algebraic
For the following exercises, identify whether the statement represents an exponential function. Explain.
4) The average annual population increase of a pack of wolves is \(25\).
5) A population of bacteria decreases by a factor of \(\frac{1}{8}\) every \(24\) hours.
 Answer:

exponential; the population decreases by a proportional rate.
6) The value of a coin collection has increased by \(3.25\%\)$\text{\hspace{0.17em}}$annually over the last \(20\) years.
7) For each training session, a personal trainer charges his clients \(\$5\) less than the previous training session.
 Answer:

not exponential; the charge decreases by a constant amount each visit, so the statement represents a linear function.
8) The height of a projectile at time \(t\) is represented by the function \(h(t)= 4.9t^2 + 18t + 40\)
For the following exercises, consider this scenario: For each year
9) Which forest’s population is growing at a faster rate?
 Answer:

The forest represented by the function
10) Which forest had a greater number of trees initially? By how many?
11) Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after \(20\) years? By how many?
 Answer:

After
12) Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after \(100\)$\text{\hspace{0.17em}}$years? By how many?
13) Discuss the above results from the previous four exercises. Assuming the population growth models continue to represent the growth of the forests, which forest will have the greater number of trees in the long run? Why? What are some factors that might influence the longterm validity of the exponential growth model?
 Answer:

Answers will vary. Sample response: For a number of years, the population of forest A will increasingly exceed forest B, but because forest B actually grows at a faster rate, the population will eventually become larger than forest A and will remain that way as long as the population growth models hold. Some factors that might influence the longterm validity of the exponential growth model are drought, an epidemic that culls the population, and other environmental and biological factors.
For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain.
14)
15)
 Answer:

exponential growth; The growth factor, \(1.06\) is greater than
16) \(y=16.5(1.025)^{\frac{1}{x}}\)
17) \(y=11,701(0.97)^t\)
 Answer:

exponential decay; The decay factor,
For the following exercises, find the formula for an exponential function that passes through the two points given.
18)
19) \((0,2000)\)$\text{\hspace{0.17em}}$and
 Answer:

\(f(x)=2000(0.1)^x\)
20) \(\left (−1,\frac{3}{2} \right )\) and \((3,24)\)
21) \(
 Answer:

\(f(x)=\left ( \frac{1}{6} \right )^{\frac{3}{5}} \left ( \frac{1}{6} \right )^{\frac{x}{5}}\approx 2.93 (0.699)^x\)
22) \((3,1)\) and \((5,4)\)
For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.
23)
1  2  3  4  
70  40  10  20 
 Answer:

Linear
24)
1  2  3  4  
70  49  34.3  24.01 
25)
1  2  3  4  
80  61  42.9  25.61 
 Answer:

Neither
26)
1  2  3  4  
10  20  40  80 
27)
1  2  3  4  
3.25  2  7.25  12.5 
 Answer:

Linear
For the following exercises, use the compound interest formula,
28) After a certain number of years, the value of an investment account is represented by the equation
29) What was the initial deposit made to the account in the previous exercise?
 Answer:

\(\$10,250\)
30) How many years had the account from the previous exercise been accumulating interest?
31) An account is opened with an initial deposit of \(\$6,500\) and earns
 Answer:

\(\$13,268.58\)
32) How much more would the account in the previous exercise have been worth if the interest were compounding weekly?
33) Solve the compound interest formula for the principal,
 Answer:

\(P=A(t)\cdot \left (1+ \frac{r}{n} \right )^{nt}\)
34) Use the formula found in the previous exercise to calculate the initial deposit of an account that is worth
35) How much more would the account in the previous two exercises be worth if it were earning interest for \(5\)
 Answer:

\(\$4,572.56\)
36) Use properties of rational exponents to solve the compound interest formula for the interest rate,
37) Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded semiannually, had an initial deposit of \(\$9,000\) and was worth \(\$13,373.53\) after \(10\) years.
 Answer:

\(4\%\)
38) Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded monthly, had an initial deposit of \(\$5,500\), and was worth \(\$38,455\) after \(30\) years.
For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain.
39) \(y=3742(e)^{0.75t}\)
 Answer:

continuous growth; the growth rate is greater than
40) \(y=150(e)^{\frac{3.25}{t}}\)
41) \(y=2.25(e)^{2t}\)
 Answer:

continuous decay; the growth rate is less than
42) Suppose an investment account is opened with an initial deposit of
43) How much less would the account from Exercise 42 be worth after
 Answer:

\(\$669.42\)
Numeric
For the following exercises, evaluate each function. Round answers to four decimal places, if necessary.
44) \(f(x)=2(5)^x\) for \(f(3)\)
45) \(f(x)=4^{2x+3}\) for \(f(1)\)
 Answer:

\(f(1)=4\)
46) \(f(x)=e^x\), for \(f(3)\)
47) \(f(x)=2e^{x1}\), for \(f(1)\)
 Answer:

\(f(1)\approx 0.2707\)
48) \(f(x)=2.7(4)^{x+1}+1.5\), for \(f(2)\)
49) \(f(x)=1.2e^{2x}0.3\), for \(f(3)\)
 Answer:

\(f(3)\approx 483.8146\)
50) \(f(x)=\frac{3}{2}(3)^{x}+\frac{3}{2}\), for \(f(2)\)
Technology
For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve.
51) \((0,3)\) and \((3,375)\)
 Answer:

\(y=3\cdot 5^x\)
52) \((3,222.62)\) and \((10,77.456)\)
53) \((20,29.495)\) and \((150,730.89)\)
 Answer:

\(y\approx 18\cdot 1.025^x\)
54) \((5,2.909)\) and \((13,0.005)\)
55) ((11,310.035)\) and \((25,356.3652)\)
 Answer:

\(y\approx 0.2\cdot 1.95^x\)
Extensions
56) The annual percentage yield (APY) of an investment account is a representation of the actual interest rate earned on a compounding account. It is based on a compounding period of one year. Show that the APY of an account that compounds monthly can be found with the formula
57) Repeat the previous exercise to find the formula for the APY of an account that compounds daily. Use the results from this and the previous exercise to develop a function \(I(n)\)$\text{\hspace{0.17em}}$for the APY of any account that compounds \(n\)$\text{\hspace{0.17em}}$times per year.
 Answer:

\(\begin{align*} APY &= \frac{A(t)a}{a}\\ &= \frac{a\left ( 1+\frac{r}{365} \right )^{365(1)}a}{a}\\ &= \frac{a\left [\left ( 1+\frac{r}{365} \right )^{365}1 \right ]}{a}\\ &= \left ( 1+\frac{r}{365} \right )^{365}1 \end{align*}\); \(I(n)=\left ( 1+\frac{r}{n} \right )^n  1\)
58) Recall that an exponential function is any equation written in the form \(f(x)=a\cdot b^x\)
59) In an exponential decay function, the base of the exponent is a value between \(0\) and \(1\). Thus, for some number
 Answer:

Let \(f\) be the exponential decay function
\(\begin{align*} f(x) &= a\cdot \left (\frac{1}{b} \right )^x \\ &= a \left (b^{1} \right )^x\\ &= a\left ( (e^n)^{1} \right )^x\\ &= a\left ( e^{n} \right )^x\\ &= a(e)^{nx} \end{align*}\)
60) The formula for the amount \(A\)
61) The fox population in a certain region has an annual growth rate of \(9\%\) per year. In the year 2012, there were \(23,900\) fox counted in the area. What is the fox population predicted to be in the year 2020?
 Answer:

\(47,622\) fox
62) A scientist begins with \(100\) milligrams of a radioactive substance that decays exponentially. After \(35\) hours, \(50\)mg of the substance remains. How many milligrams will remain after \(54\) hours?
63) In the year 1985, a house was valued at \(\$110,000\). By the year 2005, the value had appreciated to \(\$145,000\). What was the annual growth rate between 1985 and 2005? Assume that the value continued to grow by the same percentage. What was the value of the house in the year 2010?
 Answer:

\(1.39\%\); \(\$155,368.09\)
64) A car was valued at \(\$38,000\) in the year 2007. By 2013, the value had depreciated to \(\$11,000\) If the car’s value continues to drop by the same percentage, what will it be worth by 2017?
65) Jamal wants to save \(\$54,000\) for a down payment on a home. How much will he need to invest in an account with \(8.2\%\) APR, compounding daily, in order to reach his goal in \(5\) years?
 Answer:

\(\$35,838.76\)
66) Kyoko has \(\$10,000\) that she wants to invest. Her bank has several investment accounts to choose from, all compounding daily. Her goal is to have \(\$15,000\) by the time she finishes graduate school in \(6\) years. To the nearest hundredth of a percent, what should her minimum annual interest rate be in order to reach her goal? (Hint: solve the compound interest formula for the interest rate.)
67) Alyssa opened a retirement account with \(7.25\%\) APR in the year 2000. Her initial deposit was \(\$13,500\). How much will the account be worth in 2025 if interest compounds monthly? How much more would she make if interest compounded continuously?
 Answer:

\(\$82,247.78\); \(\$449.75\)
68) An investment account with an annual interest rate of \(7\%\) was opened with an initial deposit of \(\$4,000\) Compare the values of the account after \(9\) years when the interest is compounded annually, quarterly, monthly, and continuously.