5.5: Corequisite- Linear vs Exponential Models
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- 148622
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)SPECIFIC OBJECTIVES
By the end of this lesson, you should understand that
- using algebra to generalize calculations is a valuable tool.
- compounding is repeated multiplication by a compounding factor.
- compounding is best expressed in terms of exponential growth, using exponential notation.
- exponential growth models the compounding of interest on an initial investment.
By the end of this lesson, you will be able to
- write or interpret an expression that models percentage increase or decrease.
- use expressions and equations to solve contextual problems about percentage increases or decreases.
- calculate the earnings on a principal investment with annual compound interest.
- write a formula for annual compound interest.
- compare and contrast linear and exponential models.
INTRODUCTION
(1) The following question gives you practice with using percentages in expressions and equations. Match the expressions that either increase or decrease the quantity x by a given percentage with the verbal descriptions of the changes. A verbal description may be used more than once.
Answer | Column A | Column B | |
x + 0.45x | a) increase x by 45% | ||
0.45x | b) increase x by 55% | ||
x – 0.55x | c) increase x by 145% | ||
1.45x | d) increase x by 245% | ||
3.45x | e) increase x by 345% | ||
2.45x | f) decrease x by 55% | ||
x + 0.55x | g) decrease x by 45% | ||
x – 0.45x | h) decrease x by 145% | ||
0.55x |
PROBLEM SITUATION: THE FIVE-YEAR CD
Suppose you invest $1,000 principal (the original deposit in an investment) into a certificate of deposit (CD) with a five-year term (the agreed upon period of time that money is in an investment) that pays a 5% annual interest rate. The compounding period is one year. Recall that in compound interest, interest is added to the starting principal and accrues after set periods of time through the duration of the whole term.
(2) How much money will you have in your account at the end of the five-year term? (Be ready to explain your calculations.) Use the table below to find a pattern and develop a formula to model the total amount accrued in a CD with annual compounding after n years, if the principal = $1,000 and the APR = 5%. Round figures in the Amount Accrued column to two decimal places.
Term | Calculation | Amount Accrued |
1 year | $1,000 + $1,0000.05 = $1,000(1 + 0.05) = $1,000*1.05 | $1,050.00 |
2 years | ($1,000*1.05)(1.05) = $1,000(1.05)2 | |
3 years | ||
4 years | ||
5 years | ||
n years |
(3) Is your formula from the previous question linear? Explain.
(4) Using the formula and patterns you developed, fill out the following table. Round figures in the Amount Accrued column to two decimal places.
Term | Calculation | Amount Accrued |
10 years | ||
20 years | ||
30 years | ||
40 years | ||
50 years |
(5) Use the values from the table above to plot a graph of the model you developed. Note: If completing this problem online, follow the instructions given online to create your graph.
Behavior of Exponential Growth/Decay
Note from the graph, that the growth is very small at first, but then it changes dramatically. Here are some other graphs that illustrate both exponential growth and exponential decay:
Exponential growth (increasing rate of change)
Exponential decay (decreasing rate of change)
(6) Write a general formula that could be used to find the accrued amount (A) for a CD with annual compounding. Let P = the principal, r = the annual interest rate as a decimal, and t = number of years of the investment.