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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblem 55
Chapter 5, Problem 55

Use the compound interest formulas A = P (1+ r/n)nt and A =Pert to solve exercises 53-56. Round answers to the nearest cent. Suppose that you have \$12,000 to invest. Which investment yields the greater return over 3 years: 0.96% compounded monthly or 0.95% compounded continuously?

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Identify the given values: the principal amount \(P = 12000\), the time period \(t = 3\) years, the interest rates \(r_1 = 0.0096\) (0.96%) for monthly compounding, and \(r_2 = 0.0095\) (0.95%) for continuous compounding.
For the investment compounded monthly, use the formula \(A = P \left(1 + \frac{r}{n}\right)^{nt}\), where \(n = 12\) (months per year). Substitute the values to get \(A_1 = 12000 \left(1 + \frac{0.0096}{12}\right)^{12 \times 3}\).
For the investment compounded continuously, use the formula \(A = Pe^{rt}\). Substitute the values to get \(A_2 = 12000 \times e^{0.0095 \times 3}\).
Calculate the expressions inside the parentheses and exponents separately for both formulas, but do not compute the final numerical values yet.
Compare the two amounts \(A_1\) and \(A_2\) after evaluating the expressions to determine which investment yields the greater return over 3 years.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Compound Interest Formula (Periodic Compounding)

This formula, A = P(1 + r/n)^(nt), calculates the amount of money accumulated after interest is compounded periodically. Here, P is the principal, r is the annual interest rate, n is the number of compounding periods per year, and t is the time in years. It helps determine the future value of an investment with discrete compounding intervals.
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Continuous Compounding Formula

The formula A = Pe^(rt) models interest compounded continuously, where e is Euler’s number (~2.718). This represents the limit of compounding frequency increasing indefinitely. It is used to find the future value when interest is added constantly, providing a slightly higher return than periodic compounding at the same rate.
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Comparing Investment Returns

To determine which investment yields a greater return, calculate the final amounts using both compounding methods with the given rates and time. Comparing these results shows which option grows the principal more. Rounding to the nearest cent ensures practical financial interpretation.
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Example 3
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Textbook Question

Use the compound interest formulas A = P (1+ r/n)nt and A =Pert to solve exercises 53-56. Round answers to the nearest cent. Find the accumulated value of an investment of \$10,000 for 5 years at an interest rate of 1.32% if the money is d. compounded continuously.

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Begin by graphing f(x) = log₂ x. Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. h(x)=1+ log₂ x