Math Problem Statement
If $1100 is invested in an account that pays 5% compounded annually, the total amount, A(t), in the account after t years is Upper A left parenthesis t right parenthesis equals 1100 left parenthesis 1.05 right parenthesis Superscript t. a. Find the average rate of change per year of the total amount in the account for the first five years of the investment (from tequals0 to tequals5). b. Find the average rate of change per year of the total amount in the account for the second five years of the investment (from tequals5 to tequals10). c. Estimate the instantaneous rate of change for tequals5. Question content area bottom Part 1 a. Which of the following is the correct formula for finding the average rate of change from tequalsa to tequalsb? A.StartFraction Upper A left parenthesis 5 right parenthesis Over Upper A left parenthesis 0 right parenthesis EndFraction StartFraction Upper A left parenthesis 5 right parenthesis Over Upper A left parenthesis 0 right parenthesis EndFraction B.StartFraction Upper A left parenthesis 5 right parenthesis minus Upper A left parenthesis 0 right parenthesis Over 5 EndFraction StartFraction Upper A left parenthesis 5 right parenthesis minus Upper A left parenthesis 0 right parenthesis Over 5 EndFraction Your answer is correct.C.StartFraction Upper A left parenthesis 0 right parenthesis minus Upper A left parenthesis 5 right parenthesis Over 5 EndFraction StartFraction Upper A left parenthesis 0 right parenthesis minus Upper A left parenthesis 5 right parenthesis Over 5 EndFraction D.StartFraction Upper A left parenthesis 5 right parenthesis Over 5 EndFraction minus Upper A left parenthesis 0 right parenthesis StartFraction Upper A left parenthesis 5 right parenthesis Over 5 EndFraction minus Upper A left parenthesis 0 right parenthesis Part 2 The average rate of change per year of the total amount in the account for the first five years of the investment (from tequals0 to tequals5) is $ enter your response here per year. (Round to the nearest cent as needed.)
Solution
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Math Problem Analysis
Mathematical Concepts
Exponential Functions
Compound Interest
Rates of Change
Derivatives
Formulas
Average Rate of Change: (A(b) - A(a)) / (b - a)
Exponential Growth Formula: A(t) = P(1 + r)^t
Derivative of Exponential Functions: d/dt [A(t) = P(1 + r)^t] = P(1 + r)^t * ln(1 + r)
Theorems
Derivative of Exponential Functions
Mean Value Theorem
Suitable Grade Level
Grades 10-12
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