The function given by y=f(x) shows the value of 5000...

The function given by $y=f(x)$ shows the value of $ 5000$ invested at $5 \%$ interest compounded continuously, $x$ years after the money was originally invested. a. Find the average amount earned per year between the 5th year and 10 th year. b. Find the average amount earned per year between the 20th year and the 25 th year. c. Based on the answers from parts (a) and (b), does it appear that the rate at which annual income increases is increasing or decreasing with time?

The function given by $y=f(x)$ shows the value of $ 5000$ invested at $5 \%$ interest compounded continuously, $x$ years after the money was originally invested.
a. Find the average amount earned per year between the 5th year and 10 th year.
b. Find the average amount earned per year between the 20th year and the 25 th year.
c. Based on the answers from parts (a) and (b), does it appear that the rate at which annual income increases is increasing or decreasing with time?

Key Concepts

Continuous Compounding

This concept refers to the process of calculating interest such that it is added to the principal an infinite number of times per period, effectively at every possible instant. The formula P·e^(rt) is used, where P is the initial investment, r is the interest rate, and t is time. It results in exponential growth due to the continual reinvestment of interest.

Exponential Growth

Exponential growth describes processes in which the quantity increases at a rate proportional to its current value, leading to faster growth as time progresses. In the context of investments with continuously compounded interest, the balance grows exponentially, meaning that as the account value increases, the interest earned each instant also increases.

Average Rate of Change

The average rate of change is a measure of how a function's value changes between two points, typically computed by subtracting the value at the beginning of the interval from the value at the end, then dividing by the length of the interval. This calculation offers a simple way to understand growth over specified time periods, such as comparing the gains in different intervals.

Trend Analysis

This concept involves comparing the average rates of change over different intervals to determine whether growth accelerates, decelerates, or remains constant. In contexts like continuously compounded interest, trend analysis can reveal that the absolute increase in income tends to become larger over time, indicating that the rate at which annual income increases is itself increasing.

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