The Natural Exponential Function

Exponential functions are used throughout the sciences and are particular useful in business applications. For example, most banks in the world use the exponential function with simple and compound interest. In this topic we concentrate on compounding continuously and the annual percentage rate is also worked on. Attention is given to the problem of finding the doubling time for an investment.

Compounding Continuously

Definition (Continuous Compounding Interest) If dollars is invested for years at an interest rate compounded continuously, then the future value is given by

Example (Future Value for Compounding Continuously) What lump sum do parents need to deposit in an account earning 9%, compounded continuously, so that it will grow to $40,000 for their daughter's college tuition in 18 years? The future value is $40,000 and is given by the formula where and and so we have

Example (Interest for Compounding Continuously) Which investment will earn more money, a $1000 investment for 6 years at 8% compounded annually, or a $1000 investment for 6 years compounded continuously? The investment that is compounding annually will have future value of where and which is The investment that is compounding continuously will have future value where and which is Thus, the investment which is compounding continuously is the better investment.

Example (Principal for Compounding Continuously) What present value needs to be deposited to have $20,000 in 3 years with an investment that is compounded continuously at 4%? The future value is 20000 and is given by the formula where and and so we have

Definition (Annual Percentage Yield) If is the number of compounding periods per year, then is the interest rate per period and if is the annual interest rate for an investment, then the annual percentage yield is defined by the formula

For compounded continuously investment the A.P.Y. is defined by the formula

Example (Annual Percentage Yield) Suppose there are three investments to invest in (a) one at 10% compounded annually, (b) another at 9.8% compounded quarterly, and (c) a third investment at 9.65% compounded continuously. Which investment is best?

Solution. For the first investment and and so will have A.P.Y. For the second investment we have and and so we have A.P.Y. For the last investment we have A.P.Y. and so the best investment is the second.

Example (Interest Problems) What is the present value of an investment at 6% annual simple interest if it is worth $832 in 8 months?

Solution. The future value is 832 and is given by where and and so we have

Example (Interest Problems) How much more interest will be earned if $5000 is invested for 6 years at 7% compounded continuously, instead of at 7% compounded quarterly?

Solution. If we use compounding continuously then the future value is where and and so we have Thus the interest earned is If we use compounding quarterly then the future value is given by where and and so we have future value of Thus for compounding quarterly we have interest earned as Therefore, the first investment is better by

Examples (Natural Exponential Function) Solve the exponential equation .

Solution. Solving we have,

Therefore, the solutions to are and

Examples (Natural Exponential Function) Solve the exponential equation

    Solution.  Solving we have,

Therefore, the solutions to are   and

Examples (Natural Exponential Function) Solve the exponential equation

    Solution.  Solving we have,

Since, when and the solutions to is   and

Examples (Natural Exponential Function) Solve the exponential equation

    Solution.  Solving we have,


Since the only solutions come from The solution is

Examples (Graphing Natural Exponential Functions) Sketch the graph of the function

    Solution. The graph can be obtained by plotting or using and then applying the some transformations. We have,

natural exponential function _gr_120.gif]

The graph as no vertical asymptotes, the horizontal asymptote is there are no -intercepts, the -intercept is

Examples (Graphing Natural Exponential Functions) Sketch the graph of the function

    Solution. The graph can be obtained by plotting or using and then applying the some transformations. We have,

natural exponential function _gr_128.gif]

The graph as no vertical asymptotes, the horizontal asymptote is there are no -intercepts, the -intercept is

Examples (Graphing Natural Exponential Functions) Sketch the graph of the function

Solution. The graph can be obtained by plotting or using and summing point by point.. Use the fact that is symmetric and then shift. We have,

natural exponential function _gr_139.gif]

The graph as no vertical asymptotes nor horizontal asymptotes, there are no -intercepts, and the -intercept is