Chapter 4

Explain why the values of an increasing exponential function will eventually overtake the values of an increasing linear function.

The Oxford Dictionary defines the word nominal as a value that is "stated or expressed but not necessarily corresponding exactly to the real value." Develop a reasonable argument for why the term nominal rate is used to describe the annual percentage rate of an investment account that compounds interest.

For the following exercises, identify whether the statement represents an exponential function. Explain. A population of bacteria decreases by a factor of \(\frac{1}{8}\) every 24 hours.

For the following exercises, identify whether the statement represents an exponential function. Explain. The value of a coin collection has increased by 3.25\(\%\) annually over the last 20 years.

For the following exercises, identify whether the statement represents an exponential function. Explain. For each training session, a personal trainer charges his clients \(\$ 5\) less than the previous training session.

For the following exercises, identify whether the statement represents an exponential function. Explain. The height of a projectile at time \(t\) is represented by the function \(h(t)=-4.9 t^{2}+18 t+40\)

For the following exercises, consider this scenario: For each year \(t,\) the population of a forest of trees is represented by the function \(A(t)=115(1.025)^{t} .\) In a neighboring forest, the population of the same type of tree is represented by the function \(B(t)=82(1.029)^{t} .\) (Round answers to the nearest whole number.) Which forest's population is growing at a faster rate?

For the following exercises, consider this scenario: For each year \(t,\) the population of a forest of trees is represented by the function \(A(t)=115(1.025)^{t} .\) In a neighboring forest, the population of the same type of tree is represented by the function \(B(t)=82(1.029)^{t} .\) (Round answers to the nearest whole number.) Which forest had a greater number of trees initially? By how many?

For the following exercises, consider this scenario: For each year \(t,\) the population of a forest of trees is represented by the function \(A(t)=115(1.025)^{t} .\) In a neighboring forest, the population of the same type of tree is represented by the function \(B(t)=82(1.029)^{t} .\) (Round answers to the nearest whole number.) Assuming the population growth models continue to represent the growth of the forest will have a greater number of trees after 20 years? By how many?

For the following exercises, consider this scenario: For each year \(t,\) the population of a forest of trees is represented by the function \(A(t)=115(1.025)^{t} .\) In a neighboring forest, the population of the same type of tree is represented by the function \(B(t)=82(1.029)^{t} .\) (Round answers to the nearest whole number.) Assuming the population growth models continue to represent the forest will have a greater number of trees after 100 years? By how many?

For the following exercises, consider this scenario: For each year \(t,\) the population of a forest of trees is represented by the function \(A(t)=115(1.025)^{t} .\) In a neighboring forest, the population of the same type of tree is represented by the function \(B(t)=82(1.029)^{t} .\) (Round answers to the nearest whole number.) Discuss the above results from the previous four exercises. Assuming the population growth models continue to represent the growth of the forests, which forest will have the greater number of trees in the long run? Why? What are some factors that might influence the long-term validity of the exponential growth model?

For the following exercises, determine whether the equation repponential grownth, exponential decay, or neither. Explain. $$ y=300(1-t)^{5} $$

For the following exercises, determine whether the equation repponential grownth, exponential decay, or neither. Explain. $$ y=220(1.06)^{x} $$

For the following exercises, determine whether the equation repponential grownth, exponential decay, or neither. Explain. $$ y=16.5(1.025)^{\frac{1}{x}} $$

For the following exercises, determine whether the equation repponential grownth, exponential decay, or neither. Explain. $$ y=11,701(0.97)^{t} $$

For the following exercises, find the formula for an exponential function that passes through the two points given. $$ (0,6) \text { and }(3,750) $$

For the following exercises, find the formula for an exponential function that passes through the two points given. $$ (0,2000) \text { and }(2,20) $$

For the following exercises, find the formula for an exponential function that passes through the two points given. $$ \left(-1, \frac{3}{2}\right) \text { and }(3,24) $$

For the following exercises, find the formula for an exponential function that passes through the two points given. $$ (-2,6) \text { and }(3,1) $$

For the following exercises, find the formula for an exponential function that passes through the two points given. $$ (3,1) \text { and }(5,4) $$

For the following exercises, use the compound interest formula, \(A(t)=P\left(1+\frac{r}{n}\right)^{n t}\) After a certain number of years, the value of an investment account is represented by the equation \(10,250\left(1+\frac{0.04}{12}\right)^{120} .\) What is the value of the account?

For the following exercises, use the compound interest formula, \(A(t)=P\left(1+\frac{r}{n}\right)^{n t}\) An account is opened with an initial deposit of \(\$ 6,500\) and earns 3.6\(\%\) interest compounded semi-annually. What will the account be worth in 20 years?

For the following exercises, use the compound interest formula, \(A(t)=P\left(1+\frac{r}{n}\right)^{n t}\) Solve the compound interest formula for the principal, \(P\)

For the following exercises, use the compound interest formula, \(A(t)=P\left(1+\frac{r}{n}\right)^{n t}\) Use the formula found in the previous exercise the initial deposit of an account that is worth \(\$ 14,472.74\) after earning 5.5\(\%\) interest compounded monthly for 5 years. (Round to the nearest dollar.)

For the following exercises, use the compound interest formula, \(A(t)=P\left(1+\frac{r}{n}\right)^{n t}\) Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded semiannually, had an initial deposit of \(\$ 9,000\) and was worth \(\$ 13,373.53\) after 10 years.

For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain. $$ y=3742(e)^{0.75 t} $$

For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain. $$ y=2.25(e)^{-2 t} $$

For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain. Suppose an investment account is opened with an initial deposit of \(\$ 12,000\) earning 7.2\(\%\) interest compounded continuously. How much will the account be worth after 30 years?

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. $$ f(x)=2(5)^{x}, \text { for } f(-3) $$

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. $$ f(x)=-4^{2 x+3}, \quad \text { for } f(-1) $$

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. $$ f(x)=e^{x}, \text { for } f(3) $$

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. $$ f(x)=-2 e^{x-1}, \text { for } f(-1) $$

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. $$ f(x)=2.7(4)^{-x+1}+1.5, \text { for } f(-2) $$

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. $$ f(x)=1.2 e^{2 x}-0.3, \text { for } f(3) $$

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary. $$ f(x)=-\frac{3}{2}(3)^{-x}+\frac{3}{2}, \text { for } f(2) $$

For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve. $$ (0,3) \text { and }(3,375) $$

For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve. $$ (20,29.495) \text { and }(150,730.89) $$

For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve. $$ (5,2.909) \text { and }(13,0.005) $$

For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve. $$ (11,310.035) \text { and }(25,356.3652) $$

The annual percentage yield (APY) of an investment account is actual interest rate earned on a compounding account. It is based on a compounding period of one year. Show that the APY of an account that compounds monthly can be found with the formula APY \(=\left(1+\frac{r}{12}\right)^{12}-1\)

Recall that an exponential function written in the form \(f(x)=a \cdot b^{x}\) such that \(a\) and \(b\) are positive numbers and \(b \neq 1 .\) Any positive number \(b\) can be written as \(b=e^{n}\) for some value of \(n .\) Use this fact to rewrite the formula for an exponential function that uses the number \(e\) as a base.

In an exponential decay function, the base of the exponent is a value between 0 and \(1 .\) Thus, for some number \(b>1\) the exponential decay function can be written as \(f(x)=a \cdot\left(\frac{1}{b}\right)^{x}\) . Use this formula, along with the fact that \(b=e^{n},\) to show that an exponential decay function takes the form \(f(x)=a(e)^{-n x}\) for some positive number \(n\)

The formula for the amount \(A\) in an investment account with a nominal interest rate \(r\) any time \(t\) is given by \(A(t)=a(e)^{r t},\) where \(a\) is the amount of principal initially deposited into an account that compounds continuously. Prove that the percentage of interest earned to principal at any time \(t\) can be calculated with the formula \(I(t)=e^{r t}-1\)

The fox population in a certain region has an annual growth rate of 9\(\%\) per year. In the year \(2012,\) there were \(23,900\) fox counted in the area. What is the fox population predicted to be in the year 2020\(?\)

A scientist begins with 100 milligrams of a radioactive substance that decays exponentially. After 35 hours, 50 \(\mathrm{mg}\) of the substance remains. How many milligrams will remain after 54 hours?

In the year \(1985,\) a house was valued at \(\$ 110,000 .\) By the year \(2005,\) the valueciated to \(\$ 145,000\) . What was the annual growth rate between 1985 and 2005\(?\) Assume that the valued to grow by the same percentage. What was the value of the house in the year 2010\(?\)

A car was valued at \(\$ 38,000\) in the year \(2007 .\) By 2013 , the value had depreciated to \(\$ 11,000\) If the car's value continues to drop by the same percentage, what will it be worth by 2017\(?\)

Jamal wants to save \(\$ 54,000\) for a down payment on a home. How much will he need in an account with 8.2\(\%\) APR, compounding daily, in order to reach his goal in 5 years?

Kyoko has \(\$ 10,000\) that she wants to invest. Her bank has several investment accounts to choose from, all compounding daily. Her goal is to have \(\$ 15,000\) by the time she finishes graduate school in 6 years. To the nearest hundredth of a percent, what should her minimum annual interest rate be in order to reach her goal? (Hint: solve the compound interest formula for the interest rate.)

Alyssa opened a retirement account with 7.25\(\%\) APR in the year \(2000 .\) Her initial deposit was \(\$ 13,500 .\) How much will the account be worth in 2025 if interest compounds monthly? How much more would she make if interest compounded continuously?