Chapter 3

Find the payment amount p needed to amortize the given loan amount. Assume that a payment is made in each of the n compounding periods per year. \(P=\$ 7000 ; r=6 \% ; t=5\) yr, compounded monthly

Find the half-life for each situation. An element loses \(12 \%\) of its mass every year.

Differentiate. $$ y=6^{x} $$

Find the general form of \(f\) if \(f^{\prime}(x)=4 f(x)\).

Write an equivalent exponential equation. $$ \log _{3} 81=4 $$

Find the payment amount p needed to amortize the given loan amount. Assume that a payment is made in each of the n compounding periods per year. \(P=\$ 8000 ; r=5 \% ; t=8\) yr, compounded monthly

Find the half-life for each situation. An population of bacteria decreases by \(5.75 \%\) per month.

Differentiate. $$ y=7^{x} $$

Find the general form of \(g\) if \(g^{\prime}(x)=6 g(x)\)

Write an equivalent exponential equation. $$ \log _{2} 8=3 $$

Find the payment amount p needed to amortize the given loan amount. Assume that a payment is made in each of the n compounding periods per year. \(P=\$ 12,000 ; r=5.7 \% ; t=6\) yr, compounded quarterly

Differentiate. $$ f(x)=8^{x} $$

Find the general form of the function that satisfies \(d A / d t=-9 A\)

Graph. $$ y=2 \cdot 3^{x} $$

Write an equivalent exponential equation. $$ \log _{27} 3=\frac{1}{3} $$

Find the half-life for each situation. A city loses \(3.9 \%\) of its population every year.

Differentiate. $$ f(x)=15^{x} $$

Find the general form of the function that satisfies \(d P / d t=-3 P(t)\)

Write an equivalent exponential equation. $$ \log _{8} 2=\frac{1}{3} $$

Find the payment amount p needed to amortize the given loan amount. Assume that a payment is made in each of the n compounding periods per year. \(P=\$ 500 ; r=4.1 \% ; t=1 \mathrm{yr},\) compounded monthly

Differentiate. $$ g(x)=x^{5}(3.7)^{x} $$

Find the general form of the function that satisfies \(d O / d t=k O\)

Write an equivalent exponential equation. $$ \log _{a} J=K $$

Find the payment amount p needed to amortize the given loan amount. Assume that a payment is made in each of the n compounding periods per year. \(P=\$ 800 ; r=3.8 \% ; t=2 \mathrm{yr},\) compounded monthly

Find the half-life for each situation. An element loses \(1.75 \%\) of its mass every day.

Differentiate. $$ g(x)=x^{3}(5.4)^{x} $$

Graph. $$ y=4\left(\frac{1}{3}\right)^{x} $$

Find the general form of the function that satisfies \(d R / d t=k R\)

Write an equivalent exponential equation. $$ \log _{a} K=J $$

Find the payment amount p needed to amortize the given loan amount. Assume that a payment is made in each of the n compounding periods per year. \(P=\$ 150,000 ; r=5.15 \% ; t=30 \mathrm{yr},\) compounded semiannually

U.S. patents. The number of applications for patents, \(N,\) grew dramatically in recent years, with growth averaging about \(5.8 \%\) per year. That is, \(N^{\prime}(t)=0.058 N(t)\) a) Find the function that satisfies this equation Assume that \(t=0\) corresponds to \(2009,\) when approximately 483,000 patent applications were received. b) Estimate the number of patent applications in 2020 c) Estimate the doubling time for \(N(t)\).

Find the half-life for each situation. An investment loses \(1.9 \%\) of its value every week.

Differentiate. $$ y=7^{x^{4}+2} $$

Write an equivalent exponential equation. $$ -\log _{b} V=w $$

Find the payment amount p needed to amortize the given loan amount. Assume that a payment is made in each of the n compounding periods per year. \(P=\$ 150,000 ; r=5.15 \% ; t=30 \mathrm{yr},\) compounded semiannually

Pete Zah's, Inc., is selling franchises for pizza shops throughout the country. The marketing manager estimates that the number of franchises, \(N,\) will increase at the rate of \(10 \%\) per year, that is, \(\frac{d N}{d t}=0.10 \mathrm{~N}\) a) Find the function that satisfies this equation. Assume that the number of franchises at \(t=0\) is 50 b) How many franchises will there be in 20 yr? c) In what period of time will the initial number of 50 franchises double?

Differentiate. $$ y=4^{x^{2}+5} $$

Write an equivalent exponential equation. $$ -\log _{10} h=p $$

If an amount \(P_{0}\) is invested in the Mandelbrot Bond Fund and interest is compounded continuously at \(5.9 \%\) per year, the balance \(P\) grows at the rate given by \(\frac{d P}{d t}=0.059 P\) a) Find the function that satisfies the equation. Write it in terms of \(P_{0}\) and 0.059 b) Suppose \(\$ 1000\) is invested. What is the balance after I yr? After 2 yr? c) When will an investment of \(\$ 1000\) double itself?

Iodine-131 has a decay rate of \(9.6 \%\) per day. The rate of change of an amount \(N\) of iodine- 131 is given by $$\frac{d N}{d t}=-0.096 N$$ where \(t\) is the number of days since decay began. a) Let \(N_{0}\) represent the amount of iodine- 131 present at \(t=0 .\) Find the exponential function that models the situation. b) Suppose \(500 \mathrm{~g}\) of iodine- 131 is present at \(t=0\). How much will remain after 4 days? c) After how many days will half of the \(500 \mathrm{~g}\) of iodine-13l remain?

Graph. $$ y=2.6(0.8)^{x} $$

Differentiate. $$ y=e^{x^{2}} $$

Solve for \(x\). $$ \log _{7} 49=x $$

Find the payment amount p needed to amortize the given loan amount. Assume that a payment is made in each of the n compounding periods per year. \(P=\$ 90,000 ; r=7 \% ; t=12 \mathrm{yr},\) compounded annually

If an amount \(P_{0}\) is invested in a savings account and interest is compounded continuously at \(4.3 \%\) per year, the balance \(P\) grows at the rate given by \(\frac{d p}{d t}=0.043 P\) a) Find the function that satisfies the equation. Write it in terms of \(P_{0}\) and \(0.043 .\) b) Suppose \(\$ 20,000\) is invested. What is the balance after 1 yr? After 2 yr? c) When will an investment of \(\$ 20,000\) double itself?

Carbon- 14 has a decay rate of \(0.012097 \%\) per year. The rate of change of an amount \(N\) of carbon- 14 is given by $$\frac{d N}{d t}=-0.00012097 N$$ where \(t\) is the number of years since decay began. a) Let \(N_{0}\) represent the amount of carbon-l4 present at \(t=0\). Find the exponential function that models the situation. b) Suppose \(200 \mathrm{~g}\) of carbon- 14 is present at \(t=0 .\) How much will remain after 800 yr? c) After how many years will half of the \(200 \mathrm{~g}\) of carbon-l4 remain?

Differentiate. $$ y=e^{8 x} $$

Solve for \(x\). $$ \log _{5} 125=x $$

Car loans. Todd purchases a new Honda Accord LX for \(\$ 22,150 .\) He makes a \(\$ 4000\) down payment and finances the remainder through an amortized loan at an annual interest rate of \(6.5 \%\), compounded monthly for 5 yr. a) Find Todd's monthly car payment. b) Assume that Todd makes every payment for the life of the loan. Find his total payments. c) How much interest does Todd pay over the life of the loan?

Bottled water sales. Since \(2000,\) sales of bottled water have increased at the rate of approximately \(9.3 \%\) per year. That is, the volume of bottled water sold, \(G,\) in billions of gallons, \(t\) years after 2000 is growing at the rate given by \(\frac{d G}{d t}=0.093 G\). a) Find the function that satisfies the equation, given that approximately 4.7 billion gallons of bottled water were sold in 2000 . b) Predict the number of gallons of water sold in 2025 . c) What is the doubling time for \(G(t)\) ?