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For the following exercises, use the compound interest formula, $\text{\hspace{0.17em}}A(t)=P{\left(1+\frac{r}{n}\right)}^{nt}.$
After a certain number of years, the value of an investment account is represented by the equation $\text{\hspace{0.17em}}10,250{\left(1+\frac{0.04}{12}\right)}^{120}.\text{\hspace{0.17em}}$ What is the value of the account?
What was the initial deposit made to the account in the previous exercise?
$\$10,250$
How many years had the account from the previous exercise been accumulating interest?
An account is opened with an initial deposit of $6,500 and earns $\text{\hspace{0.17em}}3.6\%\text{\hspace{0.17em}}$ interest compounded semi-annually. What will the account be worth in $\text{\hspace{0.17em}}20\text{\hspace{0.17em}}$ years?
$\$13,268.58$
How much more would the account in the previous exercise have been worth if the interest were compounding weekly?
Solve the compound interest formula for the principal, $\text{\hspace{0.17em}}P$ .
$P=A(t)\cdot {\left(1+\frac{r}{n}\right)}^{-nt}$
Use the formula found in the previous exercise to calculate the initial deposit of an account that is worth $\text{\hspace{0.17em}}\$14,472.74\text{\hspace{0.17em}}$ after earning $\text{\hspace{0.17em}}5.5\%\text{\hspace{0.17em}}$ interest compounded monthly for $\text{\hspace{0.17em}}5\text{\hspace{0.17em}}$ years. (Round to the nearest dollar.)
How much more would the account in the previous two exercises be worth if it were earning interest for $\text{\hspace{0.17em}}5\text{\hspace{0.17em}}$ more years?
$\$\mathrm{4,572.56}$
Use properties of rational exponents to solve the compound interest formula for the interest rate, $\text{\hspace{0.17em}}r.$
Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded semi-annually, had an initial deposit of $9,000 and was worth $13,373.53 after 10 years.
$4\%$
Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded monthly, had an initial deposit of $5,500, and was worth $38,455 after 30 years.
For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain.
$y=3742{\left(e\right)}^{0.75t}$
continuous growth; the growth rate is greater than $\text{\hspace{0.17em}}0.$
$y=150{\left(e\right)}^{\frac{3.25}{t}}$
$y=2.25{\left(e\right)}^{-2t}$
continuous decay; the growth rate is less than $\text{\hspace{0.17em}}0.$
Suppose an investment account is opened with an initial deposit of $\text{\hspace{0.17em}}\$12,000\text{\hspace{0.17em}}$ earning $\text{\hspace{0.17em}}7.2\%\text{\hspace{0.17em}}$ interest compounded continuously. How much will the account be worth after $\text{\hspace{0.17em}}30\text{\hspace{0.17em}}$ years?
How much less would the account from Exercise 42 be worth after $\text{\hspace{0.17em}}30\text{\hspace{0.17em}}$ years if it were compounded monthly instead?
$\$669.42$
For the following exercises, evaluate each function. Round answers to four decimal places, if necessary.
$f(x)=2{\left(5\right)}^{x},$ for $\text{\hspace{0.17em}}f\left(-3\right)$
$f(x)=-{4}^{2x+3},$ for $\text{\hspace{0.17em}}f\left(-1\right)$
$f(-1)=-4$
$f(x)={e}^{x},$ for $\text{\hspace{0.17em}}f\left(3\right)$
$f(x)=-2{e}^{x-1},$ for $\text{\hspace{0.17em}}f\left(-1\right)$
$f(-1)\approx -0.2707$
$f(x)=2.7{\left(4\right)}^{-x+1}+1.5,$ for $f\left(-2\right)$
$f(x)=1.2{e}^{2x}-0.3,$ for $\text{\hspace{0.17em}}f\left(3\right)$
$f(3)\approx 483.8146$
$f(x)=-\frac{3}{2}{\left(3\right)}^{-x}+\frac{3}{2},$ for $\text{\hspace{0.17em}}f\left(2\right)$
For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve.
$(0,3)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(3,375)$
$y=3\cdot {5}^{x}$
$(3,222.62)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(10,77.456)$
$(20,29.495)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(150,730.89)$
$y\approx 18\cdot {1.025}^{x}$
$(5,2.909)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(13,0.005)$
$(\mathrm{11,310.035})\text{\hspace{0.17em}}$ and $(\mathrm{25,356.3652})$
$y\approx 0.2\cdot {1.95}^{x}$
The annual percentage yield (APY) of an investment account is a representation of the 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 $\text{\hspace{0.17em}}\text{APY}={\left(1+\frac{r}{12}\right)}^{12}-1.$
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