Section 1
Exponential Functions
In the illustration, lines $r$ and s are parallel.(graph can't copy)Find $x$.
In the illustration, lines $r$ and s are parallel.(graph can't copy)Find the measure of $\angle 1$.
In the illustration, lines $r$ and s are parallel.(graph can't copy)Find the measure of $\angle 2$.
In the illustration, lines $r$ and s are parallel.(graph can't copy)Find the measure of $\angle 3$.
Fill in the blanks.If $b>0$ and $b \neq 1, y=f(x)=b^x$ is called an _________ function.
Fill in the blanks.The _________ of an exponential function is $(-\infty, \infty)$.
Fill in the blanks.The range of an exponential function is the interval _________
Fill in the blanks.The graph of $y=f(x)=3^x$ passes through the points $(0,_________)$ and $(1,_________)$.
Fill in the blanks. If $b>1$, then $y=f(x)=b^x$ is an _________ function.
Fill in the blanks. If $0<b<1$, then $y=f(x)=b^x$ is a _________ function.
Fill in the blanks. The formula for compound interest is $A=$ _________ -
Fill in the blanks.An alternate formula for compound interest is $F V=$ _________
Find each value to four decimal places.$2^{\sqrt{2}}$
Find each value to four decimal places.$7^{\sqrt{2}}$
Find each value to four decimal places. $5^{\sqrt{5}}$
Find each value to four decimal places. $6^{\sqrt{3}}$
Simplify each expression.$\left(2^{\sqrt{3}}\right)^{\sqrt{3}}$
Simplify each expression. $3^{\sqrt{2} 3^{\sqrt{18}}}$
Simplify each expression.$7^{\sqrt{3}} 7^{\sqrt{12}}$
Simplify each expression.$\left(3^{\sqrt{5}}\right)^{\sqrt{5}}$
Graph each exponential function. Check your work with a graphing calculator:$y=f(x)=3^x$(graph can't copy)
Graph each exponential function. Check your work with a graphing calculator:$y=f(x)=5^x$(graph can't copy)
Graph each exponential function. Check your work with a graphing calculator: $y=f(x)=\left(\frac{1}{3}\right)^x$(graph can't copy)
Graph each exponential function. Check your work with a graphing calculator:$y=f(x)=\left(\frac{1}{5}\right)^x$(graph can't copy)
Graph each exponential function. Check your work with a graphing calculator:$y=f(x)=3^x-2$(graph can't copy)
Graph each exponential function. Check your work with a graphing calculator:$y=f(x)=2^x+1$(graph can't copy)
Graph each exponential function. Check your work with a graphing calculator:$y=f(x)=3^{x-1}$(graph can't copy)
Graph each exponential function. Check your work with a graphing calculator:$y=f(x)=2^{x+1}$(graph can't copy)
Find the value of b, if any, that would cause the graph of $y=b^x$ to look like the graph indicated.(graph can't copy)
Use a graphing calculator to graph each function. Tell whether the function is an increasing or a decreasing function.$f(x)=\frac{1}{2}\left(3^{x / 2}\right)$
Use a graphing calculator to graph each function. Tell whether the function is an increasing or a decreasing function. $f(x)=-3\left(2^{x / 3}\right)$
Use a graphing calculator to graph each function. Tell whether the function is an increasing or a decreasing function. $f(x)=2\left(3^{-x / 2}\right)$
Use a graphing calculator to graph each function. Tell whether the function is an increasing or a decreasing function.$f(x)=-\frac{1}{4}\left(2^{-x / 2}\right)$
An initial deposit of $$\$ 10,000$$ earns $8 \%$ interest, compounded quarterly. How much will be in the account after 10 years?
An initial deposit of $$\$ 10,000$$ carns $8 \%$ interest, compounded monthly. How much will be in the account after 10 years?
How much more interest could $$\$ 1,000$$ earn in 5 ycars, compounded quarterly, if the annual interest rate were $5 \frac{1}{2} \%$ instead of $5 \%$ ?
Which institution in the two ads provides the better investment?
If $$\$ 1$$ had been invested on July 4, 1776, at $5 \%$ interest, compounded annually, what would it be worth on July 4, 2076 ?
$$\$ 10,000$$ is invested in each of two accounts, both paying $6 \%$ annual interest. In the first account, interest compounds quarterly, and in the second account, interest compounds daily. Find the difference between the accounts after 20 years.
A radioactive material decays according to the formula $A=A_0\left(\frac{2}{3}\right)^t$, where $A_0$ is the initial amount present and $t$ is measured in years. Find an expression for the amount present in 5 years.
A colony of 6 million bacteria is growing in a culture medium. (See the illustration.) The population $P$ after $t$ hours is given by the formula $P=\left(6 \times 10^6\right)(2.3)^t$. Find the population after 4 hours.
The charge remaining in a battery decreases as the battery discharges. The charge $C$ (in coulombs) after $t$ days is given by the formula $C=\left(3 \times 10^{-4}\right)(0.7)^t$. Find the charge after 5 days.
The population of North Rivers is decreasing exponentially according to the formula $P=3,745(0.93)^t$, where $t$ is measured in years from the present date. Find the population in 6 years, 9 months.
A small business purchases a computer for $$\$ 4,700$$. It is expected that its value each year will be $75 \%$ of its value in the preceding year. If the business disposes of the computer after 5 years, find its salvage value (the value after 5 years).
In 1803, the United States acquired territory from France in the Louisiana Purchase. The country doubled its territory by adding 827,000 square miles of land for $$\$ 15$$ million. If the land has appreciated at the rate of $6 \%$ each year, what would one square mile of land be worth in 1996?
If world population is increasing exponentially, why is there cause for coneern?
How do the graphs of $y=b^x$ differ when $b>1$ and $0<b<1$ ?
In the definition of the exponential function, $b$ could not equal 0 . Why not?
In the definition of the exponential function, $b$ could not be negative. Why not?