Question 1

Solve the problem.
-Suppose $\$ 14,000$ is invested with $4 \%$ interest for 5 yr under the following compounding options. Complete the table.
$\begin{array}{l|l|l|l} 
& \text { Compounding Option } & n \text { Value } & \text { Result } \
\hline \text { a. } & \text { Daily } & & \
\text { b. } & \text { Continuously } & &
\end{array}$

Accepted Answer

For daily compounding, the formula is $A = P(1 + \frac{r}{n})^{nt}$ with $n = 365$. For continuous compounding, the formula is $A = Pe^{rt}$. Calculating both gives approximately \$17,099.45 for daily and \$17,099.64 for continuous compounding.

Question 2

Graph the function and write the domain and range in interval notation.
-$$f ( x ) = \left( \frac { 3 } { 2 } \right) ^ { x }$$

Accepted Answer

The answer of Graph the function and write the domain...

Question 3

Solve the problem.
-Use the graph of $y = \left( \frac { 1 } { 3 } \right) ^ { x }$ to graph the function. Write the domain and range in interval notation.
$f ( x ) = - \left( \frac { 1 } { 3 } \right) ^ { x } + 2$

Accepted Answer

The answer of Solve the problem.
-Use the graph of \(y...

Question 4

Graph the function and write the domain and range in interval notation.
-$$f ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x }$$

Accepted Answer

The answer of Graph the function and write the domain...

Question 5

Solve the problem.
-The atmospheric pressure on an object decreases as altitude increases. If a is the height (in km) above sea level, then the pressure P(a) (in mmHg) is approximated by P(a) = 760e-0.13a. Determine The atmospheric pressure at 7.176 km. Round to the nearest whole unit.

Accepted Answer

To find the atmospheric pressure at 7.176 km, substitute a = 7.176 into the formula P(a) = 760e^(-0.13a). Calculating this gives approximately 299 mmHg.

Question 6

Solve the problem.
-Use the graph of $y = 5 ^ { x }$ to graph the function. Write the domain and range in interval notation. $f ( x ) = 5 ^ { x } - 5$

Accepted Answer

The answer of Solve the problem.
-Use the graph of \(y...

Question 7

Use transformations of the graph $y = e ^ { x }$ to graph the function. Write the domain and range in interval
notation.
-$$f ( x ) = e ^ { x + 2 }$$

Accepted Answer

The answer of Use transformations of the graph \(y =...

Question 8

Use transformations of the graph $y = e ^ { x }$ to graph the function. Write the domain and range in interval
notation.
-$$f ( x ) = e ^ { x } + 1$$

Accepted Answer

The answer of Use transformations of the graph \(y =...

Question 9

Solve the problem.
-Use the graph of $y = 3 ^ { x }$ to graph the function. Write the domain and range in interval notation.
$$f ( x ) = 3 ^ { x + 5 } + 2$$

Accepted Answer

The answer of Solve the problem.
-Use the graph of \(y...

Question 10

Solve the problem.
-Use the graph of $y = \left( \frac { 1 } { 3 } \right) ^ { x }$ to graph the function. Write the domain and range in interval notation.
$f ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x + 1 } + 3$

Accepted Answer

The answer of Solve the problem.
-Use the graph of \(y...

Question 11

ng, in
-Scientists often use a process called carbon dating to estimate the age of archaeological finds. The process measures the amount of carbon- 14 , a radioactive isotope with a half-life of 5,730 years. If sample of wood from an ancient artifact had 20 grams of carbon-14 initially, the amount remaining grams, is given by
$$A ( t ) = 20 \left( \frac { 1 } { 2 } \right) ^ { t / 5,730 }$$
where $t$ is the number of years since the tree died. How many grams would be present after 5,730 years?

Accepted Answer

After 5,730 years, one half-life of carbon-14 has passed, so half of the original 20 grams would remain, which is 10 grams.

Question 12

Evaluate the function at the given value of x. Round to 4 decimal places if necessary.
-$$f ( x ) = e ^ { x } ; f ( - 2 )$$

Accepted Answer

To evaluate $f(x) = e^x$ at $x = -2$, substitute $-2$ for $x$ to get $f(-2) = e^{-2}$. Using a calculator, $e^{-2} \approx 0.1353$.

Question 13

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-The population of bacteria culture was 2000 at noon, and was increasing at a rate of $10 \%$ per hour. The number can be found using the function
$$P ( t ) = 2,000 ( 1.1 ) ^ { t }$$
where $t$ is the number of hours past noon. Predict the population 11 hours later, at $11 \mathrm { PM }$ to the nearest whole number.

Accepted Answer

To find the population 11 hours later, substitute $t = 11$ into the function: $P(11) = 2000 \times (1.1)^{11}$. Calculating this gives approximately 5,706.

Question 14

Solve the problem.
-Newton's law of cooling indicates that the temperature of a warm object will decrease exponentially with time and will approach the temperature of the surrounding air. The temperature $T ( t )$ is modeled by $T ( t ) = T _ { a } + \left( T _ { 0 } - T _ { a } \right) e ^ { - k t }$. In this model, $T _ { a }$ represents the temperature of the surrounding air, $T _ { 0 }$ represents the initial temperature of the object and $t$ is the time after the object starts cooling. The value of $k$ is the cooling rate and is a constant related to the physical properties of the object.
A cake comes out of the oven at $335 ^ { \circ } \mathrm { F }$ and is placed on a cooling rack in a $70 ^ { \circ } \mathrm { F }$ kitchen. After checking the temperature several minutes later, it is determined that the cooling rate $k$ is $0.050$. Write a function that models the temperature $T ( t )$ (in ${ } ^ { \circ } \mathrm { F }$ ) of the cake $t$ minutes after being removed from the oven.

Accepted Answer

The correct model is $T(t) = T_a + (T_0 - T_a)e^{-kt}$, where $T_a = 70^\circ F$, $T_0 = 335^\circ F$, and $k = 0.050$. Substituting these values gives $T(t) = 70 + (335 - 70)e^{-0.050t}$, which simplifies to $T(t) = 70 + 265e^{-0.050t}$.

Question 15

Solve the problem.
-James wants to invest $8,000. He can invest the money at 7.2% simple interest for 30 yr or he can invest at 7% with interest compounded continuously for 30 yr. Which option results in more total
Interest?

Accepted Answer

For simple interest, the formula is $A = P(1 + rt)$, where $A$ is the future value, $P$ is the principal amount, $r$ is the rate, and $t$ is the time in years. Plugging in the values for the 7.2% simple interest option: $A = 8000(1 + 0.072 \times 30) = 8000(1 + 2.16) = 8000 \times 3.16 = 25280$. The total interest earned is $25280 - 8000 = 17280$.For compound interest compounded continuously, the formula is $A = Pe^{rt}$, where $e$ is the base of the natural logarithm (approximately 2.71828). Plugging in the values for the 7% compounded continuously option: $A = 8000e^{0.07 \times 30} = 8000e^{2.1} \approx 8000 \times 8.166 = 65328$. The total interest earned is $65328 - 8000 = 57328$.Comparing the total interest earned from both options, the 7% compounded continuously yields more total interest.

Question 16

Solve the problem.
-Use the graph of $y = 3 ^ { x }$ to graph the function. Write the domain and range in interval note $f ( x ) = - 3 ^ { x }$

Accepted Answer

The answer of Solve the problem.
-Use the graph of \(y...

Question 17

Solve the problem.
-Use the graph of $y = 2 ^ { - x }$ to graph the function . Write the domain and range in interval notation $f ( x ) = 2 ^ { - x }$

Accepted Answer

The answer of Solve the problem.
-Use the graph of \(y...

Question 18

Use transformations of the graph $y = e ^ { x }$ to graph the function. Write the domain and range in interval
notation.
-$$f ( x ) = - e ^ { x } + 1$$

Accepted Answer

The answer of Use transformations of the graph \(y =...

Question 19

Solve the problem.
-Bethany needs to borrow $8,000. She can borrow money at 6.9% simple interest for 3 yr or she can borrow at 6.5% with interest compounded continuously for 3 yr. Which option results in less total
Interest?

Accepted Answer

The answer of Solve the problem.
-Bethany needs to borrow $8,000....

Question 20

Solve the problem.
-Newton's law of cooling indicates that the temperature of a warm object will decrease exponentially with time and will approach the temperature of the surrounding air. The temperature $T ( t )$ is modeled by $T ( t ) = T _ { a } + \left( T _ { 0 } - T _ { a } \right) e ^ { - k t }$. In this model, $T _ { a }$ represents the temperature of the surrounding air, $T _ { 0 }$ represents the initial temperature of the object and $t$ is the time after the object starts cooling. The value of $k$ is the cooling rate and is a constant related to the physical properties of the object.
Water in a water heater is originally $131 ^ { \circ } \mathrm { F }$. The water heater is shut off and the water cools to the temperature of the surrounding air, which is $80 ^ { \circ } \mathrm { F }$. The water cools slowly because of the insulation inside the heater, and the rate of cooling is $0.00348$. Dominic does not like to shower with water less than $116 ^ { \circ } \mathrm { F }$. If Dominic waits $24 \mathrm { hr }$ after the heater is shut off, will the water still be warm enough for a shower? How warm will the water be?

Accepted Answer

The temperature after $24 \mathrm { hr }$ will be approximately $127 ^ { \circ } \mathrm { F }$.The equation for the temperature of the water is $$T ( t ) = 80 + ( 131 - 80 ) e ^ { - 0.00348 t }$$$$T ( t ) = 80 + 51 e ^ { - 0.00348 t }$$At $t = 24$, the temperature is$$T ( 24 ) = 80 + 51 e ^ { - 0.00348 ( 24 ) }$$$$T ( 24 ) = 80 + 51 e ^ { - 0.08352 }$$$$T ( 24 ) = 80 + 51 ( 0.9203 )$$$$T ( 24 ) = 80 + 46.8153$$$$T ( 24 ) \approx 126.82 \approx 127 ^ { \circ } \mathrm { F }$$

Quiz 4: Exponential and Logarithmic Functions

Graph the function and write the domain and range in interval notation. - $f ( x ) = \left( \frac { 3 } { 2 } \right) ^ { x }$

Solve the problem. -Use the graph of $y = \left( \frac { 1 } { 3 } \right) ^ { x }$ to graph the function. Write the domain and range in interval notation. $f ( x ) = - \left( \frac { 1 } { 3 } \right) ^ { x } + 2$

Graph the function and write the domain and range in interval notation. - $f ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x }$

Solve the problem. -The atmospheric pressure on an object decreases as altitude increases. If a is the height (in km) above sea level, then the pressure P(a) (in mmHg) is approximated by P(a) = 760e-0.13a. Determine The atmospheric pressure at 7.176 km. Round to the nearest whole unit.

Solve the problem. -Use the graph of $y = 5 ^ { x }$ to graph the function. Write the domain and range in interval notation. $f ( x ) = 5 ^ { x } - 5$

Use transformations of the graph $y = e ^ { x }$ to graph the function. Write the domain and range in interval notation. - $f ( x ) = e ^ { x + 2 }$

Use transformations of the graph $y = e ^ { x }$ to graph the function. Write the domain and range in interval notation. - $f ( x ) = e ^ { x } + 1$

Solve the problem. -Use the graph of $y = 3 ^ { x }$ to graph the function. Write the domain and range in interval notation. $f ( x ) = 3 ^ { x + 5 } + 2$

Solve the problem. -Use the graph of $y = \left( \frac { 1 } { 3 } \right) ^ { x }$ to graph the function. Write the domain and range in interval notation. $f ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x + 1 } + 3$

Evaluate the function at the given value of x. Round to 4 decimal places if necessary. - $f ( x ) = e ^ { x } ; f ( - 2 )$

Solve the problem. -James wants to invest $8,000. He can invest the money at 7.2% simple interest for 30 yr or he can invest at 7% with interest compounded continuously for 30 yr. Which option results in more total Interest?

Solve the problem. -Use the graph of $y = 3 ^ { x }$ to graph the function. Write the domain and range in interval note $f ( x ) = - 3 ^ { x }$

Solve the problem. -Use the graph of $y = 2 ^ { - x }$ to graph the function . Write the domain and range in interval notation $f ( x ) = 2 ^ { - x }$

Use transformations of the graph $y = e ^ { x }$ to graph the function. Write the domain and range in interval notation. - $f ( x ) = - e ^ { x } + 1$

Solve the problem. -Bethany needs to borrow $8,000. She can borrow money at 6.9% simple interest for 3 yr or she can borrow at 6.5% with interest compounded continuously for 3 yr. Which option results in less total Interest?

Quiz 4: Exponential and Logarithmic Functions

Graph the function and write the domain and range in interval notation. - f(x) =(32) xf ( x ) = \left( \frac { 3 } { 2 } \right) ^ { x }f(x) =(23​) x

Solve the problem. -Use the graph of y=(13) xy = \left( \frac { 1 } { 3 } \right) ^ { x }y=(31​) x to graph the function. Write the domain and range in interval notation. f(x) =−(13) x+2f ( x ) = - \left( \frac { 1 } { 3 } \right) ^ { x } + 2f(x) =−(31​) x+2

Graph the function and write the domain and range in interval notation. - f(x) =(12) xf ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x }f(x) =(21​) x

Solve the problem. -The atmospheric pressure on an object decreases as altitude increases. If a is the height (in km) above sea level, then the pressure P(a) (in mmHg) is approximated by P(a) = 760e-0.13a. Determine The atmospheric pressure at 7.176 km. Round to the nearest whole unit.

Solve the problem. -Use the graph of y=5xy = 5 ^ { x }y=5x to graph the function. Write the domain and range in interval notation. f(x) =5x−5f ( x ) = 5 ^ { x } - 5f(x) =5x−5

Use transformations of the graph y=exy = e ^ { x }y=ex to graph the function. Write the domain and range in interval notation. - f(x) =ex+2f ( x ) = e ^ { x + 2 }f(x) =ex+2

Use transformations of the graph y=exy = e ^ { x }y=ex to graph the function. Write the domain and range in interval notation. - f(x) =ex+1f ( x ) = e ^ { x } + 1f(x) =ex+1

Solve the problem. -Use the graph of y=3xy = 3 ^ { x }y=3x to graph the function. Write the domain and range in interval notation. f(x) =3x+5+2f ( x ) = 3 ^ { x + 5 } + 2f(x) =3x+5+2

Solve the problem. -Use the graph of y=(13) xy = \left( \frac { 1 } { 3 } \right) ^ { x }y=(31​) x to graph the function. Write the domain and range in interval notation. f(x) =(13) x+1+3f ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x + 1 } + 3f(x) =(31​) x+1+3

Evaluate the function at the given value of x. Round to 4 decimal places if necessary. - f(x) =ex;f(−2) f ( x ) = e ^ { x } ; f ( - 2 ) f(x) =ex;f(−2)

Solve the problem. -James wants to invest $8,000. He can invest the money at 7.2% simple interest for 30 yr or he can invest at 7% with interest compounded continuously for 30 yr. Which option results in more total Interest?

Solve the problem. -Use the graph of y=3xy = 3 ^ { x }y=3x to graph the function. Write the domain and range in interval note f(x) =−3xf ( x ) = - 3 ^ { x }f(x) =−3x

Solve the problem. -Use the graph of y=2−xy = 2 ^ { - x }y=2−x to graph the function . Write the domain and range in interval notation f(x) =2−xf ( x ) = 2 ^ { - x }f(x) =2−x

Use transformations of the graph y=exy = e ^ { x }y=ex to graph the function. Write the domain and range in interval notation. - f(x) =−ex+1f ( x ) = - e ^ { x } + 1f(x) =−ex+1

Solve the problem. -Bethany needs to borrow $8,000. She can borrow money at 6.9% simple interest for 3 yr or she can borrow at 6.5% with interest compounded continuously for 3 yr. Which option results in less total Interest?

Graph the function and write the domain and range in interval notation. - $f ( x ) = \left( \frac { 3 } { 2 } \right) ^ { x }$

Solve the problem. -Use the graph of $y = \left( \frac { 1 } { 3 } \right) ^ { x }$ to graph the function. Write the domain and range in interval notation. $f ( x ) = - \left( \frac { 1 } { 3 } \right) ^ { x } + 2$

Graph the function and write the domain and range in interval notation. - $f ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x }$

Solve the problem. -Use the graph of $y = 5 ^ { x }$ to graph the function. Write the domain and range in interval notation. $f ( x ) = 5 ^ { x } - 5$

Use transformations of the graph $y = e ^ { x }$ to graph the function. Write the domain and range in interval notation. - $f ( x ) = e ^ { x + 2 }$

Use transformations of the graph $y = e ^ { x }$ to graph the function. Write the domain and range in interval notation. - $f ( x ) = e ^ { x } + 1$

Solve the problem. -Use the graph of $y = 3 ^ { x }$ to graph the function. Write the domain and range in interval notation. $f ( x ) = 3 ^ { x + 5 } + 2$

Solve the problem. -Use the graph of $y = \left( \frac { 1 } { 3 } \right) ^ { x }$ to graph the function. Write the domain and range in interval notation. $f ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x + 1 } + 3$

Evaluate the function at the given value of x. Round to 4 decimal places if necessary. - $f ( x ) = e ^ { x } ; f ( - 2 )$

Solve the problem. -Use the graph of $y = 3 ^ { x }$ to graph the function. Write the domain and range in interval note $f ( x ) = - 3 ^ { x }$

Solve the problem. -Use the graph of $y = 2 ^ { - x }$ to graph the function . Write the domain and range in interval notation $f ( x ) = 2 ^ { - x }$

Use transformations of the graph $y = e ^ { x }$ to graph the function. Write the domain and range in interval notation. - $f ( x ) = - e ^ { x } + 1$