Stewart Precalc 6e Section 4.2: The Natural Exponential Function

1 Exercise - Concepts · Level 1

The function $f(x) = e^x$ is called the _____ exponential function. The number $e$ is approximately equal to _____.

2 Exercise - Concepts · Level 1

In the formula $A(t) = P e^{r t}$ for continuously compounded interest, the letters $P$, $r$, and $t$ stand for _____, _____, and _____, respectively, and $A(t)$ stands for _____. So if \$100 is invested at an interest rate of $6%$ compounded continuously, then the amount after 2 years is _____.

3 Exercise - Skills · Level 1

Use a calculator to evaluate the function $h(x) = e^x$ at the indicated values, rounded to three decimals: $h(3)$, $h(0.23)$, $h(1)$, $h(-2)$.

4 Exercise - Skills · Level 1

Use a calculator to evaluate the function $h(x) = e^{-2 x}$ at the indicated values, rounded to three decimals: $h(1)$, $h(\sqrt{2})$, $h(-3)$, $h\left(\dfrac{1}{2}\right)$.

5 Exercise - Skills · Level 2

Complete the table of values for $f(x) = 3 e^x$, rounded to two decimal places, and sketch a graph of the function (for $x = -3, -2, -1, 0, 1, 2, 3$).

6 Exercise - Skills · Level 2

Complete the table of values for $f(x) = 2 e^{-0.5 x}$, rounded to two decimal places, and sketch a graph of the function (for $x = -3, -2, -1, 0, 1, 2, 3$).

7 Exercise - Skills · Level 2

Graph the function $f(x) = -e^x$, not by plotting points, but by starting from the graph of $y = e^x$. State the domain, range, and asymptote.

8 Exercise - Skills · Level 2

Graph the function $y = 1 - e^x$, not by plotting points, but by starting from the graph of $y = e^x$. State the domain, range, and asymptote.

9 Exercise - Skills · Level 2

Graph the function $y = e^{-x} - 1$, not by plotting points, but by starting from the graph of $y = e^x$. State the domain, range, and asymptote.

10 Exercise - Skills · Level 2

Graph the function $f(x) = -e^{-x}$, not by plotting points, but by starting from the graph of $y = e^x$. State the domain, range, and asymptote.

11 Exercise - Skills · Level 2

Graph the function $f(x) = e^{x - 2}$, not by plotting points, but by starting from the graph of $y = e^x$. State the domain, range, and asymptote.

12 Exercise - Skills · Level 2

Graph the function $y = e^{x - 3} + 4$, not by plotting points, but by starting from the graph of $y = e^x$. State the domain, range, and asymptote.

13 Exercise - Skills · Level 2

Graph the function $h(x) = e^{x + 1} - 3$, not by plotting points, but by starting from the graph of $y = e^x$. State the domain, range, and asymptote.

14 Exercise - Skills · Level 2

Graph the function $g(x) = -e^{x - 1} - 2$, not by plotting points, but by starting from the graph of $y = e^x$. State the domain, range, and asymptote.

15 Exercise - Skills · Level 3

The hyperbolic cosine function is defined by $ \cosh(x) = \dfrac{e^x + e^{-x}}{2} $

(a) Sketch the graphs of the functions $y = \dfrac{1}{2} e^x$ and $y = \dfrac{1}{2} e^{-x}$ on the same axes, and use graphical addition to sketch the graph of $y = \cosh(x)$.

Answer for 15(a)

(b) Use the definition to show that $\cosh(-x) = \cosh(x)$.

Answer for 15(b)

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16 Exercise - Skills · Level 3

The hyperbolic sine function is defined by $ \sinh(x) = \dfrac{e^x - e^{-x}}{2} $

(a) Sketch the graph of this function using graphical addition as in Exercise 15.

Answer for 16(a)

(b) Use the definition to show that $\sinh(-x) = -\sinh(x)$.

Answer for 16(b)

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17 Exercise - Skills · Level 3

(a) Draw the graphs of the family of functions $ f(x) = \dfrac{a}{2} (e^{x slash a} + e^{-x slash a}) $ for $a = 0.5, 1, 1.5,$ and $2$.

Answer for 17(a)

(b) How does a larger value of $a$ affect the graph?

Answer for 17(b)

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18 Exercise - Skills · Level 4

Find the local maximum and minimum values of the function $g(x) = x^x$ $(x > 0)$ and the value of $x$ at which each occurs. State each answer correct to two decimal places.

19 Exercise - Skills · Level 4

Find the local maximum and minimum values of the function $g(x) = e^x + e^{-3 x}$ and the value of $x$ at which each occurs. State each answer correct to two decimal places.

20 Exercise - Applications · Level 2

Medical Drugs. When a certain medical drug is administered to a patient, the number of milligrams remaining in the patient's bloodstream after $t$ hours is modeled by $ D(t) = 50 e^{-0.2 t} $ How many milligrams of the drug remain in the patient's bloodstream after 3 hours?

21 Exercise - Applications · Level 2

Radioactive Decay. A radioactive substance decays in such a way that the amount of mass remaining after $t$ days is given by $ m(t) = 13 e^{-0.015 t} $ where $m(t)$ is measured in kilograms.

(a) Find the mass at time $t = 0$.

Answer for 21(a)

(b) How much of the mass remains after 45 days?

Answer for 21(b)

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22 Exercise - Applications · Level 2

Radioactive Decay. Doctors use radioactive iodine as a tracer in diagnosing certain thyroid gland disorders. This type of iodine decays in such a way that the mass remaining after $t$ days is given by $ m(t) = 6 e^{-0.087 t} $ where $m(t)$ is measured in grams.

(a) Find the mass at time $t = 0$.

Answer for 22(a)

(b) How much of the mass remains after 20 days?

Answer for 22(b)

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23 Exercise - Applications · Level 3

Sky Diving. A sky diver jumps from a reasonable height above the ground. The air resistance she experiences is proportional to her velocity, and the constant of proportionality is $0.2$. It can be shown that the downward velocity of the sky diver at time $t$ is given by $ v(t) = 80 (1 - e^{-0.2 t}) $ where $t$ is measured in seconds and $v(t)$ is measured in feet per second (ft slash s).

(a) Find the initial velocity of the sky diver.

Answer for 23(a)

(b) Find the velocity after $5$ s and after $10$ s.

Answer for 23(b)

(c) Draw a graph of the velocity function $v(t)$.

Answer for 23(c)

(d) The maximum velocity of a falling object with wind resistance is called its terminal velocity. From the graph in part (c) find the terminal velocity of this sky diver.

Answer for 23(d)

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24 Exercise - Applications · Level 3

Mixtures and Concentrations. A 50-gallon barrel is filled completely with pure water. Salt water with a concentration of $0.3$ lb slash gal is then pumped into the barrel, and the resulting mixture overflows at the same rate. The amount of salt in the barrel at time $t$ is given by $ Q(t) = 15 (1 - e^{-0.04 t}) $ where $t$ is measured in minutes and $Q(t)$ is measured in pounds.

(a) How much salt is in the barrel after $5$ min?

Answer for 24(a)

(b) How much salt is in the barrel after $10$ min?

Answer for 24(b)

(c) Draw a graph of the function $Q(t)$.

Answer for 24(c)

(d) Use the graph in part (c) to determine the value that the amount of salt in the barrel approaches as $t$ becomes large. Is this what you would expect?

Answer for 24(d)

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25 Exercise - Applications · Level 3

Logistic Growth. Animal populations are not capable of unrestricted growth because of limited habitat and food supplies. Under such conditions the population follows a logistic growth model: $ P(t) = \dfrac{d}{1 + k e^{-c t}} $ where $c$, $d$, and $k$ are positive constants. For a certain fish population in a small pond $d = 1200$, $k = 11$, $c = 0.2$, and $t$ is measured in years. The fish were introduced into the pond at time $t = 0$.

(a) How many fish were originally put in the pond?

Answer for 25(a)

(b) Find the population after $10$, $20$, and $30$ years.

Answer for 25(b)

(c) Evaluate $P(t)$ for large values of $t$. What value does the population approach as $t \rightarrow \infty$? Does the graph shown confirm your calculations?

Answer for 25(c)

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26 Exercise - Applications · Level 3

Bird Population. The population of a certain species of bird is limited by the type of habitat required for nesting. The population behaves according to the logistic growth model $ n(t) = \dfrac{5600}{0.5 + 27.5 e^{-0.044 t}} $ where $t$ is measured in years.

(a) Find the initial bird population.

Answer for 26(a)

(b) Draw a graph of the function $n(t)$.

Answer for 26(b)

(c) What size does the population approach as time goes on?

Answer for 26(c)

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27 Exercise - Applications · Level 3

World Population. The relative growth rate of world population has been decreasing steadily in recent years. On the basis of this, some population models predict that world population will eventually stabilize at a level that the planet can support. One such logistic model is $ P(t) = \dfrac{73.2}{6.1 + 5.9 e^{-0.02 t}} $ where $t = 0$ is the year $2000$ and population is measured in billions.

(a) What world population does this model predict for the year $2200$? For $2300$?

Answer for 27(a)

(b) Sketch a graph of the function $P$ for the years $2000$ to $2500$.

Answer for 27(b)

(c) According to this model, what size does the world population seem to approach as time goes on?

Answer for 27(c)

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28 Exercise - Applications · Level 2

Tree Diameter. For a certain type of tree the diameter $D$ (in feet) depends on the tree's age $t$ (in years) according to the logistic growth model $ D(t) = \dfrac{5.4}{1 + 2.9 e^{-0.01 t}} $ Find the diameter of a 20-year-old tree.

29 Exercise - Applications · Level 2

Compound Interest. An investment of \$7,000 is deposited into an account in which interest is compounded continuously at $r = 3%$. Complete the table by filling in the amounts to which the investment grows for $t = 1, 2, 3, 4, 5, 6$ years.

30 Exercise - Applications · Level 2

Compound Interest. An investment of \$7,000 is deposited into an account in which interest is compounded continuously for $t = 10$ years. Complete the table by filling in the amounts to which the investment grows at the indicated rates: $1%$, $2%$, $3%$, $4%$, $5%$, $6%$.

31 Exercise - Applications · Level 2

Compound Interest. If \$2000 is invested at an interest rate of $3.5%$ per year, compounded continuously, find the value of the investment after the given number of years.

(a) $2$ years

Answer for 31(a)

(b) $4$ years

Answer for 31(b)

(c) $12$ years

Answer for 31(c)

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32 Exercise - Applications · Level 2

Compound Interest. If \$3500 is invested at an interest rate of $6.25%$ per year, compounded continuously, find the value of the investment after the given number of years.

(a) $3$ years

Answer for 32(a)

(b) $6$ years

Answer for 32(b)

(c) $9$ years

Answer for 32(c)

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33 Exercise - Applications · Level 3

Compound Interest. If \$600 is invested at an interest rate of $2.5%$ per year, find the amount of the investment at the end of $10$ years for the following compounding methods.

(a) Annually

Answer for 33(a)

(b) Semiannually

Answer for 33(b)

(c) Quarterly

Answer for 33(c)

(d) Continuously

Answer for 33(d)

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34 Exercise - Applications · Level 2

Compound Interest. If \$8000 is invested in an account for which interest is compounded continuously, find the amount of the investment at the end of $12$ years for the following interest rates.

(a) $2%$

Answer for 34(a)

(b) $3%$

Answer for 34(b)

(c) $4.5%$

Answer for 34(c)

(d) $7%$

Answer for 34(d)

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35 Exercise - Applications · Level 3

Compound Interest. Which of the given interest rates and compounding periods would provide the better investment?

(a) $2.5%$ per year, compounded semiannually

Answer for 35(a)

(b) $2.25%$ per year, compounded monthly

Answer for 35(b)

(c) $2%$ per year, compounded continuously

Answer for 35(c)

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36 Exercise - Applications · Level 3

Compound Interest. Which of the given interest rates and compounding periods would provide the better investment?

(a) $5.125%$ per year, compounded semiannually

Answer for 36(a)

(b) $5%$ per year, compounded continuously

Answer for 36(b)

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37 Exercise - Applications · Level 3

Investment. A sum of \$5000 is invested at an interest rate of $9%$ per year, compounded continuously.

(a) Find the value $A(t)$ of the investment after $t$ years.

Answer for 37(a)

(b) Draw a graph of $A(t)$.

Answer for 37(b)

(c) Use the graph of $A(t)$ to determine when this investment will amount to \$25,000.

Answer for 37(c)

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38 Exercise - Discovery/Writing · Level 3

The Definition of $e$. Illustrate the definition of the number $e$ by graphing the curve $y = (1 + 1 slash x)^x$ and the line $y = e$ on the same screen, using the viewing rectangle $[0, 40]$ by $[0, 4]$.

39 Example - Evaluating the Exponential Function · Level 1

Use a calculator to evaluate each expression rounded to five decimal places.

(a) $e^3$

Answer for 39(a)

(b) $2 e^{-0.53}$

Answer for 39(b)

(c) $e^{4.8}$

Answer for 39(c)

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40 Example - Transformations of the Exponential Function · Level 2

Sketch the graph of each function.

(a) $f(x) = e^{-x}$

Answer for 40(a)

(b) $g(x) = 3 e^{0.5 x}$

Answer for 40(b)

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41 Example - An Exponential Model for the Spread of a Virus · Level 3

An infectious disease begins to spread in a small city of population 10,000. After $t$ days, the number of people who have succumbed to the virus is modeled by the function $ v(t) = \dfrac{10000}{5 + 1245 e^{-0.97 t}} $

(a) How many infected people are there initially (at time $t = 0$)?

Answer for 41(a)

(b) Find the number of infected people after one day, two days, and five days.

Answer for 41(b)

(c) Graph the function $v$, and describe its behavior.

Answer for 41(c)

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42 Example - Calculating Continuously Compounded Interest · Level 2

Find the amount after 3 years if \$1000 is invested at an interest rate of $12%$ per year, compounded continuously.

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