Basic Technical Mathematics with Calculus

Allyn J. Washington, Richard S. Evans

Chapter 23

The Derivative - all with Video Answers

Educators

Section 1

Limits

Problem 1

Make the given changes in the indicated examples of this section. Then solve the resulting problems.
In Example $2,$ change the denominator to $x+2$ and then determine the continuity.

Dwijendra Rao

Dwijendra Rao

Numerade Educator

Problem 2

Make the given changes in the indicated examples of this section. Then solve the resulting problems.
In Example $8,$ change the numerator to $3 x^{2}-5 x-2$ and find the resulting limit. Disregard references to Examples 6 and 7

Dwijendra Rao

Numerade Educator

Problem 3

Make the given changes in the indicated examples of this section. Then solve the resulting problems.
In Example $11,$ change the denominator to $t+2$ and then find the limit as $t \rightarrow-2$

Dwijendra Rao

Numerade Educator

Problem 4

Make the given changes in the indicated examples of this section. Then solve the resulting problems.
In Example $14,$ change the numerator to $4 x^{2}+1$ and find the resulting limit.

Dwijendra Rao

Numerade Educator

Problem 5

Determine the values of $x$ for which the function is continuous. If the function is not continuous, determine the reason.
$$f(x)=3 x^{2}-98 x$$

Dwijendra Rao

Numerade Educator

Problem 6

Determine the values of $x$ for which the function is continuous. If the function is not continuous, determine the reason.
$$f(x)=\frac{3-x}{9+x^{2}}$$

Dwijendra Rao

Numerade Educator

Problem 7

Determine the values of $x$ for which the function is continuous. If the function is not continuous, determine the reason.
$$f(x)=\frac{x+4}{x^{2}-x}$$

Dwijendra Rao

Numerade Educator

Problem 8

Determine the values of $x$ for which the function is continuous. If the function is not continuous, determine the reason.
$$f(x)=\frac{2}{\sqrt{x+3}}$$

Dwijendra Rao

Numerade Educator

Problem 9

Determine the values of $x$ for which the function is continuous. If the function is not continuous, determine the reason.
$$f(x)=\frac{x}{\sqrt{x-2}}$$

Dwijendra Rao

Numerade Educator

Problem 10

Determine the values of $x$ for which the function is continuous. If the function is not continuous, determine the reason.
$$f(x)=\frac{3 \sqrt{x+5}}{x+8}$$

Dwijendra Rao

Numerade Educator

Problem 11

Determine the values of $x$ for which the function is continuous. If the function is not continuous, determine the reason.
(GRAPHS CANNOT COPY)

Dwijendra Rao

Numerade Educator

Problem 12

Determine the values of $x$ for which the function is continuous. If the function is not continuous, determine the reason.
(GRAPHS CANNOT COPY)

Dwijendra Rao

Numerade Educator

Problem 13

Determine the values of $x$ for which the function is continuous. If the function is not continuous, determine the reason.
(GRAPHS CANNOT COPY)

Dwijendra Rao

Numerade Educator

Problem 14

Determine the values of $x$ for which the function is continuous. If the function is not continuous, determine the reason.
(GRAPHS CANNOT COPY)

Dwijendra Rao

Numerade Educator

Problem 15

Determine the values of $x$ for which the function is continuous. If the function is not continuous, determine the reason.
(GRAPHS CANNOT COPY)

Dwijendra Rao

Numerade Educator

Problem 16

Determine the values of $x$ for which the function is continuous. If the function is not continuous, determine the reason.
(GRAPHS CANNOT COPY)

Dwijendra Rao

Numerade Educator

Problem 17

For the function shown in the graph for the indicated exercise, find $(a) f(2),$ and $(b) \lim _{x \rightarrow 2} f(x),$ if they exist.
Exercise 13

Dwijendra Rao

Numerade Educator

Problem 18

For the function shown in the graph for the indicated exercise, find $(a) f(2),$ and $(b) \lim _{x \rightarrow 2} f(x),$ if they exist.
Exercise 14

Dwijendra Rao

Numerade Educator

Problem 19

For the function shown in the graph for the indicated exercise, find $(a) f(2),$ and $(b) \lim _{x \rightarrow 2} f(x),$ if they exist.
Exercise 15

Dwijendra Rao

Numerade Educator

Problem 20

For the function shown in the graph for the indicated exercise, find $(a) f(2),$ and $(b) \lim _{x \rightarrow 2} f(x),$ if they exist.
Exercise 16

Dwijendra Rao

Numerade Educator

Problem 21

Graph the function and determine the values of $x$ for which the functions are continuous. Explain.
$$f(x)=\left\{\begin{array}{ll}
x^{2} & \text { for } x<2 \\
5 & \text { for } x \geq 2
\end{array}\right.$$

Dwijendra Rao

Numerade Educator

Problem 22

Graph the function and determine the values of $x$ for which the functions are continuous. Explain.
$$f(x)=\left\{\begin{array}{ll}
\frac{x^{3}-x^{2}}{x-1} & \text { for } x \neq 1 \\
1 & \text { for } x=1
\end{array}\right.$$

Dwijendra Rao

Numerade Educator

Problem 23

Graph the function and determine the values of $x$ for which the functions are continuous. Explain.
$$f(x)=\left\{\begin{array}{ll}
\frac{2 x^{2}-18}{x-3} & \text { for } x \neq 3 \\
12 & \text { for } x=3
\end{array}\right.$$

Dwijendra Rao

Numerade Educator

Problem 24

Graph the function and determine the values of $x$ for which the functions are continuous. Explain.
$$f(x)=\left\{\begin{array}{ll}
\frac{x+2}{x^{2}-4} & \text { for } x<-2 \\
\frac{x}{8} & \text { for } x>-2
\end{array}\right.$$

Dwijendra Rao

Numerade Educator

Problem 25

Evaluate the indicated limits by evaluating the function for values shown in the table and observing the values that are obtained. Do not change the form of the function.
$$\begin{aligned}
&\text { Find } \lim _{x \rightarrow 1} \frac{x^{3}-x}{x-1}\\
&\begin{array}{l|l|l|l|l|l|l}
x & 0.900 & 0.990 & 0.999 & 1.001 & 1.010 & 1.100 \\
\hline f(x) & & & & & &
\end{array}
\end{aligned}$$

Dwijendra Rao

Numerade Educator

Problem 26

Evaluate the indicated limits by evaluating the function for values shown in the table and observing the values that are obtained. Do not change the form of the function.
$$\text { Find } \lim _{x \rightarrow-3} \frac{x^{3}+2 x^{2}-2 x+3}{x+3}$$
$$\begin{array}{l|l|l|l|l|l|l}
x & -3.100 & -3.010 & -3.001 & -2.999 & -2.990 & -2.900 \\
\hline f(x) & & & & &
\end{array}$$

Linh Vu

Numerade Educator

Problem 27

Evaluate the indicated limits by evaluating the function for values shown in the table and observing the values that are obtained. Do not change the form of the function.
$$\begin{aligned}
&\text { Find } \lim _{x \rightarrow 2} \frac{2-\sqrt{x+2}}{x-2}\\
&\begin{array}{l|l|l|l|l|l|l}
x & 1.900 & 1.990 & 1.999 & 2.001 & 2.010 & 2.100 \\
\hline f(x) & & & & & &
\end{array}
\end{aligned}$$

Dwijendra Rao

Numerade Educator

Problem 28

Evaluate the indicated limits by evaluating the function for values shown in the table and observing the values that are obtained. Do not change the form of the function.
$$\text { Find } \lim _{x \rightarrow 0} \frac{e^{x}-1}{x}$$
$$\begin{array}{l|l|l|l|l|l|l}
x & -0.1 & -0.01 & -0.001 & 0.001 & 0.01 & .01 \\
\hline f(x) & & & & & &
\end{array}$$

Dwijendra Rao

Numerade Educator

Problem 29

Evaluate the indicated limits by evaluating the function for values shown in the table and observing the values that are obtained. Do not change the form of the function.
$$\text { Find } \lim _{x \rightarrow \infty} \frac{2 x+1}{5 x-3}$$
$$\begin{array}{l|l|l|l}
x & 10 & 100 & 1000 \\
\hline f(x) & & &
\end{array}$$

Dwijendra Rao

Numerade Educator

Problem 30

Evaluate the indicated limits by evaluating the function for values shown in the table and observing the values that are obtained. Do not change the form of the function.
$$\text { Find } \lim _{x \rightarrow \infty} \frac{1-x^{2}}{8 x^{2}+5}$$
$$\begin{array}{l|l|l|l}
x & 10 & 100 & 1000 \\
\hline f(x) & & &
\end{array}$$

Dwijendra Rao

Numerade Educator

Problem 31

Evaluate the indicated limits algebraically as in Examples $10-14 .$ Change the form of the function where necessary.
$$\lim _{x \rightarrow 3}(3 x-2)$$

Dwijendra Rao

Numerade Educator

Problem 32

Evaluate the indicated limits algebraically as in Examples $10-14 .$ Change the form of the function where necessary.
$$\lim _{x \rightarrow 4} \sqrt{x^{2}-7}$$

Dwijendra Rao

Numerade Educator

Problem 33

Evaluate the indicated limits algebraically as in Examples $10-14 .$ Change the form of the function where necessary.
$$\lim _{x \rightarrow 0} \frac{6 x^{2}+x}{x}$$

Dwijendra Rao

Numerade Educator

Problem 34

Evaluate the indicated limits algebraically as in Examples $10-14 .$ Change the form of the function where necessary.
$$\lim _{v \rightarrow 2} \frac{4 v^{2}-8 v}{v-2}$$

Dwijendra Rao

Numerade Educator

Problem 35

Evaluate the indicated limits algebraically as in Examples $10-14 .$ Change the form of the function where necessary.
$$\lim _{x \rightarrow-1} \frac{x^{2}-1}{3 x+3}$$

Dwijendra Rao

Numerade Educator

Problem 36

Evaluate the indicated limits algebraically as in Examples $10-14 .$ Change the form of the function where necessary.
$$\lim _{x \rightarrow 3} \frac{x^{2}-2 x-3}{3-x}$$

Dwijendra Rao

Numerade Educator

Problem 37

Evaluate the indicated limits algebraically as in Examples $10-14 .$ Change the form of the function where necessary.
$$\lim _{h \rightarrow 3} \frac{h^{3}-27}{h-3}$$

Dwijendra Rao

Numerade Educator

Problem 38

Evaluate the indicated limits algebraically as in Examples $10-14 .$ Change the form of the function where necessary.
$$\lim _{x \rightarrow 1 / 3} \frac{9 x-3}{3 x^{2}+5 x-2}$$

Dwijendra Rao

Numerade Educator

Problem 39

Evaluate the indicated limits algebraically as in Examples $10-14 .$ Change the form of the function where necessary.
$$\lim _{x \rightarrow 1} \frac{(2 x-1)^{2}-1}{2 x-2}$$

Dwijendra Rao

Numerade Educator

Problem 40

Evaluate the indicated limits algebraically as in Examples $10-14 .$ Change the form of the function where necessary.
$$\lim _{x \rightarrow 4} \frac{|x-4|}{x-4}$$

Dwijendra Rao

Numerade Educator

Problem 41

Evaluate the indicated limits algebraically as in Examples $10-14 .$ Change the form of the function where necessary.
$$\lim _{p \rightarrow-1} \sqrt{p}(p+1.3)$$

Dwijendra Rao

Numerade Educator

Problem 42

Evaluate the indicated limits algebraically as in Examples $10-14 .$ Change the form of the function where necessary.
$$\lim _{x \rightarrow 1}(x-1) \sqrt{x^{2}-4}$$

Dwijendra Rao

Numerade Educator

Problem 43

Evaluate the indicated limits algebraically as in Examples $10-14 .$ Change the form of the function where necessary.
$$\lim _{x \rightarrow 1} \frac{\sqrt{x}-1}{x-1}$$

Dwijendra Rao

Numerade Educator

Problem 44

Evaluate the indicated limits algebraically as in Examples $10-14 .$ Change the form of the function where necessary.
$$\lim _{x \rightarrow 8} \frac{x-8}{\sqrt[3]{x}-2}$$

Dwijendra Rao

Numerade Educator

Problem 45

Evaluate the indicated limits algebraically as in Examples $10-14 .$ Change the form of the function where necessary.
$$\lim _{h \rightarrow 0} \frac{\sqrt{9+h}-3}{h}$$

Dwijendra Rao

Numerade Educator

Problem 46

Evaluate the indicated limits algebraically as in Examples $10-14 .$ Change the form of the function where necessary.
$$\lim _{h \rightarrow 0} \frac{(4+h)^{2}-16}{h}$$

Dwijendra Rao

Numerade Educator

Problem 47

Evaluate the indicated limits algebraically as in Examples $10-14 .$ Change the form of the function where necessary.
$$\lim _{x \rightarrow \infty} \frac{3 x^{2}+4.5}{x^{2}-1.5}$$

Dwijendra Rao

Numerade Educator

Problem 48

Evaluate the indicated limits algebraically as in Examples $10-14 .$ Change the form of the function where necessary.
$$\lim _{x \rightarrow \infty} \frac{x-27}{7 x+4}$$

Dwijendra Rao

Numerade Educator

Problem 49

Evaluate the indicated limits algebraically as in Examples $10-14 .$ Change the form of the function where necessary.
$$\lim _{t \rightarrow \infty} \frac{\sqrt{t^{2}+16}}{t+1}$$

Dwijendra Rao

Numerade Educator

Problem 50

Evaluate the indicated limits algebraically as in Examples $10-14 .$ Change the form of the function where necessary.
$$\lim _{x \rightarrow \infty} \frac{1-2 x^{2}}{(4 x+3)^{2}}$$

Dwijendra Rao

Numerade Educator

Problem 51

Evaluate the function at $0.1,0.01,$ and 0.001 from both sides of the value it approaches. Evaluate the function for values of $x$ of $10,100,$ and 1000 From these values, determine the limit. Then, by using an appropriate change of form, evaluate the limit algebraically and compare values.
$$\lim _{x \rightarrow 0} \frac{x^{2}-3 x}{x}$$

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 52

Evaluate the function at $0.1,0.01,$ and 0.001 from both sides of the value it approaches. Evaluate the function for values of $x$ of $10,100,$ and 1000 From these values, determine the limit. Then, by using an appropriate change of form, evaluate the limit algebraically and compare values.
$$\lim _{x \rightarrow 3} \frac{2 x^{2}-6 x}{x-3}$$

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 53

Evaluate the function at $0.1,0.01,$ and 0.001 from both sides of the value it approaches. Evaluate the function for values of $x$ of $10,100,$ and 1000 From these values, determine the limit. Then, by using an appropriate change of form, evaluate the limit algebraically and compare values.
$$\lim _{x \rightarrow \infty} \frac{2 x^{2}+x}{x^{2}-3}$$

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 54

Evaluate the function at $0.1,0.01,$ and 0.001 from both sides of the value it approaches. Evaluate the function for values of $x$ of $10,100,$ and 1000 From these values, determine the limit. Then, by using an appropriate change of form, evaluate the limit algebraically and compare values.
$$\lim _{x \rightarrow \infty} \frac{x^{2}+5}{\sqrt{64 x^{4}+1}}$$

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 55

Solve the given problems involving limits.
Evaluate $\lim _{x \rightarrow 2} \frac{x^{3}-8}{x-2}$ by first performing long division (or synthetic division) on $\frac{x^{3}-8}{x-2}$

Dwijendra Rao

Numerade Educator

Problem 56

Solve the given problems involving limits.
Draw the graph of a function that is discontinuous at $x=2,$ has a limit of 2 as $x \rightarrow 2,$ and has a value of 3 at $x=2$

Amy Jiang

Numerade Educator

Problem 57

Solve the given problems involving limits.
A certain object, after being heated, cools at such a rate that its temperature $T$ (in $^{\circ} \mathrm{C}$ ) decreases $10 \%$ each minute. If the object is originally heated to $100^{\circ} \mathrm{C}$, $$
\text { find } \lim _{t \rightarrow 10} T \text { and } \lim _{t \rightarrow \infty} T
$$, where $t$ is the time (in min).

Gio Maya

Numerade Educator

Problem 58

Solve the given problems involving limits.
The area $A$ (in $\mathrm{mm}^{2}$ ) of the pupil of a certain person's eye is given by $A=\frac{36+24 b^{3}}{1+4 b^{3}},$ where $b$ is the brightness (in lumens) of the light source. Between what values does $A$ vary if $b \geq 0 ?$

Nick Johnson

Numerade Educator

Problem 59

Solve the given problems involving limits.
Velocity can be found by dividing the displacement $s$ of an object by the elapsed time $t$ in moving through the displacement. In a certain experiment, the following values were measured for the displacements and elapsed times for the motion of an object. Determine the limiting value of the velocity as $t \rightarrow 0$
$$\begin{array}{l|l|l|l|l|l}
s(\mathrm{cm}) & 0.480000 & 0.280000 & 0.029800 & 0.0029980 & 0.00029998 \\
\hline t(\mathrm{s}) & 0.200000 & 0.100000 & 0.010000 & 0.0010000 & 0.00010000
\end{array}$$

Dwijendra Rao

Numerade Educator

Problem 60

Solve the given problems involving limits.
A $5-\Omega$ resistor and a variable resistor of resistance $R$ are placed in parallel. The expression for the resulting resistance $R_{T}$ is given by $R_{T}=\frac{5 R}{5+R} .$ Determine the limiting value of $R_{T}$ as $R \rightarrow \infty$

Dwijendra Rao

Numerade Educator

Problem 61

Use a calculator to evaluate the indicated limits.
$$\text { Approximate } \lim _{x \rightarrow 2} \frac{2^{x}-4}{x-2}$$

Dwijendra Rao

Numerade Educator

Problem 62

Use a calculator to evaluate the indicated limits.
$$\text { Approximate } \lim _{x \rightarrow 1} \frac{4^{x}-4}{x-1}$$

Dwijendra Rao

Numerade Educator

Problem 63

Use a calculator to evaluate the indicated limits.
$$\lim _{x \rightarrow 0}(1+x)^{1 / x}$$(Do you recognize the limiting value?)

Dwijendra Rao

Numerade Educator

Problem 64

Use a calculator to evaluate the indicated limits.
$$\lim _{x \rightarrow 0} \frac{\sin x}{x} \text { (Use radian mode.) }$$

Dwijendra Rao

Numerade Educator

Problem 65

$\lim _{x \rightarrow a^{-}} f(x)$ means to find the limit as $x$ approaches a from the left only, and $\lim _{x \rightarrow a^{+}} f(x)$ means to find the limit as $x$ approaches a from the right only. These are called one-sided limits. Solve the following problems.
For the function displayed in Exercise $16,$ find:
(a) $$\lim _{x \rightarrow 2^{-}} f(x)$$
(b) $$\lim _{x \rightarrow 2^{+}} f(x)$$
(c) $$\lim _{x \rightarrow 2} f(x)$$

Dwijendra Rao

Numerade Educator

Problem 66

$\lim _{x \rightarrow a^{-}} f(x)$ means to find the limit as $x$ approaches a from the left only, and $\lim _{x \rightarrow a^{+}} f(x)$ means to find the limit as $x$ approaches a from the right only. These are called one-sided limits. Solve the following problems.
For the function displayed in Exercise $16,$ find:
(a) $$\lim _{x \rightarrow-2^{-}} f(x)$$
(b) $$\lim _{x \rightarrow-2^{+}} f(x)$$
(c) $$\lim _{x \rightarrow-2} f(x)$$

Dwijendra Rao

Numerade Educator

Problem 67

$\lim _{x \rightarrow a^{-}} f(x)$ means to find the limit as $x$ approaches a from the left only, and $\lim _{x \rightarrow a^{+}} f(x)$ means to find the limit as $x$ approaches a from the right only. These are called one-sided limits. Solve the following problems.
$$\text { Find } \lim _{x \rightarrow 4^{-}} x \sqrt{16-x^{2}}$$

Dwijendra Rao

Numerade Educator

Problem 68

$\lim _{x \rightarrow a^{-}} f(x)$ means to find the limit as $x$ approaches a from the left only, and $\lim _{x \rightarrow a^{+}} f(x)$ means to find the limit as $x$ approaches a from the right only. These are called one-sided limits. Solve the following problems.
$$\text { Explain why } \lim _{x \rightarrow 0^{+}} 2^{1 / x} \neq \lim _{x \rightarrow 0^{-}} 2^{1 / x}$$

Dwijendra Rao

Numerade Educator

Problem 69

$\lim _{x \rightarrow a^{-}} f(x)$ means to find the limit as $x$ approaches a from the left only, and $\lim _{x \rightarrow a^{+}} f(x)$ means to find the limit as $x$ approaches a from the right only. These are called one-sided limits. Solve the following problems.
$$\text { For } f(x)=\frac{x}{|x|}$$,$$\text { find } \lim _{x \rightarrow 0^{-}} f(x) \text { and } \lim _{x \rightarrow 0^{+}} f(x)$$.Is $f(x)$ continuous at $x=0 ?$ Explain.

Dwijendra Rao

Numerade Educator

Problem 70

$\lim _{x \rightarrow a^{-}} f(x)$ means to find the limit as $x$ approaches a from the left only, and $\lim _{x \rightarrow a^{+}} f(x)$ means to find the limit as $x$ approaches a from the right only. These are called one-sided limits. Solve the following problems.
In Einstein's theory of relativity, the length $L$ of an object moving at a velocity $v$ is $$L=L_{0} \sqrt{1-\frac{v^{2}}{c^{2}}}$$ where $c$ is the speed of light and $L_{0}$ is the length of the object at rest. Find $$\lim _{v \rightarrow c^{-}}$$ $L$ and explain why a limit from the left is used.

Dwijendra Rao

Numerade Educator

Problem 71

$\lim _{x \rightarrow a^{-}} f(x)$ means to find the limit as $x$ approaches a from the left only, and $\lim _{x \rightarrow a^{+}} f(x)$ means to find the limit as $x$ approaches a from the right only. These are called one-sided limits. Solve the following problems.
Is there a difference between
$$\lim _{x \rightarrow 2^{-}} \frac{1}{x-2}$$ and $$\lim _{x \rightarrow 2^{+}} \frac{1}{x-2} ?$$

Dwijendra Rao

Numerade Educator

Problem 72

Is there a difference between
$$\lim _{x \rightarrow 2^{-}} \frac{1}{\sqrt{x-2}} \text { and } \lim _{x \rightarrow 2^{+}} \frac{1}{\sqrt{x-2}} ?$$

Dwijendra Rao

Numerade Educator