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All Textbook Solutions for Calculus and Its Applications (11th Edition)
64RE65RE66REFor Exercises 1-3, consider
,
1. Numerical limits.
a. Find the limit by competing the following input-output tables.
5.9 11.9
5.99 11.99
5.999 11.999
6.1 12.1
6.01 12.01
6.001 12.001
b. b) Find if each exists.
For Exercises 1-3, consider limx6f(x),wheref(x)x236x6, Graphical limits. Find the limit by graphing the function.For Exercises 1-3, consider limx6f(x),wheref(x)x236x6, Algebraic limits. Find the limit algebraically. Show your work. limx6x236x6=limx6(x+6)(x6)x6=limx6(x+6)=12For Exercises 4-15, consider the function f graphed below. limx5f(x)For Exercises 4-15, consider the function f graphed below. limx4f(x)For Exercises 4-15, consider the function f graphed below.
6.
For Exercises 4-15, consider the function f graphed below.
7.
For Exercises 4-15, consider the function f graphed below. limx1f(x)For Exercises 4-15, consider the function f graphed below. limx1f(x)For Exercises 4-15, consider the function f graphed below. limx2f(x)For Exercises 4-15, consider the function f graphed below.
11.
For Exercises 4-15, consider the function f graphed below. Find f(2).For Exercises 4-15, consider the function f graphed below. Find f(6).For Exercises 4-15, consider the function f graphed below. State the value(s) of x at which f is not continuous.For Exercises 4-15, consider the function f graphed below.
15. State the value(s) of x for which is not defined
Determine whether each function is continuous. If a function is not continuous, state why.
16.
Determine whether each function is continuous. If a function is not continuous, state why.
17.
For Exercises 18 and 19, consider the function show in Exercises 17.
18. a. a) Find .
b. b) Find .
c. c) Is f continuous at 3?
For Exercises 18 and 19, consider the function show in Exercises 17.
19. Find .
Find each limit, if it exists. If a limit does not exist, state why.
20.
Find each limit, if it exists. If a limit does not exist, state why. limx2+x2x(x24)Find each limit, if it exists. If a limit does not exist, state why. limx07xFind the simplified difference quotient for f(x)=2x2+3x9.Find an equation of the line tangent to y=x+(4/x) at the point (4,5).25. Find the point(s) on the graph of at which the tangent line is horizontal.
Find dy/dx y=x23Find
27.
Find dy/dx y=10xFind dy/dx y=x5/4Find dy/dx y=0.5x2+0.61x+90Differentiate y=13x3x2+2x+4Differentiate
32.
Differentiate f(x)=x5xDifferentiate f(x)=(x+3)4(7x)5Differentiate y=(x54x3+x)5Differentiate
36.
Differentiate For y=x43x2 find d3ydx3.38. Social science: memory. In a certain memory experiment, a person is able to memorize m words after t minutes, where
.
a. a) Find the rate of change of the number of words memorized with respect to time.
b. b) How many words are memorized during the first 10min (at )?
c. c) At what rate are words being memorized after 10min (at )?
Business: average revenue, cost, and profit. Given revenue and cost functions R(x)=50x and C(x)=x2/3+750 where x is the number of Bluetooth speakers produced and R(x) and C(x) are in dollars, find the following: a. a) the average revenue, the average cost, and the average profit when x speakers are produced; b. b) the rate at which average cost id changing when 8 speakers are produced.For Exercises 40 and 41, let and .
40. Find
For Exercises 40 and 41, let f(x)=x2x and g(x)=2x3. Find ddx(gf)(x)A ball is placed on an inclined plane and, due to gravity alone, accelerates down the plane. Let y=s(t) represent the balls distance t seconds after starting to roll. Which below best represents s? a) b) c) d)43TFind limx3x327x3.45TFind the following limit by creating a table of values: limx05x+255x. Start with Tbl=0.1 and then go to 0.01 and 0.001. When you think you the limit, graph y=5x+255x, and use TRACE to verify your assertion.Plot the points and connect them with line segments. This can be done on many calculate by pressing STAT PLOT, turning on PLOT, and selecting the appropriate type.2. a. a) Use REGRESSION to find a cubic function
that fits the data.
b. b) Graph the function over the interval
c. c) Does the function closely model the given data?
d. d) Predict the horizontal distance from home plate at which the ball would have hit the ground had it not hit the billboard.
e. e) Find the rate of change of the ball’s height with respect to its horizontal distance from home plate.
f. Find the point(s) at which the graph has a horizontal tangent line. Explain the significance of the point(s).
3. a. a) Use REGRESSION to find a quartic function
that fits the data.
b. b) Graph the function over the interval
c. c) Does the function closely model the given data?
d. d) Predict the horizontal distance from home plate at which the ball would have hit the ground had it not hit the billboard.
e. e) Find the rate of change of the ball’s height with respect to its horizontal distance from home plate.
f. f) Find the point(s) at which the graph has a horizontal tangent line. Explain the significance of the point(s).
a. a) Although most calculate cannot fit such a function to the data, assume that the equation y=0.0015x202,500x2 has been found using a curve-fitting technique. Graph the function over the interval [0,500]. b. b) Predict the horizontal distance from home plate at which the ball would have hit the ground had it not hit the billboard. c. c) Find the rate of change of the balls height with respect to its horizontal distance from plate. d. d) Find the point(s) at which the graph has a horizontal line. Explain the significance of the point(s).5ETE6ETE7ETE8ETEComplete each of the following statements.
1. As x approaches –2, the value of –3x approaches 6.
Complete each of the following statements. As x approaches 7_, the value of x2 approaches 5.Complete each of the following statements. The notation limx4f(x) is read ______.Complete each of the following statements.
4. The notation is read _____.
Complete each of the following statements. The notation limx5f(x) is read ______.Complete each of the following statements.
6. The notation is read ______.
Complete each of the following statements.
7. The notation _____ is read “the limit, as x approaches 2 from the right.”
Complete each of the following statements. The notation ______ is read the limit, as x approaches 3 from the left,Complete each of the following statements. The notation _____ is read the limit as x approaches 5.Complete each of the following statements. The notation ______ is read the limit as x approaches 12.For Exercises 11 and 12, consider the function f given by
If a limit does not exist, states that fact.
12. find (a) ; (b) ; (c) .
For Exercises 11 and 12, consider the function f given by
If a limit does not exist, states that fact.
12. find (a) ; (b) ; (c) .
For Exercises 13 and 14, consider the function g given by g(x)={x+6,forx2,12x+1,forx2. If limit does not exist, state that fact. Find (a) limx2g(x); (b) limx2+g(x); (c) limx2g(x).For Exercises 13 and 14, consider the function g given by g(x)={x+6,forx2,12x+1,forx2. If limit does not exist, state that fact. Find (a) limx4g(x); (b) limx4+g(x); (c) limx4g(x).For Exercises 15–22, use the following graph of F to find each limit. When necessary, state that the limit does not exist.
15.
For Exercises 15–22, use the following graph of F to find each limit. When necessary, state that the limit does not exist.
16.
For Exercises 15–22, use the following graph of F to find each limit. When necessary, state that the limit does not exist.
17.
For Exercises 15–22, use the following graph of F to find each limit. When necessary, state that the limit does not exist.
18.
For Exercises 1522, use the following graph of F to find each limit. When necessary, state that the limit does not exist. limx6F(x)For Exercises 15–22, use the following graph of F to find each limit. When necessary, state that the limit does not exist.
20.
For Exercises 15–22, use the following graph of F to find each limit. When necessary, state that the limit does not exist.
21.
For Exercises 1522, use the following graph of F to find each limit. When necessary, state that the limit does not exist. limx2+F(x)For Exercises 23-30, use the following graph of G to find each limit. When necessary, state that limit does not exist. limx0G(x)For Exercises 23-30, use the following graph of G to find each limit. When necessary, state that limit does not exist. limx2G(x)For Exercises 23-30, use the following graph of G to find each limit. When necessary, state that limit does not exist. limx1+G(x)For Exercises 23-30, use the following graph of G to find each limit. When necessary, state that limit does not exist. limx1G(x)For Exercises 23-30, use the following graph of G to find each limit. When necessary, state that limit does not exist.
27.
For Exercises 23-30, use the following graph of G to find each limit. When necessary, state that limit does not exist.
28.
For Exercises 23-30, use the following graph of G to find each limit. When necessary, state that limit does not exist. limx3G(x)For Exercises 23-30, use the following graph of G to find each limit. When necessary, state that limit does not exist.
30.
For Exercises 31–40, use the following graph of H to find each limit. When necessary, state that the limit does not exist.
31.
For Exercises 3140, use the following graph of H to find each limit. When necessary, state that the limit does not exist. limx3H(x)For Exercises 3140, use the following graph of H to find each limit. When necessary, state that the limit does not exist. limx2H(x)For Exercises 3140, use the following graph of H to find each limit. When necessary, state that the limit does not exist. limx2+H(x)For Exercises 3140, use the following graph of H to find each limit. When necessary, state that the limit does not exist. limx1+H(x)For Exercises 31–40, use the following graph of H to find each limit. When necessary, state that the limit does not exist.
36.
For Exercises 3140, use the following graph of H to find each limit. When necessary, state that the limit does not exist. limx1H(x)For Exercises 31–40, use the following graph of H to find each limit. When necessary, state that the limit does not exist.
38.
For Exercises 31–40, use the following graph of H to find each limit. When necessary, state that the limit does not exist.
39.
For Exercises 3140, use the following graph of H to find each limit. When necessary, state that the limit does not exist. limx3+H(x)For Exercises 41-50, use the following graph of f to find each limit. When necessary, state that the limit does not exist. limx2F(x)For Exercises 41-50, use the following graph of f to find each limit. When necessary, state that the limit does not exist. limx1f(x)For Exercises 41-50, use the following graph of f to find each limit. When necessary, state that the limit does not exist.
43.
For Exercises 41-50, use the following graph of f to find each limit. When necessary, state that the limit does not exist. limx3f(x)For Exercises 41-50, use the following graph of f to find each limit. When necessary, state that the limit does not exist.
45.
For Exercises 41-50, use the following graph of f to find each limit. When necessary, state that the limit does not exist. limx3f(x)For Exercises 41-50, use the following graph of f to find each limit. When necessary, state that the limit does not exist. limx2f(x)For Exercises 41-50, use the following graph of f to find each limit. When necessary, state that the limit does not exist.
48.
For Exercises 41-50, use the following graph of f to find each limit. When necessary, state that the limit does not exist. limxf(x)For Exercises 41-50, use the following graph of f to find each limit. When necessary, state that the limit does not exist.
50.
For Exercises 51-68, graph each function and then find the specified limits. When necessary, state that the limit does not exist. f(x)=|x|; find limx0f(x) and limx2f(x).For Exercises 51-68, graph each function and then find the specified limits. When necessary, state that the limit does not exist. f(x)=x2; find limx1f(x) and limx0f(x).For Exercises 51-68, graph each function and then find the specified limits. When necessary, state that the limit does not exist. g(x)= x25; find limx0g(x) and limx1g(x).For Exercises 51-68, graph each function and then find the specified limits. When necessary, state that the limit does not exist. g(x)=|x|+1; find limx3g(x) and limx0g(x).For Exercises 51-68, graph each function and then find the specified limits. When necessary, state that the limit does not exist.
55. ; find and.
For Exercises 51-68, graph each function and then find the specified limits. When necessary, state that the limit does not exist. F(x)=1x3; find limx3F(x) and limx4F(x).For Exercises 51-68, graph each function and then find the specified limits. When necessary, state that the limit does not exist. f(x)=1x2;findlimxf(x)andlimx0f(x).For Exercises 51-68, graph each function and then find the specified limits. When necessary, state that the limit does not exist. f(x)=1x+3;findlimxf(x)andlimx0f(x).For Exercises 51-68, graph each function and then find the specified limits. When necessary, state that the limit does not exist.
59. find and .
For Exercises 51-68, graph each function and then find the specified limits. When necessary, state that the limit does not exist. g(x)=1x2+4; find limxg(x) and limx2g(x).For Exercises 51-68, graph each function and then find the specified limits. When necessary, state that the limit does not exist. F(x)={2x+1,forx1,x,forx1. Find limx1F(x),limx1+F(x),andlimx1F(x).For Exercises 51-68, graph each function and then find the specified limits. When necessary, state that the limit does not exist.
62.
Find .
For Exercises 51-68, graph each function and then find the specified limits. When necessary, state that the limit does not exist.
63.
Find .
For Exercises 51-68, graph each function and then find the specified limits. When necessary, state that the limit does not exist.
64.
Find .
For Exercises 51-68, graph each function and then find the specified limits. When necessary, state that the limit does not exist. F(x)={2x3,forx1,x3,forx1. Findlimx1F(x).For Exercises 51-68, graph each function and then find the specified limits. When necessary, state that the limit does not exist. G(x)={x2,forx1,x+2,forx1. Findlimx1G(x).For Exercises 51-68, graph each function and then find the specified limits. When necessary, state that the limit does not exist.
67.
Find and .
For Exercises 51-68, graph each function and then find the specified limits. When necessary, state that the limit does not exist.
68.
Find .
Business and Economics
Taxicab fares. In New York City, taxicabs change passengers $2.50 for entering a cab and then $050 for each one-fifth of a mile (or fraction thereof) traveled. (There are additional charge for slow traffic and idle times, but these are not considered in this problem.) If x presents the distance traveled in miles, then is the cost of the taxi fare, where
and so on. The graph of C is show below. (Source; New York City Taxi and Limousine Commission.)
Using the graph of the taxicab fare function, find each of the following limits. if it exists.
69.
Taxicab fares. In New York City, taxicabs change passengers $2.50 for entering a cab and then $050 for each one-fifth of a mile (or fraction thereof) traveled. (There are additional charge for slow traffic and idle times, but these are not considered in this problem.) If x presents the distance traveled in miles, then C(x) is the cost of the taxi fare, where C(x)=2.50,ifx=0,C(x)=3.00,if0x0.2,C(x)=3.50,if0.2x0.4,C(x)=4.00,if0.4x0.6, and so on. The graph of C is show below. (Source; New York City Taxi and Limousine Commission.) Using the graph of the taxicab fare function, find each of the following limits. if it exists. limx0.2C(x),limx0.2+C(x),limx0.2C(x)Taxicab fares. In New York City, taxicabs change passengers $2.50 for entering a cab and then $050 for each one-fifth of a mile (or fraction thereof) traveled. (There are additional charge for slow traffic and idle times, but these are not considered in this problem.) If x presents the distance traveled in miles, then C(x) is the cost of the taxi fare, where C(x)=2.50,ifx=0,C(x)=3.00,if0x0.2,C(x)=3.50,if0.2x0.4,C(x)=4.00,if0.4x0.6, and so on. The graph of C is show below. (Source; New York City Taxi and Limousine Commission.) Using the graph of the taxicab fare function, find each of the following limits. if it exists. limx0.6C(x),limx0.6+C(x),limx0.6C(x)The Postage function. The cost of sending a large envelope via U.S. first-class mail in 2014 was $0.98 for the first ounce and $0.21 for each additional ounce (or fraction thereof). (Source; www.usps.com.) If x represents the weight of a large envelope, in ounces, then p(x) is the cost of mailing it, where p(x)=0.98,if0x1,p(x)=1.19,if1x2,p(x)=1.40,if2x3, And so on, up through 13 ounce. The graph of p is show below. Using the graph of the postage function, find each of the following limit, if it exists. limx1p(x),limx1+p(x),limx1p(x)The Postage function.
The cost of sending a large envelope via U.S. first-class mail in 2014 was $0.98 for the first ounce and $0.21 for each additional ounce (or fraction thereof). (Source; www.usps.com.) If x represents the weight of a large envelope, in ounces, then is the cost of mailing it, where
And so on, up through 13 ounce. The graph of p is show below.
Using the graph of the postage function, find each of the following limit, if it exists.
73.
The Postage function. The cost of sending a large envelope via U.S. first-class mail in 2014 was $0.98 for the first ounce and $0.21 for each additional ounce (or fraction thereof). (Source; www.usps.com.) If x represents the weight of a large envelope, in ounces, then p(x) is the cost of mailing it, where p(x)=0.98,if0x1,p(x)=1.19,if1x2,p(x)=1.40,if2x3, And so on, up through 13 ounce. The graph of p is show below. Using the graph of the postage function, find each of the following limit, if it exists. limx2.6p(x),limx2.6+p(x),limx2.6p(x)The Postage function.
The cost of sending a large envelope via U.S. first-class mail in 2014 was $0.98 for the first ounce and $0.21 for each additional ounce (or fraction thereof). (Source; www.usps.com.) If x represents the weight of a large envelope, in ounces, then is the cost of mailing it, where
And so on, up through 13 ounce. The graph of p is show below.
Using the graph of the postage function, find each of the following limit, if it exists.
75.
The Postage function.
The cost of sending a large envelope via U.S. first-class mail in 2014 was $0.98 for the first ounce and $0.21 for each additional ounce (or fraction thereof). (Source; www.usps.com.) If x represents the weight of a large envelope, in ounces, then is the cost of mailing it, where
And so on, up through 13 ounce. The graph of p is show below.
Using the graph of the postage function, find each of the following limit, if it exists.
76.
Tax Rate Schedule. The federal tax rate for single files is given as a percentage of taxable income in the graph below. (Source; troweprice.com, 2013.) Use the graph for Exercises 77-79, Find limx8925r(x),limx8925+r(x)andlimx8925r(x).Tax Rate Schedule. The federal tax rate for single files is given as a percentage of taxable income in the graph below. (Source; troweprice.com, 2013.) Use the graph for Exercises 77-79, Find limx10,000r(x),limx10,000+r(x)andlimx10,000r(x).Tax Rate Schedule. The federal tax rate for single files is given as a percentage of taxable income in the graph below. (Source; troweprice.com, 2013.) Use the graph for Exercises 77-79, Find limx50,000r(x)andlimx87,850r(x).Tax Rate Schedule.
The federal tax rate for heads of household is given in the graph below.
(Source; troweprice.com, 2013.)
Use the graph for Exercises 80-82.
80. Find .
Tax Rate Schedule.
The federal tax rate for heads of household is given in the graph below.
(Source; troweprice.com, 2013.)
Use the graph for Exercises 80-82.
81. Find .
Tax Rate Schedule.
The federal tax rate for heads of household is given in the graph below.
(Source; troweprice.com, 2013.)
Use the graph for Exercises 80-82.
82. Find .
In Exercises 83-58, fill in each blank so that exists.
83.
In Exercises 83-58, fill in each blank so that limx2f(x) exists. f(x)={12x+1,forx2,32x+,forx2In Exercises 83-58, fill in each blank so that limx2f(x) exists. f(x)={x29,forx2,x2+,forx2Graph the function f given by f(x)={3,forx=2,x2forx2. Use GRAPH and TRACE to find each of the following limits. When necessary, state that the limit does not exists. a. limx2+f(x) b. limx2f(x) c. limx2f(x) d. limx2+f(x) e. limx2f(x) f. Does limx2f(x)=f(2)? g. Does limx2f(x)=f(2)?In Exercises 87-89, use GRAFH and TRACE to find each limit. When necessary, state that the limit does not exist.
87. For .
In Exercises 87-89, use GRAFH and TRACE to find each limit. When necessary, state that the limit does not exist.
88. For .
In Exercises 87-89, use GRAFH and TRACE to find each limit. When necessary, state that the limit does not exist. For f(x)=x5x24x5,findlimx1f(x)andlimx5f(x).Classify each statement as either true or false.
1.
2EClassify each statement as either true or false. If limx1g(x)=5,thenlimx1[g(x)]2=25.Classify each statement as either true or false.
4. If .
Classify each statement as either true or false. If g is discontinuous at x=3,theng(3) must not exist.Classify each statement as either true or false. If f is continuous at x=2,thenf(2) must exist.Classify each statement as either true or false.
7. If exists, then F must be continuous at.
Classify each statement as either true or false. If limx7G(x) equals G(7), then G must be continuous at x=7.Use the theorem on limits of rational functions to find each limit. When necessary, state that the limit does not exist. limx2(4x5)Use the theorem on limits of rational functions to find each limit. When necessary, state that the limit does not exist. limx1(3x+2)Use the theorem on limits of rational functions to find each limit. When necessary, state that the limit does not exist. limx1(x24)Use the theorem on limits of rational functions to find each limit. When necessary, state that the limit does not exist. limx2(x2+3)Use the theorem on limits of rational functions to find each limit. When necessary, state that the limit does not exist. limx5(x26x+9)Use the theorem on limits of rational functions to find each limit. When necessary, state that the limit does not exist. limx3(x24x+7)Use the theorem on limits of rational functions to find each limit. When necessary, state that the limit does not exist.
15.
Use the theorem on limits of rational functions to find each limit. When necessary, state that the limit does not exist. limx1(3x5+4x43x+6)Use the theorem on limits of rational functions to find each limit. When necessary, state that the limit does not exist. limx3x225x25Use the theorem on limits of rational functions to find each limit. When necessary, state that the limit does not exist. limx3x28x2For Exercises 19-30, the initial substitution of x=a yields the form 0/0. Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist. limx3x29x3For Exercises 19-30, the initial substitution of yields the form 0/0. Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist.
20.
For Exercises 19-30, the initial substitution of yields the form 0/0. Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist.
21.
For Exercises 19-30, the initial substitution of yields the form 0/0. Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist.
22.
For Exercises 19-30, the initial substitution of x=a yields the form 0/0. Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist. limx23x2+x14x24For Exercises 19-30, the initial substitution of yields the form 0/0. Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist.
24.
For Exercises 19-30, the initial substitution of yields the form 0/0. Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist.
25.
For Exercises 19-30, the initial substitution of x=a yields the form 0/0. Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist. limx1x31x1For Exercises 19-30, the initial substitution of x=a yields the form 0/0. Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist. limx25x5x25For Exercises 19-30, the initial substitution of yields the form 0/0. Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist.
28.
For Exercises 19-30, the initial substitution of x=a yields the form 0/0. Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist. limx2x2+3x10x24x+4For Exercises 19-30, the initial substitution of yields the form 0/0. Look for ways to simplify the function algebraically, or use a table or graph to determine the limit. When necessary, state that the limit does not exist.
30.
Use the Limit Properties to find the following limits. If a limit does not exist, state that fact. limx5x216Use the Limit Properties to find the following limits. If a limit does not exist, state that fact.
32.
Use the Limit Properties to find the following limits. If a limit does not exist, state that fact. limx2x29Use the Limit Properties to find the following limits. If a limit does not exist, state that fact.
34.
Use the Limit Properties to find the following limits. If a limit does not exist, state that fact.
35.
Use the Limit Properties to find the following limits. If a limit does not exist, state that fact.
36.
Determine whether each of the function show in Exercises 37-41, is continuous over the interval
.
37.
Determine whether each of the function show in Exercises 37-41, is continuous over the interval (6,6).Determine whether each of the function show in Exercises 37-41, is continuous over the interval
.
39.
Determine whether each of the function show in Exercises 37-41, is continuous over the interval
.
40.
Determine whether each of the function show in Exercises 37-41, is continuous over the interval
.
41.
Use the graphs and functions in Exercises 37-41 to answer each of the following. If an expression does not exist, state that fact. a. Find limx1+g(x), limx1g(x), and limx1g(x). b. Find g(1). c. Is g continuous at x=1? Why or why not? d. Find limx2g(x). e. Find g(2). f. Is g continuous at x=2? why or why not?Use the graphs and functions in Exercises 37-41 to answer each of the following. If an expression does not exist, state that fact. a. Find limx1+f(x),limx1f(x),andlimx1f(x). b. Find f(1). c. Is f continuous at x=1? why or why not? d. Find limx2f(x). e. find f(2). f. Is f Continuous at x=2? Why or why not?Use the graphs and functions in Exercises 37-41 to answer each of the following. If an expression does not exist, state that fact. a. Find limx1h(x). b. Find h(1). c. Is h continuous at x=1? Why or why not? d. Find limx2h(x). e. Find h(2). f. Is h continuous at x=2? why or why not?Use the graphs and functions in Exercises 37-41 to answer each of the following. If an expression does not exist, state that fact.
45. a. Find .
b. Find .
c. Is k continuous at ? why or why not?
d. Find .
e. Find .
f. Is k continuous at ? Why or why not?
Use the graphs and functions in Exercises 37-41 to answer each of the following. If an expression does not exist, state that fact. a. Find limx1t(x). b. Find t(1). c. Is t continuous at x=1? Why or why not? d. Find limx2t(x). e. Find t(2). f. Is t continuous at x=2? Why or why not?Answer Exercises 47-48 using the graph provided.
47.
a. Find .
b. Find .
c. Find .
d. Find .
e. Is G continuous at ? Why or why not?
f. Is G continuous at ? Why or why not?
g. Is G continuous at ? Why or why not?
Answer Exercises 47-48 using the graph provided.
48. Consider the function
Find .
Find .
c. .
d. Find .
e. Is C continuous at ? Why or why not?
f. Is C continuous at ? Why or why not?
49. Is the function given by continuous at ? Why or why not?
Is the function given by f(x)=3x2 continuous at x=5? Why or why not?Is the function given by G(x)=1x continuous at x=0? Why or why not?Is the function given by F(x)=x continuous at x=1? Why or why not?Is the function given by f(x)={12x+1,forx4,x+7,forx4, Continuous at x=4? Why or why not?Is the function given by g(x)={13x+4,forx3,2x1,forx3, continuous at x=3? Why or why not?Is the function given by F(x)={13x+4,forx3,2x5,forx3, Continuous at x=3? Why or why not?56. Is the function given by
Continuous at? Why or why not?
Is the function given by g(x)={12x+1,forx4,x+7,forx4, Continuous at x=4? Why or why not?Is the function given by f(x)={13x+4,forx3,2x1,forx3, Continuous at x=3? Why or why not?59. Is the function given by
Continuous at? Why or why not?
Is the function given by F(x)={x21x1,forx1,4,forx=1, Continuous at x=1? Why or why not?Is the function given by G(x)={x23x4x4,forx42x3,forx4, Continuous at x=4? Why or why not?62. Is the following given by
Continuous at? Why or why not?
Is the function given by g(x)=1x27x+10 continuous at x=5? Why or why not?64. Is the function given by continuous at? Why or why not?
Is the function given by G(x)=1x26x+8 continuous at x=2? Why or why not?66. Is the function given by continuous at? Why or why not?
67. Is the function given by continuous over the interval? Why or not why?
68. Is the function given by continuous over the interval why or why not?
Is the function given by G(x)=1x1 continuous over the interval (0,)? why or why not?Is the function given by f(x)=1x+3 continuous over the interval (7,7)? Why or why not?71. Is the function given by continuous on?
72. Is the function given by continuous on?
Business and Economics
73. The candy factory sells candy by the pound, charging $1.50 per pound for quantities up to and including 20 pound. Above 20 pound, the candy factory changes $ 1.25 per pound for the entire quantify. If x represents the number of pound, the price function is
Find .
Business and Economics The candy Shoppe charge $0.08 per copy for quantities up to and including 100 copies. For quantities above 100, the charge is $0.06 per copy. If x represents the number of copies, the price function is p(x)={0.08x,forx100,0.06x,forx100. Find limx100p(x),limx100+p(x),andlimx100p(x).A lab technician controls the temperature T inside a kiln. From an initial temperature of 0 degrees Celsius (C), he allows the temperature to increase by 2C per minute for the next 60 minutes. After the 60th minute, he allows the temperature to cool by 3C per minute. If t is the number of minutes, the temperature T is given by T(t)={2t,fort60,3003t,fort60. Find limt60T(t),limt60+T(t),andlimt60T(t).76. In Exercises 73, let
Find k such that the price function p is continuous at.
In Exercises 74, let p(x)={0.08x,forx100,0.06x+k,forx100. Find k such that the price function p is continuous at x=100.Find each limit, if it exists. If a limit does not exist, state that fact. a. limx0|x|x b. limx2x2+8x24In Exercises 7986, find each limit. Use TABLE and start with Tbl=0.1 Then use 0.01, 0.001, and 0.0001. When you think you know the limit, graph and use TRACE to verify your assertion. Then try to verify it algebraically. limx1x1x1In Exercises 7986, find each limit. Use TABLE and start with Tbl=0.1 Then use 0.01, 0.001, and 0.0001. When you think you know the limit, graph and use TRACE to verify your assertion. Then try to verify it algebraically. lima2a24a2+53In Exercises 7986, find each limit. Use TABLE and start with Tbl=0.1 Then use 0.01, 0.001, and 0.0001. When you think you know the limit, graph and use TRACE to verify your assertion. Then try to verify it algebraically. lima04+x4xxIn Exercises 79–86, find each limit. Use TABLE and start with Then use 0.01, 0.001, and 0.0001. When you think you know the limit, graph and use TRACE to verify your assertion. Then try to verify it algebraically.
82.
In Exercises 7986, find each limit. Use TABLE and start with Tbl=0.1 Then use 0.01, 0.001, and 0.0001. When you think you know the limit, graph and use TRACE to verify your assertion. Then try to verify it algebraically. lima07+2x7xIn Exercises 79–86, find each limit. Use TABLE and start with Then use 0.01, 0.001, and 0.0001. When you think you know the limit, graph and use TRACE to verify your assertion. Then try to verify it algebraically.
84.
In Exercises 7986, find each limit. Use TABLE and start with Tbl=0.1 Then use 0.01, 0.001, and 0.0001. When you think you know the limit, graph and use TRACE to verify your assertion. Then try to verify it algebraically. limx0749x2xIn Exercises 79–86, find each limit. Use TABLE and start with Then use 0.01, 0.001, and 0.0001. When you think you know the limit, graph and use TRACE to verify your assertion. Then try to verify it algebraically.
86.
For each function in Exercises 1-16, (a) find the simplified from of the different of the different quotient and then (b) complete the following table, x h f(x+h)f(x)h 5 2 5 1 5 0.1 5 0.01 f(x)=5x2For each function in Exercises 1-16, (a) find the simplified from of the different of the different quotient and then (b) complete the following table,
x h
5 2
5 1
5 0.1
5 0.01
2.
For each function in Exercises 1-16, (a) find the simplified from of the different of the different quotient and then (b) complete the following table,
x h
5 2
5 1
5 0.1
5 0.01
3.
For each function in Exercises 1-16, (a) find the simplified from of the different of the different quotient and then (b) complete the following table, x h f(x+h)f(x)h 5 2 5 1 5 0.1 5 0.01 f(x)=4x2For each function in Exercises 1-16, (a) find the simplified from of the different of the different quotient and then (b) complete the following table,
x h
5 2
5 1
5 0.1
5 0.01
5.
For each function in Exercises 1-16, (a) find the simplified from of the different of the different quotient and then (b) complete the following table, x h f(x+h)f(x)h 5 2 5 1 5 0.1 5 0.01 f(x)=x2+xFor each function in Exercises 1-16, (a) find the simplified from of the different of the different quotient and then (b) complete the following table, x h f(x+h)f(x)h 5 2 5 1 5 0.1 5 0.01 f(x)=9xFor each function in Exercises 1-16, (a) find the simplified from of the different of the different quotient and then (b) complete the following table, x h f(x+h)f(x)h 5 2 5 1 5 0.1 5 0.01 f(x)=2xFor each function in Exercises 1-16, (a) find the simplified from of the different of the different quotient and then (b) complete the following table,
x h
5 2
5 1
5 0.1
5 0.01
9.
a. (a) 2;
b. (b) 2, 2, 2, 2
For each function in Exercises 1-16, (a) find the simplified from of the different of the different quotient and then (b) complete the following table,
x h
5 2
5 1
5 0.1
5 0.01
10.
a. (a) –2;
b. (b) –2, –2, –2, –2
For each function in Exercises 1-16, (a) find the simplified from of the different of the different quotient and then (b) complete the following table,
x h
5 2
5 1
5 0.1
5 0.01
11.
For each function in Exercises 1-16, (a) find the simplified from of the different of the different quotient and then (b) complete the following table,
x h
5 2
5 1
5 0.1
5 0.01
12.
For each function in Exercises 1-16, (a) find the simplified from of the different of the different quotient and then (b) complete the following table,
x h
5 2
5 1
5 0.1
5 0.01
13.
For each function in Exercises 1-16, (a) find the simplified from of the different of the different quotient and then (b) complete the following table, x h f(x+h)f(x)h 5 2 5 1 5 0.1 5 0.01 f(x)=x23x a. (a) 2x+h3 b. (b) 9,8,7.1,7.01For each function in Exercises 1-16, (a) find the simplified from of the different of the different quotient and then (b) complete the following table, x h f(x+h)f(x)h 5 2 5 1 5 0.1 5 0.01 f(x)=x23x+5 a. (a) 2x+h3 b. (b) 9,8,7.1,7.01For each function in Exercises 1-16, (a) find the simplified from of the different of the different quotient and then (b) complete the following table,
x h
5 2
5 1
5 0.1
5 0.01
16.
a.
b.
For Exercises 17-24, use each graph to estimate the average rate of change of the percentage of new employees in that type of employees from 2003 to 2007, from 2003 to 2012. (Source: Bureau of labor statistics.) Total employment.For Exercises 17-24, use each graph to estimate the average rate of change of the percentage of new employees in that type of employees from 2003 to 2007, from 2003 to 2012. (Source: Bureau of labor statistics.) Construction.For Exercises 17-24, use each graph to estimate the average rate of change of the percentage of new employees in that type of employees from 2003 to 2007, from 2003 to 2012. (Source: Bureau of labor statistics.)
19. Professional services.
For Exercises 17-24, use each graph to estimate the average rate of change of the percentage of new employees in that type of employees from 2003 to 2007, from 2003 to 2012. (Source: Bureau of labor statistics.) Health care.For Exercises 17-24, use each graph to estimate the average rate of change of the percentage of new employees in that type of employees from 2003 to 2007, from 2003 to 2012. (Source: Bureau of labor statistics.) Education.For Exercises 17-24, use each graph to estimate the average rate of change of the percentage of new employees in that type of employees from 2003 to 2007, from 2003 to 2012. (Source: Bureau of labor statistics.)
22. Government.
For Exercises 17-24, use each graph to estimate the average rate of change of the percentage of new employees in that type of employees from 2003 to 2007, from 2003 to 2012. (Source: Bureau of labor statistics.)
23. Mining and logging.
For Exercises 17-24, use each graph to estimate the average rate of change of the percentage of new employees in that type of employees from 2003 to 2007, from 2003 to 2012. (Source: Bureau of labor statistics.) Manufacturing.25. Use the following graph to find the average rate of change in U.S. energy consumption from 1982 to 1992, from 1992 to 2002, and from 2002 to 2012.
U.S. ENERGY CONSUMPTION
(Source U.S. energy information administration.)
26. Use the following graph to find the average rate of change of the U.S. trade deficit with Japan from 2007 to 2009, from 2009 to 2011, and 2013.
U.S. TRADE DEFICIT WITH JAPAN
(Source; U.S Census Bureau, Statistical Abstract of the United States, 2012.)
(Source; U.S Census Bureau, Statistical Abstract of the United States, 2012.)
27. Utility. Utility is a type of function that occurs in economics. When a consumer receives x units of a product, a certain amount of pleasure, or utility U, is gained. The following represents a typical utility function.
a. a) Find the average rate of change of U as x change from 0 to 1; from 1 to 2; from 2 to 3; from 3 to 4.
b. b) Why do you think the average rates of change are decreases?
28. Advertising results. The following graph shows a typical response to advertising. When amount a is spent on advertising, units of a product are sold.
a. a) find the average rate of change of N as a change from 0 to 1; from to 1 to 2; from 2 to 3; from 3 to 4.
b. b) Why do you think the average rates of change are decreasing as x increases?
29E30. Compound interest. The amount of money, in a saving account that pays 6% interest, compounded quarterly for t years, when an initial investment of is made, is given by
a. a) Find
b. b) Find
c. c) Find
d. d) Find what rate of change does this represent?
31. Population change. The population of payton country was 5400 at the last census and decreasing at the rate of 2.5% per year. The total population of the county after t years, , is given by
.
Find. What rate of change does this represent?
Population change. The undergraduate population at harbor Collage was 17,000 and increasing at the rate of 4.2% per year. The undergraduate Population after t years, p(t), is given by P(t)=17,000(1.042)t. Find P(6)P(2)62. What rate of change does this represent?Total cost. Suppose Fast Trends determines that the cost, in dollars, of producing x iPod holders is given by C(x)=0.05x2+50x. Find C(305)C(300)305300, and interpret the significance of this result to the company.Total revenue. Suppose Fast Trends determines that the revenue, in dollars, from the sale of x iPod holders is given by R(x)=0.001x2+150x. R(x)=0.01x2+150x. Find R(305)R(300)305300, and interpret the significance of this result to the company.35. Growth of a baby. The median weights of babies at age t months are graphed below.
(Source; Developed by the National Center for Health Statistics in collaboration with the national Center for Chronic Disease Prevention and Health promotion, 2000; checked in 2004,)
Use the graph of girls’ median weight to estimate;
a. a) The average growth rate of a girl during her first 12 months. (Your answer should be in pounds per month.)
b. b) The average growth rate of a girl during her second 12 month.
c. c) The average growth rate of a girl during her first 24 months.
d. d) Based on your answers in parts (a)-(c) and the graph, estimate the growth rate of a typical 12-month-old girl, and explain how you arrived at this figure.
e. e) In What-three-month period does a baby girl grow fastest?
36. Growth of a baby. Use the graph of boys’ median weight in Exercises 35 to estimate;
a. a) the average growth rate of a boy during his first 15 months. (Your answer should be in pounds per month.)
b. b) the average growth rate of a boy during his second 15 months. (Your answer should be in pounds per month.)
c. c) The average growth rate of a boy during his first 30 months. (Your answer should be in pounds per month.)
d. d) Based on your answer in part (a)-(c) and the graph, estimate the growth rate of a typical boy at exactly 15 months, and explain how you arrived at this figure.
Home range. It has been show that the home range, in hectares, of a carnivorous mammal weighing w grams can be approximated by H(w)=0.11w1.36. (Source: Based on information in Emlen, J. M., Ecology: An Evolutionary Approach, p. 200, Reading, MA: Addison-Wesley, 1973; and Harestad, A. S., and Bunnel, F. L., Home Range and Body Weight-A Reevaluation, Ecology, Vol. 60, No. 2, pp. 389-402.) a. a) Find the average rate at which a carnivorous mammals home range increases as the animals weight grows from 500 g to 700 g. b. b) Find H(300)H(200)300200. What does this rate represent?38. Reader range. The function given by can be used to approximate the maximum range, in miles, of an ARSR-3 surveillance radar with a peak power of x watts (W). (Source: Introduction to RADAR Techniques, federal Aviation Administration, 1988.)
a. a) Find the rate at which maximum radar range as peak power increases from 40,000 W to 60,000 W.
b. b) Find. What does this rate represent?
Memory. The total numbers of words, M(t), that a person can memorize in t is show in the following graph. a) Find the average rate of change of M as t changes from 0 to 8; from 8 to 16; from 16 to 24; from 24 to 32; from 32 to 36. b) Why do the average rates of changes become 0 when t exceeds 24 min?Gas mileage. At the beginning of a trip, the odometer on a car reads 30,680, and the car has a full tank of gas. At the end of the trip, the odometer reads 31,077. It takes 13.5 gal of gas to refill the tank. a. a) What is the average rate at which the car was travelling, in miles per gallon? b. b) What is the average rate of gas consumption in gallons per mile?Average velocity. In second, an object dropped from a certain height will fall s(t) feet, where s(t)=16t2. a. a) Find s(t)s(3). b. b) What is the average rate of change of distance with respect to time during the period from 3 to 5 sec? This known as average velocity, or speed.42E43. Population growth. The two curves below describe the numbers of people in two countries at time t, in years.
a. a) Find the average rate of changes of each population with respect to time t as t changes from 0 to 4. This is often called the average growth rate.
b. b) If the calculations in part (a) were the only ones made, would we detect the fact that the populations were growing differently? Why or why not?
c. c) Find the average rates of changes of each population as t changes from 0 to 1; from 1 to 2; from 2 to 3; from 3 to 4.
d. For which population does the statement “the population grew consistently at a rate of 125 million per year” convey accurate information? Why?
44. Business: comparing rate of changes. The following two graphs show the number of federally insured banks from 1987 to 2013 and the Nasdaq Composite Stock Index over a 6-month period in 2013.
(Source: Federal Deposit Insurance Corp., 12/26/13.)
(Source: Federal Deposit Insurance Corp., 12/26/13.)
(Source: Nasdeq.)
Explain the difference between these graphs in as ways as you can. Be sure to mention
average rates of change.
45. Rising cost of collage. Like most things, the cost of a collage education has gone up over time. The graph below display the yearly costs of 4-year collages in 2010 dollars-indicating that the cost prior to 2010 have been adjusted for information.
(Source: National center Education Statistics, Annual of Educational, Statistics, 2012.)
a. In what school year(s) did the cost of a private 4-years appear to increases the most?
b. In what school year(s) the cost of a public 4-year collage appear to increases the most?
c. Assuming an annual inflation rate of 3% calculate the cost of a year at a public and at a private 4-year collage in 2000. Express the cost in 2000 dollars.
Find the simplified difference quotient for each function listed. f(x)=mx+bFind the simplified difference quotient for each function listed.
47.
Find the simplified difference quotient for each function listed.
48.
Find the simplified difference quotient for each function listed.
49.
Find the simplified difference quotient for each function listed.
50.
Find the simplified difference quotient for each function listed.
51.
Find the simplified difference quotient for each function listed. f(x)=1x2Find the simplified difference quotient for each function listed.
53.
54. Below are the steps in the simplification of the difference quotient for (see Example 8).
Provide a brief justification for each step.
a. a) Multiplying by 1
b. b) Expanding (multiplying) the numerator
c. c)
d. d)
e. e) Assuming
For Exercises 55 and 56, find the simplified different quotient. f(x)=2x+1For Exercises 55 and 56, find the simplified different quotient.
56.
In Exercises 1-16;
a. a) Graph the function.
b. b) Draw tangent lines to the graph at point whose x-coordinates are –2, 0, and 1.
c. c) Find by determining .
d. d) Find. These slopes should match those of the lines you drew in part (b).
1.
In Exercises 1-16;
a. a) Graph the function.
b. b) Draw tangent lines to the graph at point whose x-coordinates are –2, 0, and 1.
c. c) Find by determining .
d. d) Find. These slopes should match those of the lines you drew in part (b).
2.
In Exercises 1-16;
a. a) Graph the function.
b. b) Draw tangent lines to the graph at point whose x-coordinates are –2, 0, and 1.
c. c) Find by determining .
d. d) Find. These slopes should match those of the lines you drew in part (b).
3.
a.Graph the function. b.Draw tangent lines to the graph at point whose x-coordinates are 2, 0, and 1. c.Find f(x) by determining limx0f(x+h)f(x)h. d. d) Find f(2),f(0,)andf(1). These slopes should match those of the lines you drew in part (b). f(x)=3x2In Exercises 1-16;
a. a) Graph the function.
b. b) Draw tangent lines to the graph at point whose x-coordinates are –2, 0, and 1.
c. c) Find by determining .
d. d) Find. These slopes should match those of the lines you drew in part (b).
5.
In Exercises 1-16;
a. a) Graph the function.
b. b) Draw tangent lines to the graph at point whose x-coordinates are –2, 0, and 1.
c. c) Find by determining .
d. d) Find. These slopes should match those of the lines you drew in part (b).
6.
In Exercises 1-16;
a. a) Graph the function.
b. b) Draw tangent lines to the graph at point whose x-coordinates are –2, 0, and 1.
c. c) Find by determining .
d. d) Find. These slopes should match those of the lines you drew in part (b).
7.
a.Graph the function. b.Draw tangent lines to the graph at point whose x-coordinates are 2, 0, and 1. c.Find f(x) by determining limx0f(x+h)f(x)h. d. d) Find f(2),f(0,)andf(1). These slopes should match those of the lines you drew in part (b). f(x)=2x+5a.Graph the function. b.Draw tangent lines to the graph at point whose x-coordinates are 2, 0, and 1. c.Find f(x) by determining limx0f(x+h)f(x)h. d. d) Find f(2),f(0,)andf(1). These slopes should match those of the lines you drew in part (b). f(x)=34x2In Exercises 1-16;
a. a) Graph the function.
b. b) Draw tangent lines to the graph at point whose x-coordinates are –2, 0, and 1.
c. c) Find by determining .
d. d) Find. These slopes should match those of the lines you drew in part (b).
10.
a. a) Graph the function. b. b) Draw tangent lines to the graph at point whose x-coordinates are 2, 0, and 1. c. c) Find f(x) by determining limx0f(x+h)f(x)h. d. d) Find f(2),f(0,)andf(1). These slopes should match those of the lines you drew in part (b). f(x)=x2+xa. a) Graph the function. b. b) Draw tangent lines to the graph at point whose x-coordinates are 2, 0, and 1. c. c) Find f(x) by determining limx0f(x+h)f(x)h. d. d) Find f(2),f(0,)andf(1). These slopes should match those of the lines you drew in part (b). f(x)=x2x