1. A curve has equation .
(a) Write an expression for the slope of the secant line
through the points and .
(b) Write an expression for the slope of the tangent line at P.
; 2. Graph the curve in the viewing rectangles
by , by , and by
y苷 f 共x兲
P共3, f 共3兲兲 Q共x, f 共x兲兲
y苷 sin x 关⫺2, 2兴
关⫺2, 2兴 关⫺1, 1兴 关⫺1, 1兴 关⫺0.5, 0.5兴
. What do you notice about the curve as you zoom in toward the origin?
3. (a) Find the slope of the tangent line to the parabola at the point
( i ) using Definition 1 ( ii ) using Equation 2 (b) Find an equation of the tangent line in part (a).
共1, 3兲 y苷 4x ⫺ x2
关⫺0.5, 0.5兴
2.1 Exercises
; Graphing calculator or computer required 1.Homework Hints available at stewartcalculus.com 110 CHAPTER 2 DERIVATIVES
In the following example we estimate the rate of change of the national debt with respect to time. Here the function is defined not by a formula but by a table of values.
Let be the Canadian gross public debt at time t. The table in the margin gives approximate values of this function by providing midyear estimates, in billions of dollars, from 1994 to 2002. Interpret and estimate the value of .
SOLUTION
The derivative means the rate of change of D with respect to t when , that is, the rate of increase of the national debt in 1998.
According to Equation 5,
So we compute and tabulate values of the difference quotient (the average rates of change) as shown in the table at the left. From this table we see that lies some- where between and billion dollars per year. [Here we are making the reason- able assumption that the debt didn’t fluctuate wildly between 1998 and 2002.] We estimate that the rate of change of the Canadian debt in 1998 was the average of these two numbers, namely
The minus sign means that the debt was decreasing at that time.
Another method would be to plot the debt function and estimate the slope of the tangent line when .
In Examples 3, 6, and 7 we saw three specific examples of rates of change: the veloc- ity of an object is the rate of change of displacement with respect to time; marginal cost is the rate of change of production cost with respect to the number of items produced; the rate of change of the debt with respect to time is of interest in economics. Here is a small sample of other rates of change: In physics, the rate of change of work with respect to time is called power. Chemists who study a chemical reaction are interested in the rate of change in the concentration of a reactant with respect to time (called the rate of reaction).
A biologist is interested in the rate of change of the population of a colony of bacteria with respect to time. In fact, the computation of rates of change is important in all of the natu- ral sciences, in engineering, and even in the social sciences. Further examples will be given in Section 2.7.
All these rates of change are derivatives and can therefore be interpreted as slopes of tangents. This gives added significance to the solution of the tangent problem. Whenever we solve a problem involving tangent lines, we are not just solving a problem in geome- try. We are also implicitly solving a great variety of problems involving rates of change in science and engineering.
EXAMPLE 7 D
共t兲
D
⬘共1998兲
D⬘共1998兲
t
苷 1998
D
⬘共1998兲 苷 lim
t l1998 D共t兲 ⫺ D共1998兲
t⫺ 1998
D
⬘共1998兲
⫺1.1 ⫺5.5
D
⬘共1998兲 ⬇ ⫺3.3 billion dollars per year
t
苷 1998
A Note on Units
The units for the average rate of change are the units for divided by the units for , namely, billions of dollars per year. The instan- taneous rate of change is the limit of the aver- age rates of change, so it is measured in the same units: billions of dollars per year.
⌬t
⌬D ⌬D兾⌬t
t
1994 414.0
1996 469.5
1998 467.3
2000 456.4
2002 442.3
D共t兲
t
1994 13.3
1996 2000
2002 ⫺6.3⫺5.5⫺1.1 D共t兲 ⫺ D共1998兲
t⫺ 1998
98845_02_ch02_p103-111.qk_98845_02_ch02_p103-111 8/16/11 9:52 AM Page 110
SECTION 2.1 DERIVATIVES AND RATES OF CHANGE 111
; (c) Graph the parabola and the tangent line. As a check on your work, zoom in toward the point until the parabola and the tangent line are indistinguishable.
4. (a) Find the slope of the tangent line to the curve at the point
( i ) using Definition 1 ( ii ) using Equation 2 (b) Find an equation of the tangent line in part (a).
; (c) Graph the curve and the tangent line in successively smaller viewing rectangles centered at until the curve and the line appear to coincide.
5–8 Find an equation of the tangent line to the curve at the given point.
5. , 6. ,
7. 8. ,
9. (a) Find the slope of the tangent to the curve at the point where . (b) Find equations of the tangent lines at the points
and .
; (c) Graph the curve and both tangents on a common screen.
10. (a) Find the slope of the tangent to the curve at the point where .
(b) Find equations of the tangent lines at the points and .
; (c) Graph the curve and both tangents on a common screen.
11. (a) A particle starts by moving to the right along a horizontal line; the graph of its position function is shown. When is the particle moving to the right? Moving to the left?
Standing still?
(b) Draw a graph of the velocity function.
12. Shown are graphs of the position functions of two runners, and , who run a 100-m race and finish in a tie.
(a) Describe and compare how the runners run the race.
y苷 1兾sx x苷 a
共1, 1兲
(
4, 12)
s (meters)
0 2 4 6
4 2
t (seconds)
A B
s (meters)
0 4 8 12
80 40
t (seconds) A
B
共1, 3兲
y苷 x ⫺ x3 共1, 0兲
共1, 0兲
共2, 3兲 y苷 x3⫺ 3x ⫹ 1 共2, ⫺4兲
y苷 4x ⫺ 3x2
共1, 1兲 y苷 2x⫹ 1
x⫹ 2
(
1, 1兲 y苷 sx ,x苷 a y苷 3 ⫹ 4x2⫺ 2x3
共1, 5兲 共2, 3兲
(b) At what time is the distance between the runners the greatest?
(c) At what time do they have the same velocity?
13. If a ball is thrown into the air with a velocity of 10 m兾s, its height ( in meters) after seconds is given by
. Find the velocity when .
14. If a rock is thrown upward on the planet Mars with a velocity of , its height ( in meters) after seconds is given by
.
(a) Find the velocity of the rock after one second.
(b) Find the velocity of the rock when . (c) When will the rock hit the surface?
(d) With what velocity will the rock hit the surface?
15. The displacement ( in meters) of a particle moving in a straight line is given by the equation of motion , where is measured in seconds. Find the velocity of the par -
ticle at times , and .
16. The displacement ( in meters) of a particle moving in a straight line is given by , where is mea- sured in seconds.
(a) Find the average velocity over each time interval:
( i ) ( ii )
( iii ) ( iv)
(b) Find the instantaneous velocity when .
(c) Draw the graph of as a function of and draw the secant lines whose slopes are the average velocities in part (a) and the tangent line whose slope is the instantaneous velocity in part (b).
17. For the function t whose graph is given, arrange the follow- ing numbers in increasing order and explain your reasoning:
18. Find an equation of the tangent line to the graph of
at if and .
19. If an equation of the tangent line to the curve at the
point where is , find and .
20. If the tangent line to at (4, 3) passes through the point (0, 2), find and .
10 m兾s t
H苷 10t ⫺ 1.86t2
t苷 a
s苷 1兾t2 t
t苷 a, t 苷 1, t 苷 2 t苷 3
s苷 t2⫺ 8t ⫹ 18 t
关3, 4兴 关3.5, 4兴
关4, 5兴 关4, 4.5兴
t苷 4
s t
0 t⬘共⫺2兲 t⬘共0兲 t⬘共2兲 t⬘共4兲
y=©
1 3 4
_1 0 2 x
y
y苷 t共x兲 x苷 5 t共5兲 苷 ⫺3 t⬘共5兲 苷 4
y苷 f 共x兲 a苷 2 y 苷 4x ⫺ 5 f共2兲 f⬘共2兲
y苷 f 共x兲 f共4兲 f⬘共4兲
t
y苷 10t ⫺ 4.9t2 t苷 2
98845_02_ch02_p103-111.qk_98845_02_ch02_p103-111 8/16/11 9:52 AM Page 111
112 CHAPTER 2 DERIVATIVES
21. Sketch the graph of a function for which ,
, and .
22. Sketch the graph of a function for which
, , ,
, , and .
23. If , find and use it to find an equation of the tangent line to the curve at the point . 24. If , find and use it to find an equation of
the tangent line to the curve at the point . 25. (a) If , find and use it to find an
equation of the tangent line to the curve at the point .
; (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.
26. (a) If , find and use it to find equations of the tangent lines to the curve at the points and .
; (b) Illustrate part (a) by graphing the curve and the tangent lines on the same screen.
27–32 Find .
27. 28.
29. 30.
31. 32.
33–38 Each limit represents the derivative of some function at some number . State such an and in each case.
33. 34.
35. 36.
37. 38.
39–40 A particle moves along a straight line with equation of motion , where is measured in meters and in seconds.
Find the velocity and the speed when .
39. 40.
41. A warm can of soda is placed in a cold refrigerator. Sketch the graph of the temperature of the soda as a function of time. Is the initial rate of change of temperature greater or less than the rate of change after an hour?
42. A roast turkey is taken from an oven when its temperature has reached 85°C and is placed on a table in a room where the temperature is 24°C. The graph shows how the temperature
f共0兲 苷 0 f
f⬘共2兲 苷 ⫺1 f⬘共1兲 苷 0
f⬘共0兲 苷 3,
t
t⬘共1兲 苷 t⬘共3兲 苷 0 t共0兲 苷 t共2兲 苷 t共4兲 苷 0
limx l 5⫺t共x兲 苷 ⬁ t⬘共2兲 苷 ⫺1
t⬘共0兲 苷 t⬘共4兲 苷 1 limx l⫺1⫹t共x兲 苷 ⫺⬁
f⬘共1兲 f共x兲 苷 3x2⫺ x3
共1, 2兲 y苷 3x2⫺ x3
t⬘共1兲 t共x兲 苷 x4⫺ 2
共1, ⫺1兲 y苷 x4⫺ 2
F⬘共2兲 F共x兲 苷 5x兾共1 ⫹ x2兲
y苷 5x兾共1 ⫹ x2兲 共2, 2兲
G⬘共a兲 G共x兲 苷 4x2⫺ x3
y苷 4x2⫺ x3 共3, 9兲
共2, 8兲
f⬘共a兲
f共t兲 苷 2t3⫹ t f共x兲 苷 3x2⫺ 4x ⫹ 1
f共x兲 苷 x⫺2 f共t兲 苷 2t⫹ 1
t⫹ 3
f共x兲 苷 4 s1⫺ x f共x兲 苷 s1 ⫺ 2x
f a
f a
lim
h l0
s416⫹ h ⫺ 2 lim h
h l0
共1 ⫹ h兲10⫺ 1 h
x llim兾4
tan x⫺ 1 x⫺兾4 limx l5
2x⫺ 32 x⫺ 5
lim
t l1
t4⫹ t ⫺ 2 t⫺ 1 lim
h l0
cos共 ⫹ h兲 ⫹ 1 h
t s
s苷 f 共t兲
t苷 5
f共t兲 苷 100 ⫹ 50t ⫺ 4.9t2 f共t兲 苷 t⫺1⫺ t
of the turkey decreases and eventually approaches room tem- perature. By measuring the slope of the tangent, estimate the rate of change of the temperature after an hour.
43. The table shows the number of passengers that arrived in Ireland by air, in millions.
(a) Find the average rate of increase of
( i ) from 2001 to 2005 ( ii ) from 2003 to 2005 ( iii ) from 2005 to 2007
In each case, include the units.
(b) Estimate the instantaneous rate of growth in 2005 by taking the average of two average rates of change. What are its units?
44. The number of locations of a popular coffeehouse chain is given in the table. (The numbers of locations as of October 1 are given.)
(a) Find the average rate of growth
( i ) from 2006 to 2008 ( ii ) from 2006 to 2007 ( iii ) from 2005 to 2006
In each case, include the units.
(b) Estimate the instantaneous rate of growth in 2006 by taking the average of two average rates of change. What are its units?
(c) Estimate the instantaneous rate of growth in 2006 by mea- suring the slope of a tangent.
(d) Estimate the intantaneous rate of growth in 2007 and com- pare it with the growth rate in 2006. What do you conclude?
45. The cost ( in dollars) of producing units of a certain com-
modity is .
(a) Find the average rate of change of with respect to when the production level is changed
( i ) from to ( ii ) from to
(b) Find the instantaneous rate of change of with respect to when . (This is called the marginal cost. Its sig- nificance will be explained in Section 2.7.)
T (°C)
0 30 60 90 120 150
40 80
t (min) P
N
x C共x兲 苷 5000 ⫹ 10x ⫹ 0.05x2
x C
x苷 105 x苷 100
x苷 101 x苷 100
C x苷 100
x
P
P
Year 2004 2005 2006 2007 2008
N 8569 10,241 12,440 15,011 16,680
Year 2001 2003 2005 2007 2009
P 8.49 9.65 11.78 14.54 12.84
98845_02_ch02_p112-121.qk_98845_02_ch02_p112-121 8/16/11 9:53 AM Page 112
SECTION 2.1 DERIVATIVES AND RATES OF CHANGE 113 46. If a cylindrical tank holds 100,000 liters of water, which can be
drained from the bottom of the tank in an hour, then Torricelli’s Law gives the volume of water remaining in the tank after minutes as
Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of with respect to ) as a func- tion of t. What are its units? For times t苷 0, 10, 20, 30, 40, 50, and 60 min, find the flow rate and the amount of water remain- ing in the tank. Summarize your findings in a sentence or two.
At what time is the flow rate the greatest? The least?
47. The cost of producing x kilograms of gold from a new gold mine is dollars.
(a) What is the meaning of the derivative ? What are its units?
(b) What does the statement mean?
(c) Do you think the values of will increase or decrease in the short term? What about the long term? Explain.
48. The number of bacteria after t hours in a controlled laboratory experiment is .
(a) What is the meaning of the derivative ? What are its units?
(b) Suppose there is an unlimited amount of space and nutrients for the bacteria. Which do you think is larger,
or ? If the supply of nutrients is limited, would that affect your conclusion? Explain.
49. Let be the temperature (in ) in Manila hours after noon on July 19, 2011. The table shows values of this function recorded every two hours. What is the meaning of ? Estimate its value.
50. The quantity (in kilograms) of a gourmet ground coffee that is sold by a coffee company at a price of p dollars per kilogram is .
(a) What is the meaning of the derivative ? What are its units?
(b) Is positive or negative? Explain.
51. The quantity of oxygen that can dissolve in water depends on the temperature of the water. (So thermal pollution influences
n苷 f 共t兲
f⬘共5兲
f⬘共10兲 f⬘共5兲
Q苷 f 共 p兲
f⬘共8兲 f⬘共8兲
V t
C苷 f 共x兲
f⬘共x兲 f⬘共50兲 苷 36
f⬘共x兲
V t
0艋 t 艋 60 V共t兲 苷 100,000
(
1⫺601t)
2T共t兲 ⬚C t
T⬘共5兲
the oxygen content of water.) The graph shows how oxygen solubility varies as a function of the water temperature . (a) What is the meaning of the derivative ? What are its
units?
(b) Estimate the value of and interpret it.
52. The graph shows the influence of the temperature on the maximum sustainable swimming speed of Coho salmon.
(a) What is the meaning of the derivative ? What are its units?
(b) Estimate the values of and and interpret them.
53–54 Determine whether exists.
53.
54.
T S
S⬘共T兲 S⬘共16兲
(mg/ L)
4 8 12 16 S
0 T (°C)
Adapted from Kupchella & Hyland, Environmental Science: Living Within the System of Nature, 2d ed.; © 1989. Printed and electronically reproduced by permission of Pearson Education, Inc., Upper Saddle River, NJ.
8 16 24 32 40
T S
S⬘共T兲 S⬘共25兲 S⬘共15兲
0 10 20 T (°C)
S (cm/s) 20
f⬘共0兲
f共x兲 苷
再
x sin 0 1x if xif x苷 0苷 0f共x兲 苷
再
x02 sin 1x if xif x苷 0苷 0t 1 3 5 7 9 11
T 32 32 31 27 26 25
98845_02_ch02_p112-121.qk_98845_02_ch02_p112-121 8/16/11 9:53 AM Page 113
122 CHAPTER 2 DERIVATIVES
1–2 Use the given graph to estimate the value of each derivative.
Then sketch the graph of .
1. (a) (b) (c)
(d) (e) (f)
(g)
2. (a) (b) (c)
(d) (e) (f)
(g) (h)
f⬘
y
x 1
1
y
0 x
1 1
f⬘共⫺3兲 f⬘共⫺2兲 f⬘共⫺1兲
f⬘共0兲 f⬘共1兲 f⬘共2兲
f⬘共3兲
f⬘共0兲 f⬘共1兲 f⬘共2兲
f⬘共3兲 f⬘共4兲 f⬘共5兲
f⬘共6兲 f⬘共7兲
3. Match the graph of each function in (a)–(d) with the graph of its derivative in I–IV. Give reasons for your choices
.
y
0 y
0
y
0
y
0
x x
x x
(b) (a)
(c) (d)
II I
III IV
y
0
y
0
y
0
x
x y
0 x
x
2.2 Exercises
the slope of , we have
for all values of . So is a constant function and its graph is a horizontal line. There- fore, for all values of ,
We can also interpret the third derivative physically in the case where the function is the position function of an object that moves along a straight line. Because
, the third derivative of the position function is the derivative of the accel- eration function and is called the jerk:
Thus the jerk j is the rate of change of acceleration. It is aptly named because a large jerk means a sudden change in acceleration, which causes an abrupt movement in a vehicle.
We have seen that one application of second and third derivatives occurs in analyzing the motion of objects using acceleration and jerk. We will investigate another applica- tion of second derivatives in Section 3.3, where we show how knowledge of gives us information about the shape of the graph of . In Chapter 11 we will see how second and higher derivatives enable us to represent functions as sums of infinite series.
f
⬙共x兲
f
共x兲 苷 6
x f
x
f共4兲
共x兲 苷 0
s
苷 s共t兲
s 苷 共s⬙兲⬘ 苷 a⬘
j
苷
da dt苷
d3sdt3
f
⬙
f; Graphing calculator or computer required 1.Homework Hints available at stewartcalculus.com 98845_02_ch02_p122-131.qk_98845_02_ch02_p122-131 8/16/11 9:53 AM Page 122
SECTION 2.2 THE DERIVATIVE AS A FUNCTION 123 4–11 Trace or copy the graph of the given function . (Assume that
the axes have equal scales.) Then use the method of Example 1 to sketch the graph of below it.
4.
5. 6.
7. 8.
9. 10.
11.
12. Shown is the graph of the population function for yeast cells in a laboratory culture. Use the method of Example 1 to graph the derivative . What does the graph of tell us about the yeast population?
f f⬘
0 x
y
x y
0 0 x
y
x y
0 0 x
y
0 x
y
x y
0
0 x
y
P共t兲
(yeast cells)
t (hours) P
0 5 10 15
500
P⬘共t兲 P⬘
13. A rechargeable battery is plugged into a charger. The graph shows , the percentage of full capacity that the battery reaches as a function of time elapsed ( in hours).
(a) What is the meaning of the derivative ?
(b) Sketch the graph of . What does the graph tell you?
14. The graph (from the US Department of Energy) shows how driving speed affects gas mileage. Fuel economy is measured in miles per gallon and speed is measured in miles per hour.
(a) What is the meaning of the derivative ? (b) Sketch the graph of .
(c) At what speed should you drive if you want to save on gas?
15. The graph shows how the average age of first marriage of Japanese men varied in the last half of the 20th century.
Sketch the graph of the derivative function . During which years was the derivative negative?
16. Make a careful sketch of the graph of the sine function and below it sketch the graph of its derivative in the same manner as in Exercises 4 –11. Can you guess what the derivative of the sine function is from its graph?
C共t兲
t
C⬘共t兲 C⬘共t兲
t (hours)
2 4 6 8 10 12
20 40 60 80 100
percentage of full charge
C
F v
F⬘共v兲 F⬘共v兲
√ (mi/ h) 0
10 30 20
70 60 40
20 30 50
10 F (mi/ gal)
M⬘共t兲
1990 2000 25
M
1960 1970 1980 27
t 98845_02_ch02_p122-131.qk_98845_02_ch02_p122-131 8/16/11 9:53 AM Page 123
124 CHAPTER 2 DERIVATIVES
;17. Let .
(a) Estimate the values of , , , and by using a graphing device to zoom in on the graph of . (b) Use symmetry to deduce the values of , ,
and .
(c) Use the results from parts (a) and (b) to guess a formula for .
(d) Use the definition of derivative to prove that your guess in part (c) is correct.
;18. Let .
(a) Estimate the values of , , , , and by using a graphing device to zoom in on the graph of f.
(b) Use symmetry to deduce the values of , , , and .
(c) Use the values from parts (a) and (b) to graph . (d) Guess a formula for .
(e) Use the definition of derivative to prove that your guess in part (d) is correct.
19–29 Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.
19. 20.
21. 22.
23. 24.
25. 26.
27. 28.
29.
30. (a) Sketch the graph of by starting with the graph of and using the transformations of Sec - tion 1.3.
(b) Use the graph from part (a) to sketch the graph of . (c) Use the definition of a derivative to find . What are
the domains of f and ?
; (d) Use a graphing device to graph and compare with your sketch in part (b).
31. (a) If , find .
; (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of and .
32. (a) If , find .
; (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of and .
f共x兲 苷 x2
f⬘共2兲 f⬘共1兲
f⬘
(
12)
f⬘共0兲
f f⬘共⫺1兲 f⬘
(
⫺12)
f⬘共⫺2兲 f⬘共x兲
f共x兲 苷 x3
f⬘共3兲 f⬘共2兲
f⬘共1兲 f⬘
(
12)
f⬘共0兲
f⬘共⫺1兲 f⬘
(
⫺12)
f⬘共⫺3兲 f⬘共⫺2兲
f⬘ f⬘共x兲
f共x兲 苷 mx ⫹ b f共x兲 苷12x⫺13
f共x兲 苷 1.5x2⫺ x ⫹ 3.7 f共t兲 苷 5t ⫺ 9t2
f共x兲 苷 x2⫺ 1 2x⫺ 3 t共x兲 苷 s9 ⫺ x
f共x兲 苷 x3兾2 G共t兲 苷 1⫺ 2t
3⫹ t f共x兲 苷 x4
f共x兲 苷 s6 ⫺ x y苷 sx
f⬘ f⬘共x兲 f⬘
f⬘
f⬘共x兲 f共x兲 苷 x4⫹ 2x
f⬘ f
f⬘共x兲 f共x兲 苷 x ⫹ 1兾x
f⬘ f
f共x兲 苷 x3⫺ 3x ⫹ 5 f共x兲 苷 x ⫹ sx
33. The unemployment rate varies with time. The table gives the percentage of unemployed in the Australian labor force measured at midyear from 1995 to 2004.
(a) What is the meaning of ? What are its units?
(b) Construct a table of estimated values for .
34. Let be the percentage of the population of the Philippines over the age of 60 at time . The table gives projections of values of this function from 1995 to 2020.
(a) What is the meaning of ? What are its units?
(b) Construct a table of estimated values for . (c) Graph and .
35–38 The graph of is given. State, with reasons, the numbers at which is not differentiable.
35. 36.
37. 38.
;39. Graph the function . Zoom in repeatedly, first toward the point (⫺1, 0) and then toward the origin.
What is different about the behavior of in the vicinity of these two points? What do you conclude about the differen- tiability of f ?
;40. Zoom in toward the points (1, 0), (0, 1), and (⫺1, 0) on the graph of the function . What do you notice? Account for what you see in terms of the differen- tiability of t.
U⬘共t兲
U⬘共t兲
P⬘共t兲
P⬘共t兲 P⬘
P f f
2 4 x
y
0
_2 2 x
y
0
_2 2 x
y
_2 4 x 0
y
0
f共x兲 苷 x ⫹ s
ⱍ
xⱍ
f
t共x兲 苷 共x2⫺ 1兲2兾3 U共t兲
P共t兲
t
t t
1995 8.1 2000 6.2
1996 8.0 2001 6.9
1997 8.2 2002 6.5
1998 7.9 2003 6.2
1999 6.7 2004 5.6
U共t兲 U共t兲
t t
1995 5.2 2010 6.7
2000 5.5 2015 7.7
2005 6.1 2020 8.9
P共t兲 P共t兲
98845_02_ch02_p122-131.qk_98845_02_ch02_p122-131 8/16/11 9:53 AM Page 124
SECTION 2.2 THE DERIVATIVE AS A FUNCTION 125 41. The figure shows the graphs of , , and . Identify each
curve, and explain your choices.
42. The figure shows graphs of , , and . Identify each curve, and explain your choices.
43. The figure shows the graphs of three functions. One is the posi- tion function of a car, one is the velocity of the car, and one is its acceleration. Identify each curve, and explain your choices.
44. The figure shows the graphs of four functions. One is the position function of a car, one is the velocity of the car, one is its acceleration, and one is its jerk. Identify each curve, and explain your choices.
f⬙ f⬘ f
x
y a
b
c
f f⬙ f⬘ f,
x
y a b c d
t
y a
b c
0
0 t
y a
b c
d
;45–46 Use the definition of a derivative to find and . Then graph , , and on a common screen and check to see if your answers are reasonable.
45.
46.
;47. If , find , , , and .
Graph , , , and on a common screen. Are the graphs consistent with the geometric interpretations of these derivatives?
48. (a) The graph of a position function of a car is shown, where s is measured in meters and t in seconds. Use it to graph the velocity and acceleration of the car. What is the acceleration at seconds?
(b) Use the acceleration curve from part (a) to estimate the jerk at seconds. What are the units for jerk?
49. Let .
(a) If , use Equation 2.1.5 to find . (b) Show that does not exist.
(c) Show that has a vertical tangent line at . (Recall the shape of the graph of . See Figure 13 in Section 1.2.)
50. (a) If , show that does not exist.
(b) If , find .
(c) Show that has a vertical tangent line at .
; (d) Illustrate part (c) by graphing .
51. Show that the function is not differentiable at 6. Find a formula for and sketch its graph.
52. Where is the greatest integer function not differen- tiable? Find a formula for and sketch its graph.
53. (a) Sketch the graph of the function . (b) For what values of is differentiable?
(c) Find a formula for .
f⬙共x兲 f⬘共x兲
f⬙ f⬘ f
f共x兲 苷 3x2⫹ 2x ⫹ 1 f共x兲 苷 x3⫺ 3x
f共4兲共x兲 f共x兲
f⬙共x兲 f⬘共x兲 f共x兲 苷 2x2⫺ x3
f f⬙ f⬘ f
0 10 t
s
100
20
t苷 10
f共x兲 苷 s3x
f⬘共a兲 a苷 0
f⬘共0兲
共0, 0兲 y苷 s3x
f
t⬘共0兲 t共x兲 苷 x2兾3
t⬘共a兲 a苷 0
共0, 0兲 y苷 x2兾3
y苷 x2兾3
f共x兲 苷
ⱍ
x⫺ 6ⱍ
f⬘
f共x兲 苷 冀 x冁 f⬘
f共x兲 苷 x
ⱍ
xⱍ
f x f⬘ t苷 10
98845_02_ch02_p122-131.qk_98845_02_ch02_p122-131 8/16/11 9:53 AM Page 125
126 CHAPTER 2 DERIVATIVES
54. The left-hand and right-hand derivatives of at are defined by
and
if these limits exist. Then exists if and only if these one- sided derivatives exist and are equal.
(a) Find and for the function
(b) Sketch the graph of . f共x兲 苷
0 5⫺ x
if x艋 0 if 0⬍ x ⬍ 4 1
5⫺ x if x艌 4
f a
f⬘⫺共a兲 苷 limh l0⫺ f共a ⫹ h兲 ⫺ f 共a兲 h f⬘⫹共a兲 苷 limh l0⫹ f共a ⫹ h兲 ⫺ f 共a兲
h f⬘共a兲 f⬘⫹共4兲 f⬘⫺共4兲
f
(c) Where is discontinuous?
(d) Where is not differentiable?
55. Recall that a function is called even if for all in its domain and odd if for all such . Prove
each of the following.
(a) The derivative of an even function is an odd function.
(b) The derivative of an odd function is an even function.
56. When you turn on a hot-water faucet, the temperature of the water depends on how long the water has been running.
(a) Sketch a possible graph of as a function of the time that has elapsed since the faucet was turned on.
(b) Describe how the rate of change of with respect to varies as increases.
(c) Sketch a graph of the derivative of .
57. Let be the tangent line to the parabola at the point . The angle of inclination of is the angle that makes with the positive direction of the -axis. Calculate correct to the nearest degree.
ᐍ y苷 x2
共1, 1兲 ᐍ ᐍ
x
T t T
t T
t
T f
f共⫺x兲 苷 f 共x兲 f
x f共⫺x兲 苷 ⫺f 共x兲
x
f
If it were always necessary to compute derivatives directly from the definition, as we did in the preceding section, such computations would be tedious and the evaluation of some limits would require ingenuity. Fortunately, several rules have been developed for finding derivatives without having to use the definition directly. These formulas greatly simplify the task of differentiation.
Let’s start with the simplest of all functions, the constant function . The graph of this function is the horizontal line y 苷 c, which has slope 0, so we must have . (See Figure 1.) A formal proof, from the definition of a derivative, is also easy:
In Leibniz notation, we write this rule as follows.
Derivative of a Constant Function
Power Functions
We next look at the functions , where n is a positive integer. If , the graph of is the line y 苷 x, which has slope 1. (See Figure 2.) So
f
共x兲 苷 c
f
⬘共x兲 苷 0
f
⬘共x兲 苷 lim
h l 0
f
共x ⫹ h兲 ⫺ f 共x兲
h
苷 lim
h l 0
c
⫺ c
h苷 lim
h l 0
0 苷 0
d
dx
共c兲 苷 0
f
共x兲 苷 x
n n苷 1
f
共x兲 苷 x
d
dx
共x兲 苷 1
12.3 Differentiation Formulas
FIGURE 1
The graph of ƒ=c is the line y=c, so fª(x)=0.
y
c
0 x
y=c slope=0
y
0
x y=x
slope=1
FIGURE 2
The graph of ƒ=x is the line y=x, so fª(x)=1.
98845_02_ch02_p122-131.qk_98845_02_ch02_p122-131 8/16/11 9:53 AM Page 126
Let the -coordinate of one of the points in question be . Then the slope of the tangent line at that point is . This tangent line will be parallel to the line , or
, if it has the same slope, that is, . Equating slopes, we get
Therefore the required points are and . The hyperbola and the tangents are shown in Figure 6.
We summarize the differentiation formulas we have learned so far as follows.
Table of Differentiation Formulas
共2, 6兲 共⫺2, ⫺6兲
x a
⫺12兾a
23x ⫹ y 苷 0
y
苷 ⫺3x ⫺3
⫺ 12
a2
苷 ⫺3 or
a2苷 4 or
a苷 ⫾2
d
dx
共c兲 苷 0
ddx
共x
n兲 苷 nx
n⫺1共cf 兲⬘ 苷 cf⬘ 共 f ⫹ t兲⬘ 苷 f⬘⫹ t⬘ 共 f ⫺ t兲⬘ 苷 f⬘⫺ t⬘
共 ft兲⬘ 苷 ft⬘ ⫹ tf⬘ 冉 t
f冊 ⬘ 苷 t
f⬘ ⫺ ft⬘ t
2136 CHAPTER 2 DERIVATIVES
FIGURE 6
3x+y=0 xy=12
0 x
y
(_2, _6)
(2, 6)
1–22 Differentiate the function.
1. 2.
3. 4.
5. 6.
7. 8.
9. 10.
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.
21. 22.
f共x兲 苷 x3⫺ 4x ⫹ 6 f共t兲 苷12t6⫺ 3t4⫹ t t共x兲 苷 x2共1 ⫺ 2x兲 h共x兲 苷 共x ⫺ 2兲共2x ⫹ 3兲
B共y兲 苷 cy⫺6 A共s兲 苷 ⫺12
s5 y苷 x5兾3⫺ x2兾3
S共 p兲 苷 sp ⫺ p y苷 sx 共x ⫺ 1兲
R共a兲 苷 共3a ⫹ 1兲2 S共R兲 苷 4R2
y苷 x2⫹ 4x ⫹ 3
sx y苷 sx ⫹ x
x2
H共x兲 苷 共x ⫹ x⫺1兲3 t共u兲 苷 s2 u ⫹ s3u
u苷 s5t ⫹ 4st5 v苷冉sx ⫹ s31x冊2
f共x兲 苷 186.5 f共x兲 苷 s30
f共x兲 苷 5x ⫺ 1 F共x兲 苷 ⫺4x10
y苷 x⫺2兾5
23. Find the derivative of in two ways:
by using the Product Rule and by performing the multiplication first. Do your answers agree?
24. Find the derivative of the function
in two ways: by using the Quotient Rule and by simplifying first. Show that your answers are equivalent. Which method do you prefer?
25–44 Differentiate.
25.
26.
27.
28.
29. 30.
V共x兲 苷 共2x3⫹ 3兲共x4⫺ 2x兲 L共x兲 苷 共1 ⫹ x ⫹ x2兲共2 ⫺ x4兲
F共y兲 苷冉y12 ⫺ y34冊共y ⫹ 5y3兲
J共v兲 苷 共v3⫺ 2v兲共v⫺4⫹v⫺2兲
F共x兲 苷 x4⫺ 5x3⫹ sx x2
f共x兲 苷 共1 ⫹ 2x2兲共x ⫺ x2兲
t共x兲 苷 3x⫺ 1
2x⫹ 1 f共t兲 苷 2t
4⫹ t2
2.3 Exercises
; Graphing calculator or computer required 1.Homework Hints available at stewartcalculus.com 98845_02_ch02_p132-141.qk_98845_02_ch02_p132-141 8/16/11 9:54 AM Page 136
SECTION 2.3 DIFFERENTIATION FORMULAS 137
31. 32.
33. 34.
35. 36.
37. 38.
39. 40.
41. 42.
43. 44.
45. The general polynomial of degree has the form
where . Find the derivative of
;46–48 Find . Compare the graphs of and and use them to explain why your answer is reasonable.
46.
47. 48.
;49. (a) Use a graphing calculator or computer to graph the func-
tion in the viewing
rectangle by .
(b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of . (See Example 1 in Section 2.2.)
(c) Calculate and use this expression, with a graphing device, to graph . Compare with your sketch in part (b).
;50. (a) Use a graphing calculator or computer to graph the func- tion in the viewing rectangle by .
(b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of . (See Example 1 in Section 2.2.)
(c) Calculate and use this expression, with a graphing device, to graph . Compare with your sketch in part (b).
y苷 v
3⫺ 2vsv
v y苷 t
共t ⫺ 1兲2 y苷 x3
1⫺ x2 y苷 x⫹ 1
x3⫹ x ⫺ 2
t共t兲 苷 t⫺ st t1兾3 y苷 t2⫹ 2
t4⫺ 3t2⫹ 1
y苷 A ⫹ B x ⫹ C
x2 y苷 ax2⫹ bx ⫹ c
y苷 cx 1⫹ cx f共t兲 苷 2t
2⫹ st
y苷 u6⫺ 2u3⫹ 5 u2 y苷 s3t共t2⫹ t ⫹ t⫺1兲
f共x兲 苷 ax⫹ b cx⫹ d f共x兲 苷 x
x⫹ c x
n
P共x兲 苷 anxn⫹ an⫺1xn⫺1⫹ ⭈ ⭈ ⭈ ⫹ a2x2⫹ a1x⫹ a0 P.
an苷 0
f⬘ f f⬘共x兲
f共x兲 苷 x兾共x2⫺ 1兲
f共x兲 苷 x ⫹ 1 f共x兲 苷 3x15⫺ 5x3⫹ 3 x
f共x兲 苷 x4⫺ 3x3⫺ 6x2⫹ 7x ⫹ 30 关⫺10, 50兴
关⫺3, 5兴
f⬘ f⬘共x兲
f⬘
关⫺4, 4兴 t共x兲 苷 x2兾共x2⫹ 1兲
关⫺1, 1.5兴
t⬘ t⬘共x兲
t⬘
51–52 Find an equation of the tangent line to the curve at the given point.
51. ,
52. ,
53. (a) The curve is called a witch of Maria Agnesi. Find an equation of the tangent line to this curve at the point .
; (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.
54. (a) The curve is called a serpentine.
Find an equation of the tangent line to this curve at the point .
; (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.
55–58 Find equations of the tangent line and normal line to the curve at the given point.
55. , 56. ,
57. , 58. ,
59–62 Find the first and second derivatives of the function.
59. 60.
61. 62.
63. The equation of motion of a particle is , where is in meters and is in seconds. Find
(a) the velocity and acceleration as functions of , (b) the acceleration after 2 s, and
(c) the acceleration when the velocity is 0.
64. The equation of motion of a particle is
where is in meters and is in seconds.
(a) Find the velocity and acceleration as functions of . (b) Find the acceleration after 1 s.
; (c) Graph the position, velocity, and acceleration functions on the same screen.
共1, 2兲 y苷 x4⫹ 2x2⫺ x
共1, 1兲 y苷 2x
x⫹ 1
y苷 1兾共1 ⫹ x2兲
(
⫺1, 12)
y苷 x兾共1 ⫹ x2兲 共3, 0.3兲
共1, 9兲 y苷 共1 ⫹ 2x兲2 共1, 2兲
y苷 x ⫹ sx
共4, 0.4兲 y苷 sx
x⫹ 1 共1, 2兲
y苷 3x⫹ 1 x2⫹ 1
G共r兲 苷 sr ⫹ s3r f共x兲 苷 x4⫺ 3x3⫹ 16x
f共x兲 苷 1 3⫺ x f共x兲 苷 x2
1⫹ 2x
s s苷 t3⫺ 3t t
t
t s
s苷 t4⫺ 2t3⫹ t2⫺ t
t 98845_02_ch02_p132-141.qk_98845_02_ch02_p132-141 8/16/11 9:54 AM Page 137
138 CHAPTER 2 DERIVATIVES
65. Boyle’s Law states that when a sample of gas is compressed at a constant pressure, the pressure of the gas is inversely proportional to the volume of the gas.
(a) Suppose that the pressure of a sample of air that occupies at is . Write as a function of . (b) Calculate when . What is the meaning
of the derivative? What are its units?
;66. Car tires need to be inflated properly because overinflation or underinflation can cause premature treadware. The data in the table show tire life ( in thousands of kilometers) for a cer- tain type of tire at various pressures ( in ).
(a) Use a graphing calculator or computer to model tire life with a quadratic function of the pressure.
(b) Use the model to estimate when and when . What is the meaning of the derivative?
What are the units? What is the significance of the signs of the derivatives?
67. Suppose that , , , and .
Find the following values.
(a) (b)
(c)
68. Find , given that , , ,
and .
(a) (b)
(c) (d)
69. If , where and , find .
70. If and , find
71. If and are the functions whose graphs are shown, let and .
(a) Find (b) Find P
V
P V
50 kPa 25⬚C
0.106 m3
P苷 50 kPa dV兾dP
t⬘共5兲 苷 2 t共5兲 苷 ⫺3
f⬘共5兲 苷 6 f共5兲 苷 1
共 f兾t兲⬘共5兲 共 ft兲⬘共5兲
共t兾f 兲⬘共5兲
f⬘共2兲 苷 ⫺2 t共2兲 苷 4
f共2兲 苷 ⫺3 h⬘共2兲
t⬘共2兲 苷 7
h共x兲 苷 f 共x兲 t共x兲 h共x兲 苷 5f 共x兲 ⫺ 4t共x兲
h共x兲 苷 t共x兲 1⫹ f 共x兲 h共x兲 苷 f共x兲
t共x兲
f⬘共4兲 t⬘共4兲 苷 7
t共4兲 苷 8 f共x兲 苷 sx t共x兲
h⬘共2兲 苷 ⫺3 h共2兲 苷 4
d
dx 冉h共x兲x 冊冟x苷2
f t
v共x兲 苷 f 共x兲兾t共x兲 u共x兲 苷 f 共x兲t共x兲
v⬘共5兲.
u⬘共1兲.
f
g
x y
0 1
1 L
P kPa
dL兾dP P苷 200 P苷 300
72. Let and , where and
are the functions whose graphs are shown.
(a) Find . (b) Find .
73. If is a differentiable function, find an expression for the derivative of each of the following functions.
(a) (b) (c)
74. If is a differentiable function, find an expression for the derivative of each of the following functions.
(a) (b)
(c) (d)
75. Find the points on the curve where the tangent is horizontal.
76. For what values of does the graph of
have a horizontal tangent?
77. Show that the curve has no tangent line with slope 4.
78. Find an equation of the tangent line to the curve that is parallel to the line .
79. Find equations of both lines that are tangent to the curve and are parallel to the line . 80. Find equations of the tangent lines to the curve
that are parallel to the line .
81. Find an equation of the normal line to the parabola that is parallel to the line . 82. Where does the normal line to the parabola at the
point (1, 0) intersect the parabola a second time? Illustrate with a sketch.
83. Draw a diagram to show that there are two tangent lines to the parabola that pass through the point . Find the coordinates of the points where these tangent lines inter- sect the parabola.
84. (a) Find equations of both lines through the point that are tangent to the parabola .
F
G x y
0 1
1
t
y苷 t共x兲 y苷 x x
t共x兲 y苷 xt共x兲
f
y苷 f共x兲 x2 y苷 x2f共x兲
y苷 1⫹ xf 共x兲 y苷 x2 sx
f共x兲
y苷 2x3⫹ 3x2⫺ 12x ⫹ 1
x f共x兲 苷 x3⫹ 3x2⫹ x ⫹ 3
y苷 6x3⫹ 5x ⫺ 3
y苷 xsx y苷 1 ⫹ 3x
12x⫺ y 苷 1 y苷 1 ⫹ x3
y苷 x⫺ 1 x⫹ 1 x⫺ 2y 苷 2
x⫺ 3y 苷 5 y苷 x2⫺ 5x ⫹ 4
y苷 x ⫺ x2
共0, ⫺4兲 y苷 x2
G F Q共x兲 苷 F共x兲兾G共x兲
P共x兲 苷 F共x兲 G共x兲
Q⬘共7兲 P⬘共2兲
共2, ⫺3兲 y苷 x2⫹ x
P 179 193 214 242 262 290 311
L 80 106 126 130 119 113 95
98845_02_ch02_p132-141.qk_98845_02_ch02_p132-141 8/16/11 9:54 AM Page 138
SECTION 2.3 DIFFERENTIATION FORMULAS 139 (b) Show that there is no line through the point that is
tangent to the parabola. Then draw a diagram to see why.
85. (a) Use the Product Rule twice to prove that if , , and are
differentiable, then .
(b) Taking in part (a), show that
(c) Use part (b) to differentiate .
86. Find the derivative of each function by calculating the first few derivatives and observing the pattern that occurs.
(a) (b)
87. Find a second-degree polynomial such that ,
, and .
88. The equation is called a differential equation because it involves an unknown function and its derivatives and . Find constants such that the function satisfies this equation. (Differen- tial equations will be studied in detail in Chapter 9.)
89. Find a cubic function whose graph
has horizontal tangents at the points and . 90. Find a parabola with equation that has
slope 4 at , slope at , and passes through the point .
91. In this exercise we estimate the rate at which the total personal income is rising in the Richmond-Petersburg, Virginia, metro- politan area. In 1999, the population of this area was 961,400, and the population was increasing at roughly 9200 people per year. The average annual income was $30,593 per capita, and this average was increasing at about $1400 per year (a little above the national average of about $1225 yearly). Use the Product Rule and these figures to estimate the rate at which total personal income was rising in the Richmond-Petersburg area in 1999. Explain the meaning of each term in the Product Rule.
92. A manufacturer produces bolts of a fabric with a fixed width.
The quantity q of this fabric (measured in meters) that is sold is a function of the selling price p ( in dollars per meter), so we can write . Then the total revenue earned with selling price p is .
(a) What does it mean to say that and
?
(b) Assuming the values in part (a), find and interpret your answer.
93. Let
Is differentiable at 1? Sketch the graphs of and . h f t 共 fth兲⬘ 苷 f ⬘th ⫹ ft⬘h ⫹ fth⬘
f苷 t 苷 h d
dx 关 f 共x兲兴3苷 3关 f 共x兲兴2f⬘共x兲
y苷 共x4⫹ 3x3⫹ 17x ⫹ 82兲3 nth
f共x兲 苷 1兾x f共x兲 苷 xn
P共2兲 苷 5 P
P⬙共2兲 苷 2 P⬘共2兲 苷 3
y⬙ ⫹ y⬘ ⫺ 2y 苷 x2
y A, B, and C y⬙
y⬘
y苷 Ax2⫹ Bx ⫹ C
y苷 ax3⫹ bx2⫹ cx ⫹ d 共2, 0兲 共⫺2, 6兲
y苷 ax2⫹ bx ⫹ c x苷 ⫺1
⫺8 x苷 1
共2, 15兲
f共20兲 苷 10,000 f⬘共20兲 苷 ⫺350
R⬘共20兲
f共x兲 苷再xx2⫹ 1⫹ 1 if xif x⬍ 1艌 1
f⬘ f f
共2, 7兲
q苷 f 共p兲 R共p兲 苷 pf 共p兲
94. At what numbers is the following function differentiable?
Give a formula for and sketch the graphs of and . 95. (a) For what values of is the function
differentiable? Find a formula for . (b) Sketch the graphs of and .
96. Where is the function differenti -
able? Give a formula for and sketch the graphs of and . 97. For what values of and is the line tangent to
the parabola when ?
98. (a) If , where and have derivatives of all
orders, show that .
(b) Find similar formulas for and . (c) Guess a formula for .
99. Find the value of such that the line is tangent to the curve .
100. Let
Find the values of and that make differentiable everywhere.
101. An easy proof of the Quotient Rule can be given if we make the prior assumption that exists, where . Write
; then differentiate using the Product Rule and solve the resulting equation for .
102. A tangent line is drawn to the hyperbola at a point . (a) Show that the midpoint of the line segment cut from this
tangent line by the coordinate axes is .
(b) Show that the triangle formed by the tangent line and the coordinate axes always has the same area, no matter where
is located on the hyperbola.
103. Evaluate .
104. Draw a diagram showing two perpendicular lines that intersect on the -axis and are both tangent to the parabola . Where do these lines intersect?
105. If , how many lines through the point are normal lines to the parabola ? What if ?
106. Sketch the parabolas and . Do you
think there is a line that is tangent to both curves? If so, find its equation. If not, why not?
P
lim
x l 1
x1000⫺ 1 x⫺ 1
y y苷 x2
c⬎12 共0, c兲
y苷 x2 c艋12
y苷 x2 y苷 x2⫺ 2x ⫹ 2 P
F苷 f兾t F⬘共x兲
f苷 Ft
F⬘
P xy苷 c
f b
m
f共x兲 苷再xmx2 ⫹ b if xif x艋 2⬎ 2
y苷 csx
F⬙ 苷 f ⬙t ⫹ 2f ⬘t⬘ ⫹ ft⬙
F共4兲 F F共n兲
y苷32x⫹ 6 c
f t F共x兲 苷 f 共x兲t共x兲
x苷 2 y苷 ax2
2x⫹ y 苷 b b
a
h共x兲 苷
ⱍ
x⫺ 1ⱍ
⫹ⱍ
x⫹ 2ⱍ
h⬘ h h⬘
f⬘ f
f⬘
f共x兲 苷
ⱍ
x2⫺ 9ⱍ
x
t共x兲 苷
再
2x2x2⫺ x⫺ x2 if xif 0if x艋 0艌 2⬍ x ⬍ 2t⬘ t t⬘
t 98845_02_ch02_p132-141.qk_98845_02_ch02_p132-141 8/16/11 9:54 AM Page 139
146 CHAPTER 2 DERIVATIVES
If we let , then as , so by Equation 2 we have
Calculate .
SOLUTION
Here we divide numerator and denominator by x :
(by the continuity of cosine and Equation 2)
lim
x l 0sin 7x 4x 苷 7
4 lim
x l 0
冉 sin 7x 7x 冊
苷 7 4 lim
l 0
sin
苷 7
4 ⴢ 1 苷 7 4
v
EXAMPLE 6lim
x l 0x cot x
lim
x l 0x cot x苷 lim
x l 0
x cos x
sin x 苷 lim
x l 0
cos x sin x
x苷 lim
x l 0
cos x lim
x l 0sin x
x苷 cos 0
1 苷 1
x l 0
l 0
苷 7x
1–16 Differentiate.
1. 2.
3. 4.
5. 6.
7. 8.
9. 10.
11. 12.
13. 14.
15. 16.
17. Prove that .
18. Prove that .
19. Prove that .
f共x兲 苷 sx sin x f共x兲 苷 3x2 2 cos x
y苷 2 sec x csc x f共x兲 苷 sin x 12cot x
y苷 u共a cos u b cot u兲 y苷 c cos t t2sin t
y苷 sin cos
y苷 x
2 tan x
y苷 cos x 1 sin x f共兲 苷 sec
1 sec
y苷 1 sec x tan x y苷 t sin t
1 t
y苷 x2 sin x tan x h共兲 苷 csc cot
d
dx 共csc x兲 苷 csc x cot x d
dx 共sec x兲 苷 sec x tan x d
dx 共cot x兲 苷 csc2x
t共t兲 苷 t3cos t t共t兲 苷 4 sec t tan t
20. Prove, using the definition of derivative, that if , then .
21–24 Find an equation of the tangent line to the curve at the given point.
21. 22. ,
23. , 24. ,
25. (a) Find an equation of the tangent line to the curve at the point .
; (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.
26. (a) Find an equation of the tangent line to the curve
at the point .
; (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.
27. (a) If , find .
; (b) Check to see that your answer to part (a) is reasonable by graphing both and for .
28. (a) If , find .
; (b) Check to see that your answer to part (a) is reasonable by graphing both and for .
f共x兲 苷 cos x f共x兲 苷 sin x
y苷 sec x, 共兾3, 2兲 y苷 共1 x兲 cos x 共0, 1兲 y苷 cos x sin x 共, 1兲 y苷 x tan x 共, 兲
y苷 2x sin x 共兾2, 兲
y苷 3x 6 cos x 共兾3, 3兲
f共x兲 苷 sec x x f共x兲 f f
ⱍ
xⱍ
兾2f共x兲 苷 sx sin x f共x兲
f f 0 x 2
2.4 Exercises
; Graphing calculator or computer required 1.Homework Hints available at stewartcalculus.com 98845_02_ch02_p142-151.qk_98845_02_ch02_p142-151 8/16/11 9:55 AM Page 146
SECTION 2.4 DERIVATIVES OF TRIGONOMETRIC FUNCTIONS 147
29. If , find .
30. If , find .
31. (a) Use the Quotient Rule to differentiate the function
(b) Simplify the expression for by writing it in terms of and , and then find .
(c) Show that your answers to parts (a) and (b) are equivalent.
32. Suppose and , and let
and . Find
(a) (b) 33. For what values of does the graph of
have a horizontal tangent?
34. Find the points on the curve at which the tangent is horizontal.
35. A mass on a spring vibrates horizontally on a smooth level surface (see the figure). Its equation of motion is
, where is in seconds and in centimeters.
(a) Find the velocity and acceleration at time .
(b) Find the position, velocity, and acceleration of the mass at time . In what direction is it moving at that time?
;36. An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the mass is pulled downward and then released, it vibrates vertically. The equation of
motion is , , where is measured
in centi meters and in seconds. (Take the positive direction to be downward.)
(a) Find the velocity and acceleration at time . (b) Graph the velocity and acceleration functions.
(c) When does the mass pass through the equilibrium position for the first time?
(d) How far from its equilibrium position does the mass travel?
(e) When is the speed the greatest?
37. A ladder 6 m long rests against a vertical wall. Let be the angle between the top of the ladder and the wall and let be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does
change with respect to when ? H共兲 苷 sin H共兲 and H 共兲 f共t兲 苷 csc t f 共兾6兲
f共x兲 苷 tan x 1 sec x
f共x兲
sin x cos x f共x兲
f共兾3兲 苷 4 f共兾3兲 苷 2 t共x兲 苷 f 共x兲 sin x h共x兲 苷 共cos x兲兾f 共x兲
t共兾3兲 h共兾3兲
x f共x兲 苷 x 2 sin x
y苷 共cos x兲兾共2 sin x兲
x共t兲 苷 8 sin t t x
t t苷 2兾3
x x
0
equilibrium position
s苷 2 cos t 3 sin t t 0 s t
t
x
x 苷 兾3
38. An object with mass is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle with the plane, then the magnitude of the force is
where is a constant called the coefficient of friction.
(a) Find the rate of change of with respect to . (b) When is this rate of change equal to 0?
; (c) If , , and , draw the graph
of as a function of and use it to locate the value of for which . Is the value consistent with your answer to part (b)?
39–48 Find the limit.
39. 40.
41. 42.
43. 44.
45. 46.
47. 48.
49–50 Find the given derivative by finding the first few deriva- tives and observing the pattern that occurs.
49. 50.
51. Find constants such that the function
satisfies the differential equation .
52. (a) Evaluate .
(b) Evaluate .
; (c) Illustrate parts (a) and (b) by graphing . 53. Differentiate each trigonometric identity to obtain a new
(or familiar) identity.
(a) (b)
(c) lim
x l 0
sin 3x
x lim
x l 0
sin 4x sin 6x lim
t l 0
tan 6t
sin 2t lim
l 0
cos 1 sin lim
x l 0
sin 3x
5x3 4x lim
x l 0
sin 3x sin 5x x2
lim l 0
sin
tan lim
x l 0
sin共x2兲 x
lim
x l兾4
1 tan x
sin x cos x lim
x l 1
sin共x 1兲 x2 x 2
d99
dx99共sin x兲 d35
dx35共x sin x兲
A and B y苷 A sin x B cos x y y 2y 苷 sin x
lim
x l x sin 1 x lim
x l 0x sin 1 x
y苷 x sin共1兾x兲
tan x苷 sin x
cos x sec x苷 1
cos x sin x cos x 苷 1 cot x
csc x m
F苷 mt
sin cos
F
m苷 20 kg t 苷 9.8 m兾s2 苷 0.6
F
dF兾d 苷 0 98845_02_ch02_p142-151.qk_98845_02_ch02_p142-151 8/16/11 9:55 AM Page 147
