Page 10 - 2024-calc4e-SE proofs-4e.indd

Sullivan
9:25
04 apcalc4e 45342 ch02 166 233 3pp
June 19, 2023
Section 2.3 • The Derivative of a Polynomial Function; The Derivative of y = e and y = ln x
x
In Chapter 1, we found that the natural logarithm function y = ln x is continuous
NEED TO REVIEW? The natural logarithm
on its domain {x|x > 0}. Below we give the rule for finding the derivative of y = ln x.
function is defined in Section P.5, p. 47.
THEOREM Derivative of the Natural Logarithm Function y = ln x
The derivative of the natural logarithm function y = ln x, x > 0, is 199
d 1
�
y = ln x = (2)
dx x
We do not have the necessary mathematics to prove (2) now. We will prove the
theorem in Chapter 3.
EXAMPLE 8 Differentiating a Function Involving y = ln x
Practice, Practice, Practice: Hone your skills
Sullivan 04 apcalc4e 45342 ch02 166 233 3pp June 19, 2023 9:25 Find the derivative f (x) = 3 ln x − 5x . 2
Solution
2
The function f is the difference between 3 ln x and 5x . Then using (2), we find that
176 Chapter 2 • The Derivative and Its Properties
d d d d 3
with exercises at every turn. � 2 2
f (x) = dx (3 ln x − 5x ) = dx (3 ln x) − dx (5x ) = 3 dx (ln x) − 5 · 2x = x − 10x
s (miles)
PAGE
172 31. Approximating Velocity An object moves on a line according (4, 2.7) (7, 2.7) Difference Rule Constant Multiple Rule; Use (2).
↑
↑
↑
2
to the position function s(t) = 10t (s in centimeters and t in Princeton 2.7 Simple Power Rule
Junction
seconds). Approximate the velocity of the object at time t 0 = 3s RR station
by letting �t first equal 0.1 s, then 0.01 s, and finally 0.001 s. R
What limit does the velocity appear to be approaching? Organize NOW WORK Problem 23 and AP Practice Problems 7 and 9.
the results in a table. 1 (10, 1) (11, 1) Want to master calculus concepts?
32. Approximating Velocity An object moves on a line according 2.3 Assess Your Understanding
2
to the position function s(t) = 5 − t (s in centimeters and t in Princeton (0, 0) (13, 0) Assess Your Understanding exercises
0
seconds). Approximate the velocity of the object Concepts and Vocabulary 0 9:41 4 7 10 13 17 t (minutes) appear at the end of each section. Selected
Start with the at time t 0 = 1 by
Sullivan
July 13,
05 apcalc4e 45342 ch03 234 283 3pp
University 2023
Concepts and
letting �t first equal 0.1, then 0.01, and finally 0.001. What limit
7
2
t + 2
3
PAGE
does the velocity appear to be approaching? Organize the results 194 1. d π = ; d x = . answers appear in the back of the book.
x − 5x
3
dx walks to the deli, which is six blocks east of her house. After
38. Jen
dx
in a table. Vocabulary walking two blocks, she realizes she left her phone on her desk, so 17. f (t) = 5 18. f (x) = 9
to check your
PAGE
2. When n is a positive integer, the Simple Power Rule
173 33. Motion on a Line As an object moves on a line, its signed 3 • The Derivative of Composite, Implicit, and Inverse Functions x + 2x + 1 1 2
264
Chapter
3
she runs home. After getting the phone, and closing and locking
distance s (in meters) from the origin after t seconds is given by the door, d x = . 19. f (x) = 7 20. f (x) = (ax + bx + c), a �= 0
n
a
states that Jen starts on her way again. At the deli, she waits in line
comprehension of
dx
2
the position function s = f (t) = 3t + 4t. Find the velocity v to buy a bottle of vitaminwater TM , and then she jogs home. The x 1 x
In Problems 9–16, f and g are inverse functions. For each function f,
50. Tangent Line
22. f (x) =− e
the section’s main
at t 0 = 0. At t 0 = 2. At any time t 0 . find g (y 0 ). 3. True or False The derivative of a power function of degree 21. f (x) = 4e Easy cross-references
graph below represents Jen’s journey. The time t is in minutes,
2
(a) Find an equation for the tangent line to the graph
�
greater than 1 is also a power function.
34. Motion on a Line As an object moves on a line, its signed and s is Jen’s distance, in blocks, from home. of y = tan PAGE −1 x at x = 1. 24. f (x) = 5 ln x + 8
3
5
10. f (x) = x ; y 0 = 27
points. after t seconds is given by
distance s (in meters) from the origin 9. f (x) = x ; y 0 = 32 4. If k is a constant and f is a differentiable function, 199 23. f (x) = x − ln x back to text.
(a) At what times is she headed toward the deli? PAGE −1 u 26. f (u) = 3e + 10 ln u
u
198 25. f (u) = 5 ln u − 2e x and the tangent line
d
3
2
the position function s = f (t) = 2t + 4. Find the velocity v at 2, x ≥ 0; y 0 = 6 (b) [kf (x)] = . (b) Use technology to graph y = tan In the practice problems, look for
then At what times is she headed home?
11. f (x) = x +
found in (a).
dx
t 0 = 0. At t 0 = 3. At any time t 0 . 2 (c) When is the graph horizontal? What does this indicate? In Problems 27–32, find each derivative.
the red icon with page number
x
12. f (x) = x − 5, x ≥ 0; y 0 = 4
35. Motion on a Line As an object moves on a line, its signed 5. The derivative of f (x) = e is . 260 51. Tangent Line d √ f (x) = 2x − x, x ≥ 1, is 2t − 5
PAGE
3
(d) Find Jen’s average velocity from home until she starts back to The function

4
1
d
that directs you back to the
28.
27.
1/3
distance s from the origin at time t is given by the position ; y 0 = 2 6. True or False The derivative of an exponential one-to-one and has an inverse function g. Find an equation 8 of the
get her phone. = x
; y 0 = 4
2/5
13. f (x) = x
14. f (x)
3 t +
Skill Building
dt
dt
2
x
1
x velocity from home to the deli after line to the graph of g at the point (14, 2) on g.
tangent
2
function s = s(t) = 3t − , where s is in centimeters and t is in function f (x) = a , where a > 0 and a �= 1, is always d A corresponding worked example in
(e) Find Jen’s average 2/3
dC
4/3
+ 1; y 0 = 49
+ 5; y 0 = 6
15. f (x) = 3x
a constant multiple of a . x
problems help getting her phone. 29. if A(R) = π R 2 30. if C = 2π R
t
16. f (x) =
d R
d R is
the chapter text.
5
seconds. Find the velocity v of the object at t 0 = 1 and t 0 = 4. 52. Tangent Line The function f (x) = x − 3x, x ≤ −1,
(f) Find her average velocity from the deli to home.
In Problems 17–42, find the derivative of dV 4 d P
3g. Find an equation of the
PAGE has an inverse function
one-to-one and
36. Motion on a Line As an object moves on a line, Skill Building each function. sec −1 x 2NOT FINAL - For Review Purposes Only - Do Not Copy.
you to develop its signed
if P = 0.2T
32.
if V = πr
195 31.
s (blocks)
3
−1
−1
tangent line to the graph
17. f (x) = sin
18. f (x) = sin (3x − 2)
computation skills
distance s from the origin at time t is given by the position (4x) In Problems 7–26, find the derivative of each function using the dr of g at the point (−2, 1) on g. dT
Deli 45342 ch03 234 283 3pp
√ Sullivan 05 apcalc4e 6 (16, 6) (19, 6) July 13, 2023 9:41 In Problems 33–36:
function s = s(t) = 2 t, where s is in centimeters and t is in formulas of this section. (a, b, c, and d, when they appear, are 1/3 is one-to-one and
53. Normal Line The function f (x) = x + 2x
PAGE
−1
−1
263 19. g(x) = sec (3x)
(a) Find the slope of the tangent line to the graph of each
seconds. Find the velocity v of the object at t 0 = 1 and constants.) 20. g(x) = cos (2x) has an inverse function g. Find an equation of the normal line to
and the ability to t 0 = 4.
function f at the indicated point.
PAGE
37. The Princeton Dinky is the shortest rail line in −1 195 7. f (x) = 3x + √ 2 −1 8. f (x) = 5x − π the graph of g at the point (3, 1) on g.
t
select the best the country. t
21. s(t) = tan
22. s(t) = sec
(b) Find an equation of the tangent line at the point.
3
It runs for 2.7 miles, connecting Princeton University to the 2 9. f (x) = x + 3x + 4 (5, 2) 10. f (x) = 4x + 2x − 2 4 Section 3.1 • The Chain Rule 245
2
4
2
Icons identify problems
(c) Find an equation of the normal line at the point.
approach to solve from the
Princeton Junction railroad station. The Dinky starts −1 2 5 24. f (x) = sin (1 − x ) 3 54. Normal Line The function f (x) = x + x, x > 0, is one-to-one
2
−1
2
23. f (x) = tan (1 − 2x )
2
04 apcalc4e 45342 ch02 166 233 3pp
Sullivan
12. f (u) = 9u − 2u + 4u + 4 an inverse function g. Find an equation of the normal line
a problem.
university and moves north toward Princeton Junction. Its 11. f (u) = 8u − 5u + 1 June 19, 2023 9:25 and has (d) Graph f and the tangent line and normal line found
104. Median Earnings that require a graphing
3 on a Line An object moves along a line so that
98. Motion
2 (25, 0)to the graph of g at the point (2, 1) on g.
−1
2
2
−1
(7, 0) x
25. f (x) = sec (x
3 26. f (x) = cos
2
distance from Princeton is shown in the graph (top, right), where+ 2) 13. f (s) = as + s 0 t ≥ 0 seconds, its position from the origin in (b) and (c) on the same set of axes. The median earnings E, in dollars, of
14. f (s) = 4 − πs
House time
workers 18 years and over are given in the table below:
at
the time t is in minutes and the distance s of the Dinky 1 2 0 5 t 10 15 20 1 8 55. Motion on a Line An object moves along the x-axis calculator or computer
t (minutes)
PAGE
25
4
3
197 33. f (x) = x + 3x − 1 at (0, −1) 34. f (x) = x + 2x − 1 at (1, 2)
is
PAGE
6 s(t) = sin e , in feet.
−1 x
from Princeton University is in miles. 261 27. F(x) = sin −1 x 15. f (t) = (t − 5t)(0, 0) (6, 0) e 8 2 so that its position x from the x origin (in meters) 36. f (x) = 4 − e at (0, 3) CAS .
28. F(x) = tan
e
16. f (x) = (x − 5x + 2)
6
35. f (x) = e + 5x at (0, 1) is given
x
200 Chapter 2 • The Derivative and Its Propertiesacceleration a of the object at any Year 1980 1985 algebra system 2010 2015 2020
(a) Find the velocity
√ v and
1
2000
1995
2005
1990
−1
29. g(x) = tan −1 Applications and Extensions x by x(t) = sin −1 1 , t > 0, where t is the time in seconds.
30. g(x) = sec
time t.
x t Median
(b) At what time
−1 does the object first have zero velocity?
−1
of
the
x
32. g(x) = x tan
In Problems 37–42: 39. Slope of a Tangent Line An equation of the tangent line to thethe velocity of the object at t = 2 s. 17,181 21,793 26,792 32,604 41,231 49,733 48,000 50,295
line
to
equation
31. g(x) = x sin
In Problems 49 and 50, for each function f:
tangent Find the
(a)
Earnings 12,665
An (x + 1)
� � the time t found
(c) What is the acceleration of the object at
graph of a function f at (2, 8) is y =−5x + 12. What is f (2)?(2)?
12. What is f
(a) Find the points, if any, at which the graph of each function f (b) Find the acceleration of the object at t = 2 s.
(a) Find f (x) by expanding f (x) and differentiating the polynomial.
in (b)?
34. s(t) = t sin
−1 2
2
2
Source: U.S. Bureau of the Census, Current Population Survey.
�
−1 3
33. s(t) = t sec
t
t
40. Slope of a Tangent Line An equation of the tangent line of a a
An equation of the tangent line of
has a horizontal tangent line.
Once you’ve mastered 239 99. Resistance The −1 resistance R (measured in ohms) of � � sin .
CAS (b) Find f (x) using a CAS. x
4
−1
PAGE
−1
PAGE
262 35. f (x) = tan (cos x)
� 56. If g(x) = cos (cos x), show that g (x) =
function f at (3, 4) is y = x + 1. What is f (3)?
36. f (x) = sin (sin x)
| sin x| the exponential function of best fit and show that it
3 line.
(b) Find an equation for each horizontal tangent
(c) Show that the results found in parts (a) and (b) are equivalent.
computational −1 x) 41. Tangent LineG(x) = cos(tan −1 x) 57. Show . The radius x is (a) Find 50. f (x) = (x + x) 4
an 80-meter-long electric wire of radius x (in centimeters)
2 2 d
t
equals E = E(t) = 12, 376 (1.036) , where t is the number
(c) Solve the inequality f (x)> 0. 38.
37. G(x) = sin(tan
−1
tan (cot x) =−1.
Does the tangent line to the graph of y = x x that
y =
�
© 2024 BFW Publishers PAGES
0.0048
skills, tackle the
of years since 1979.
MancusoMichael/AP Images Applications and sin 1 − x 2 170 43. Respiration Rate Does the tangent line to the graph of y = x x 3 3 d d Applications and Extensions rate of change at t = 36 (year 2015).
dx
2
3
49. f (x) = (2x − 1)
is given by the formula R
at (1, 1) pass through the point (2, 5)? = R(x) =
x 2
�
(d) Solve the inequality f (x)< 0.
39. f (x) = e
d
40.
tan −1 (3x)
42. Tangent Line f (x) = e
1 (b) Find the rate of change of E with respect to t.
−1
cot
for all x �= 0.
−1
tan
y =
given by x = 0.1991 + 0.000003T where T is
58. Show that the temperature
x =
at (1, 1) pass through the point (2, 5)? the same set
(e) Graph f and any horizontal lines found in (b) on
dx
dx
x (c) Find the rate of change at t = 26 (year 2005).
−1
−1
x
sec
x
in Kelvin. How fast is R changing with respect to T
Extensions problems,
PAGE
2
42. g(x) =√
2x the
of axes. √
A human being’s respiration rate R R
(d) Find
41. g(x) =

−1
x 2 − 1
when T = 320 K?
sin
x − x
59. Show that
.
+ 0 59
(f) Describe
which are applied or the graph of f for the results obtained in parts (c)
1 − x 2 =
1 − x 2 rate of change at t = 41 (year 2020).
100. Pendulum Motion in a Car The motion of a pendulum
where p is the partial pressure of carbon dioxide in the lungs. In Problems 51–56, find each limit.
is the partial pressure of carbon dioxide in the lungs.
carbon
in
dioxide
of
is
the
pressure
lungs.
the
partial dy
and (d).
extend the concepts of (in breaths per minute) is given by R = R (p) = 10.35 + 0.59p, p dx (e) Find the
(f) Compare the answers to (c), (d), and (e). Interpret each
Find the rate of change in respiration when p = 50. a car moving at a low,
the swinging in the direction of motion of
in .
In Problems 43–46, use implicit differentiation
of to find
rate
when
Find the rate of change in respiration when
Find
change
respiration
(a) When is the Dinky headed toward Princeton University? constant speed can be modeled by Challenge Problem 8 answer
dx
8 and explain the differences.
1
1
PAGE
197 37.
38. f (x) = x + 4x − 3
the section… f (x) = 3x − 12x + 4
2
2
(b) When is it headed toward Princeton Junction? −1 y = 2x 44. Instantaneous Rate of Change The 60. Another way of finding 4 the derivative of y = x is to use inverse
− 4
√
n
+ h
−1
2
43. y + sin
volume V of the right circular cylinder of ofof + tan
right y = 3
44. xy
105. Motion on a Line
An object moves along
cylinder
the
of the right circular cylinder of
5a line so that at
2
circular = 0.05 sin(2t) + 3t
2
0 ≤ t ≤ π
s = s(t)
(c) When is the Dinky stopped? x height 5 m and radius r m shown in the functions. The function y = f (x) = x , n a positive integer, has 5(2 + h) − 5 · 2 5
n
51. lim
40. f (x) = 2e − 1
time t > 0 its position s from
x π
−1
−1
39. f (x) = x + e3
2
46. sec
45. 40 tan
y − πx
where s is the
(d) Find its average velocity on a trip from Princeton to Princeton y = 2π figure is V = V (r) = 5πr . Find the meters and t is the time in seconds. n − 1 . So, if h x �= 0, then f (x) �= 0. 52. lim the origin is s = s(t). The
h
y − xy = distance in
h→0
2
h→0 = nx
�
�
the derivative f (x)
3
5 m
Junction. 3 instantaneous rate of change of the 3 π π The inverse function of f, namely, velocity v of the object ds , and its acceleration
√
√
√
4
π
42. f (x) = x − 4x
41. f (x) = x − 3x + 2
5
5 x = g(y) = n y, is definedis v =
10
3 · 8
.
3
47. The function
dt
(e) Find its average velocity for the round trip shown in the f (x) = x + 2x has an inverse function g. 8 , t = 4 for all y if n is odd and for all y ≥ 0 if n is even. Since this 54. lim π(1 + h) − π
3(8 + h) −
(a) Find the velocity v at t =
, and t =
volume with respect to the radius
53. lim
2
d s
2
dv
�
�
r
h→0 v = v(s) is expressed as a
(b)
graph, that is, from t = 0 to t = 13. Find g (0) and g (3). when r = 3 m. Find the acceleration a at the times given in (a). h→0 h all y �= 0, we have 2 . If the velocity h
=
inverse function is differentiable for is a =
dt
dt
43. Motion on a Line At t seconds, an object moving on a line is s 3 3 n n
3
48. The function f (x) = 2x + x − 3 has an inverse function g. = v(t), and a = a(t) on the same screen.
a(x + h) − ax
b(x + h) − bx
(c) Graph s = s(t), v
1
3
meters from the origin, where s(t) = t − t + 1.
56. lim
Find g (−3) and g (0). Source: Mathematics students at Trine University, Angola, Indiana. d √ n y = 1 function of s, show that the acceleration a can be expressed
�
�
g (y)
�
=
55. lim =
nx n − 1
dv
Find the velocity of the object at t = 0 and at t = 5. h→0 dy h f � (x) . h→0 h
as a = v
ds
Applications and Extensions 101. Economics The function A(t) = 102 − 90 e −0.21t In Problems 57–62, find an equation of the tangent line(s) to the graph rating
n − 1 represents
√ n − 1− 1
1
n
(n − 1)/nn
1
n
1
, we have
= ny
ny
,
)
= n
(
/
Since nx
/
n y
ny
(
have
1 − (1/n))
= ny 106. Student Approval
−
we Professor Miller’s student approval
−
=
=
the relationship between A, the percentage of the market
44. Motion on a Line At t seconds, an object moving on a line is s
49. Tangent Line
of the function f that is (are) parallel to the line L.
4 by the latest generation smart phones, and t, the time
penetrated
3
…and the Challenge problems—more 1. d √ n y = d y y 1 1/n n = = 1 1 = = 1 1 y y (1/n) − 11 10 sin 2πt
1
n
meters from the origin, where s(t) = t − t +
(a) Find an equation for the tangent line to the graph
)
/
/
(
−
in years, where t = 0 corresponds to the year 2025.
7
n n
dy
ny
2ny 1 − (1/n)1 − (1/n)
L: y = 5x
difficult, thought-provoking extensions t = 1. dy 57. f (x) = 3x − x; is modeled by the function Q(t) = 21 + √ t − √ 20 ,
Find the velocity of the object at t = 0 and at
−1 x
of y = sin
at the origin.
(a) Find lim A(t) and interpret the result.
2
and
3 the Chain Rule to prove the formulaChain Rule to prove the formula
from
Use the result from above and the
above
of the section material — often combine Use the result 58. f (x) = 2x + 1;where 0 ≤ t ≤ 16 is the number of weeks since the semester
L: y = 6x − 1
t→∞
−1 x
Motion on a Line In Problems 45 and 46, each position function explain how the graph d p p began.
and the tangent line
(b) Use technology to graph y = sin (b) Graph the function A = A(t), and
x
2
p
q
p/qq
p
(p/q) − 11
)
x x
(
/
x x
/
concepts learned in previous chapters. t of an object 59. f (x) = e ; = q q L: y − x − 5 = 0
−
gives the signed distance s from the origin at time
supports the answer in (a).
=
dx
found in (a).
(a) Find Q (t).
�
60. f (x) =−2e ;
moving on a line: (c) Find the rate of change of A with respect to time. x (b) Evaluate Q (1), Q (5), and Q (10) .
�
�
L: y + 2x − 8 = 0 �
(d) Evaluate A (5) and A (10) and interpret these results. (c) Interpret the
�
�
61. f (x) = 3 ln x;
L: y = 3x − 2 results obtained in (b).
(a) Find the velocity v of the object at any time t. � � (d) Use technology to graph Q(t) and Q (t).
(e) Graph the function A = A (t), and explain how the graph
�
supports the answers in (d).
(b) When is the velocity of the object 0? tion-lev el pr oblem 62. f (x) = ln x − 2x; L:3x − y = 4
T
Throughout the section-level problem
oughout the sec
hr
(e) How would you explain the results in (d) to Professor
63. Tangent Lines Let f (x) = 4x − 3x − 1.
102. Meteorology The 9 atmospheric pressure at a height of x meters Miller? 3
identifies
sets, a Group Work icon 2
3
45. s(t) = 2 − 5t + t 2 46. s(t) = t − t + 6t + 4.00012x kg/m . What is the Source: Mathematics students at Millikin University, Decatur,
2
4 −0
above sea level is P(x) = 10 e
2
rate of change of the pressure with respect
problems selected by the authors as to the height (a) Find an equation of the tangent line to the graph of f
Illinois.
potential candidates for group projects
at x = 500 m? At x = 750 m?
In Problems 47 and 48, use the graphs to find each derivative. at x = 2. 107. Angular Velocity If the disk in the figure is
103. Hailstones Hailstones originate at an altitude of
rotated about a vertical line through an angle θ,
or assignments. (b) Find the coordinates of any points on the graph of f where
about 3000 m, although this varies. As they fall, air resistance
torsion in the wire attempts to turn
47. Let u(x) = f (x) + g(x) and v(x) = f (x) − g(x). the tangent line is parallel to y = x + 12. the disk in the
slows down the hailstones considerably. In one model of air
xxiv resistance, the speed of a hailstone of mass m as a function opposite direction. The motion θ at time t
(c) Find an equation of the tangent line to the graph of f at any
© 2024 BFW Publishers PAGES NOT FINAL
(assuming no friction or air resistance) obeys the
y mg points found in (b). θ
For Review Purposes Only, all other uses prohibited
equation
� −kt/m
(1−e
( 4, 5) of time t is given by �v(t) = k (b) u (4) ) m/s, (d) Graph f, the tangent line found in (a), the line y = x + 12,
(a) u (0)
� Do Not Copy or Post in Any Form.

2
(c) v (−2) acceleration due to gravity and k
�
4 y f (x) where g = 9.8 m/s is the (d) v (6) and any tangent lines found in (c) on the same screen.
2k
1
π
(6, 4) is a constant that depends on the size of the hailstone θ(t) = cos t
(1, 2) (e) 3u (5) (f) −2v (3) 3 2 5
�
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and the conditions of the air. 3 2
64. Tangent Lines Let f (x) = x + 2x + x − 1.
( 1, 2)
01_apcalc4e_45342_fm_i_xxix_3pp.indd 24 (1, 1) (a) Find the acceleration a(t) of a hailstone as a function where k is the coefficient of torsion of the wire. 10/11/23 2:38 PM
dθ
of time t. (a) Find an equation of the tangent line to the graph of f at any time t.
of the disk
(a) Find the angular velocity ω =
4 2 2 4 6 x at x = 0. dt
(b) Find lim v(t). What does this limit say about the
(b) What is the angular velocity at t = 3?
t→∞
2 (5, 2) speed of the hailstone? (b) Find the coordinates of any points on the graph of f where
( 4, 3) y g(x) (c) Find lim a(t). What does this limit say about the the tangent line is parallel to y = 3x − 2.
t→∞
acceleration of the hailstone? (c) Find an equation of the tangent line to the graph of f at any
48. Let F(t) = f (t) + g(t) and G(t) = g(t) − f (t). points found in (b).
(d) Graph f, the tangent line found in (a), the line y = 3x − 2,
y y g(t) (7, 6) and any tangent lines found in (c) on the same screen.
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6 (4, 6) (a) F (0) (b) F (3)
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(c) F (−4) (d) G (−2) 65. Tangent Line Show that the y
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( 3, 4) y e x
4 (e) G (−1) (f) G (6) line perpendicular to the x-axis
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( 5, 3)
and containing the point (x, y)
x
2 y f(t) on the graph of y = e and the (x, y)
( 5, 2) (2, 2)
tangent line to the graph
x
of y = e at the point (x, y)
4 2 4 6 t
intersect the x-axis 1 unit apart.
2 (5, 2) See the figure.
x 1 x x
66. Tangent Line Show that the tangent line to the graph
n
of y = x , n ≥ 2 an integer, at (1, 1) has y-intercept 1 − n.