Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
Hi I just need help answering number 13, and 19 only thank you.
![Terms and Concepts
25.
dx
1. Define the term "antiderivative" in your own words.
x – 7x
dx
26.
2. Is it more accurate to refer to "the" antiderivative of f(x)
or "an" antiderivative of f(x)?
5 -
3
27.
xp X +X
3. Use your own words to define the indefinite integral of f(x).
2
4. Fill in the blanks: "Inverse operations do the
28.
u° – 2u° – u +:
7
du
things in the
order."
5. What is an "initial value problem"?
29.
+ 4)(2u + 1) du
6. The derivative of a position function is a
func-
30.
+ 3t + 2) dt
tion.
7. The antiderivative of an acceleration function is a
1+ Vx+ x
31.
dx
function.
| sin' x+ cos? x dx
Problems
32.
In Exercises 8-40, evaluate the given indefinite integral.
33.
2 + tan? 0 do
8.
3x dx
34.
sec t(sec t + tan t) dt
1 - sin? t
dt
sin? t
9.
35.
xp
|(10 – 2) dx
sin 2x
dx
sin x
10.
36.
4 + 6u
du
37.
11.
dt
Vu
sin 0 + sin 0 tan? 0
do
12.
dt
38.
sec2
39.
2 +t
dt
13.
dt
| V + V* dx
40.
14.
dx
41. This problem investigates why Theorem 32 states that
15.
sec 0
9 de
dx = In |x| + C.
(a) What is the domain of y = In x?
16.
sin 0 de
(b) Find (In x).
17.
(sec x tan x + cSc x cot x) dx
(c) What is the domain of y = In(-x)?
(d) Find (In(-x)).
18.
dt
2
(e) You should find that 1/x has two types of antideriva-
tives, depending on whether x > 0 or x < 0. In
|(2+ + 3)° dt
20. ( + 3)(? – 21) dt
19.
one expression, give a formula for
dx that takes
these different domains into account, and explain
your answer.
In Exercises 42-52, find f(x) described by the given initial value
problem.
21.
dx
22.
e* dx
42. f'(x) = sin x and f(0) = 2
43. f'(x) = 5e* and f(0) = 10
44. f'(x) = 4x³ – 3x and f(-1) = 9
45. f'(x) = sec² x and f(/4) = 5
23.
dx
4x° – 7
dx
24.
46. f"(x) = 5 and f'(0) = 7, f(0) = 3
227](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf266520-407e-4b51-839c-e1d92d759be6%2F0bc82338-182d-4243-b712-f3dccb5a40e0%2Fphn5g4c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Terms and Concepts
25.
dx
1. Define the term "antiderivative" in your own words.
x – 7x
dx
26.
2. Is it more accurate to refer to "the" antiderivative of f(x)
or "an" antiderivative of f(x)?
5 -
3
27.
xp X +X
3. Use your own words to define the indefinite integral of f(x).
2
4. Fill in the blanks: "Inverse operations do the
28.
u° – 2u° – u +:
7
du
things in the
order."
5. What is an "initial value problem"?
29.
+ 4)(2u + 1) du
6. The derivative of a position function is a
func-
30.
+ 3t + 2) dt
tion.
7. The antiderivative of an acceleration function is a
1+ Vx+ x
31.
dx
function.
| sin' x+ cos? x dx
Problems
32.
In Exercises 8-40, evaluate the given indefinite integral.
33.
2 + tan? 0 do
8.
3x dx
34.
sec t(sec t + tan t) dt
1 - sin? t
dt
sin? t
9.
35.
xp
|(10 – 2) dx
sin 2x
dx
sin x
10.
36.
4 + 6u
du
37.
11.
dt
Vu
sin 0 + sin 0 tan? 0
do
12.
dt
38.
sec2
39.
2 +t
dt
13.
dt
| V + V* dx
40.
14.
dx
41. This problem investigates why Theorem 32 states that
15.
sec 0
9 de
dx = In |x| + C.
(a) What is the domain of y = In x?
16.
sin 0 de
(b) Find (In x).
17.
(sec x tan x + cSc x cot x) dx
(c) What is the domain of y = In(-x)?
(d) Find (In(-x)).
18.
dt
2
(e) You should find that 1/x has two types of antideriva-
tives, depending on whether x > 0 or x < 0. In
|(2+ + 3)° dt
20. ( + 3)(? – 21) dt
19.
one expression, give a formula for
dx that takes
these different domains into account, and explain
your answer.
In Exercises 42-52, find f(x) described by the given initial value
problem.
21.
dx
22.
e* dx
42. f'(x) = sin x and f(0) = 2
43. f'(x) = 5e* and f(0) = 10
44. f'(x) = 4x³ – 3x and f(-1) = 9
45. f'(x) = sec² x and f(/4) = 5
23.
dx
4x° – 7
dx
24.
46. f"(x) = 5 and f'(0) = 7, f(0) = 3
227
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