Q11,13, 17,19,21 and Q23 needed Needed to be solved Q11,13, 17,19,21 and Q23 these are very easy questions solve all in the order to get positive feedback
Q11,13, 17,19,21 and Q23 needed Needed to be solved Q11,13, 17,19,21 and Q23 these are very easy questions solve all in the order to get positive feedback
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Q11,13, 17,19,21 and Q23 needed
Needed to be solved Q11,13, 17,19,21 and Q23
these are very easy questions solve all in the order to get positive feedback
![EXAMPLE 10
SOLUTION
INCORPORATING
(a)
TECHNOLOGY
1. f(x) = x
4. f(x)=e-3x
Determine the following:
7. [4x³ dx
Verifying and Completing an Integral Formula Find the value of k that makes the fol-
lowing antidifferentiation formula true:
8.
3. [/dx
d
dx
Check Your Understanding 6.1
1. Determine the following:
[17/2 di
The given formula is equivalent to saying that the derivative of k(1-2x) + C is
(1-2x)³. Differentiating, we obtain
EXERCISES 6.1
Find all antiderivatives of each following
2. f(x)=9x8
5. f(x)=3
-(k(1-2x) + C) =
(b)
» / ( 3 ² + ² + ³ ) dx
fa-
function:
=(k(1-
3. f(x) = ³x
6. f(x)=-4x
1-2x)³ dx = k(1-2x)4+ C.
9. [7 dx
(k(1-2x)) +
=k(4)(1-2x)³(-2) = -8k(1-2x)³.
This is supposed to equal (1-2x)³. Therefore, -8k(1-2x)³ = (1-2x)³, which implies
-8k = 1, so k = -. Setting k=-, we obtain the integral formula
d
=k (1-2x)4+0
dx
=k(4)(1-2x)³(1-2x)
dx
fa- -2x)³ dx = -
If you wish, you can verify this formula by taking the derivative of both sides, as we did
in Example 9.
» Now Try Exercise 29
Graphing an Antiderivative and Solving a Differential Equation To graph the solu-
tion of the differential equation in Example 7, proceed as follows. First, press [Y-];
then set Y₁ = fnInt(X²-2, X, 1, X) + 4/3. To display fnInt, press [MATH] 9 in classic
mode. Upon pressing [GRAPH, you will notice that graphing Y₁ proceeds very slowly.
You can speed it by setting xRes (in the WINDOW] screen) to a higher value. Figure 3
shows the graph of the solution to Example 7.
Figure 3
10.
(c)
-D²-=
11.
X=1
6.1 Antidifferentiation 299
[1² dx (ka
- 2x) + C.
NORMAL FLOAT FRAC REAL RADIAN MP 0
V₁=fnInt(X-2.X.1.X+4/3
Solutions can be found following the section exercises.
2. Find a function f(t) that satisfies f'(t) = 3t+ 5 and f(0) = 5.
f=/dx (
Generalized power rule.
Y=403
(k a constant)
12.
2. [x.x² dx
dx (c a constant = 0)
13. / (+) de](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbfb5a78-2b03-4613-890c-0754521d637e%2F583ac373-02e8-41ee-9da9-6f7acaeef379%2Fvtiq1es_processed.jpeg&w=3840&q=75)
Transcribed Image Text:EXAMPLE 10
SOLUTION
INCORPORATING
(a)
TECHNOLOGY
1. f(x) = x
4. f(x)=e-3x
Determine the following:
7. [4x³ dx
Verifying and Completing an Integral Formula Find the value of k that makes the fol-
lowing antidifferentiation formula true:
8.
3. [/dx
d
dx
Check Your Understanding 6.1
1. Determine the following:
[17/2 di
The given formula is equivalent to saying that the derivative of k(1-2x) + C is
(1-2x)³. Differentiating, we obtain
EXERCISES 6.1
Find all antiderivatives of each following
2. f(x)=9x8
5. f(x)=3
-(k(1-2x) + C) =
(b)
» / ( 3 ² + ² + ³ ) dx
fa-
function:
=(k(1-
3. f(x) = ³x
6. f(x)=-4x
1-2x)³ dx = k(1-2x)4+ C.
9. [7 dx
(k(1-2x)) +
=k(4)(1-2x)³(-2) = -8k(1-2x)³.
This is supposed to equal (1-2x)³. Therefore, -8k(1-2x)³ = (1-2x)³, which implies
-8k = 1, so k = -. Setting k=-, we obtain the integral formula
d
=k (1-2x)4+0
dx
=k(4)(1-2x)³(1-2x)
dx
fa- -2x)³ dx = -
If you wish, you can verify this formula by taking the derivative of both sides, as we did
in Example 9.
» Now Try Exercise 29
Graphing an Antiderivative and Solving a Differential Equation To graph the solu-
tion of the differential equation in Example 7, proceed as follows. First, press [Y-];
then set Y₁ = fnInt(X²-2, X, 1, X) + 4/3. To display fnInt, press [MATH] 9 in classic
mode. Upon pressing [GRAPH, you will notice that graphing Y₁ proceeds very slowly.
You can speed it by setting xRes (in the WINDOW] screen) to a higher value. Figure 3
shows the graph of the solution to Example 7.
Figure 3
10.
(c)
-D²-=
11.
X=1
6.1 Antidifferentiation 299
[1² dx (ka
- 2x) + C.
NORMAL FLOAT FRAC REAL RADIAN MP 0
V₁=fnInt(X-2.X.1.X+4/3
Solutions can be found following the section exercises.
2. Find a function f(t) that satisfies f'(t) = 3t+ 5 and f(0) = 5.
f=/dx (
Generalized power rule.
Y=403
(k a constant)
12.
2. [x.x² dx
dx (c a constant = 0)
13. / (+) de
![300 CHAPTER 6 The Definite Integral
- 17/1dx
16. / (+1/2+2√x) dx
14.
18. /(277-Vx) dx
20.
· fex dx
•1222, dx
24. /(-36²" + 2x^²) de
22.
-3e-x -
dx
25.
27.
28.
29.
30.
31.
32.
In Exercises 25-36, find the value of k that makes the antidifferen-
tiation formula true. [Note: You can check your answer without
looking in the answer section. How?]
33.
34.
35.
[5e-2
36.
Se-21 dt = ke-21+ C
2e4x-1 dx = ke4x-1 + C
4
Ta
e³x+1 dx =
Ja
- dx =
(8 - x)4
15.
3. [x√x dx
+C
3
17. [(x-20²¹ + 1) de
dx
k
(8-x)³
2+x
k
3x+1
(5x-7)-2 dx = k(5x-7) + C
2-3x
19.
f3e-2x dx
fedx
· S-26²²² +1
21.
23.
√x + 1 dx = k(x + 1)³/² + C
[(4-
(4x) dx = k ln 14-x + C
[(3x + 2)¹ dx = k(3x + 2)5+ C
・/120
26.
[30/10
+ C
(2x - 1)³ dx = k(2x - 1)4 + C
-dx = k ln |2 + x + C
do khi
dx = k ln |2-3x + C
+ 1)dx
3e¹/10 dt = ke¹/10 + C
Find all functions f(1) that satisfy the given condition.
37. f'(1) = 13/2
4
38. f(1) =
6+1
39. f'(t)=0
40. f'(t)=1²-5t-7
Find all functions f(x) that satisfy the given conditions.
41. f'(x) = .5e-0.2x, f(0) = 0 42. f'(x)=2x-ex, f(0) = 1
43. f'(x)= x, f(0) = 3 44. f'(x) = 8x¹/3, f(1) = 4
45. f'(x)=√x + 1, f(4)=0 46. f'(x) = x² + √x, f(1) = 3
47. Figure 4 shows the graphs of several functions f(x) for which
f'(x)=. Find the expression for the function f(x) whose
graph passes through (1, 2).
(1, 2)
(a)
8
6
6+
Figure 4
48. Figure 5 shows the graphs of several functions f(x) for which
f'(x)=3. Find the expression for the function f(x) whose
graph passes through (6, 3).
+ C
1
(c) • (In x)² + C
Figure 5
49. Which of the following is In x dx?
1
(6,3)
4
Figure 6
2
6
Y
6
-2-
50. Which of the following is fxVx+ 1 dx?
(a) (x + 1)/2-(x + 1)³/² + C
(b) x²(x+1)/2 + C
8
8
2
51. Figure 6 contains the graph of a function F(x). On the same
coordinate system, draw the graph of the function G(x) hav-
ing the properties G(0)= 0 and G'(x) = F'(x) for each .x.
10 12
(b) x lnx-x+ C
A
4
+x
10 12
y = F(x)
6](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbfb5a78-2b03-4613-890c-0754521d637e%2F583ac373-02e8-41ee-9da9-6f7acaeef379%2F2vuu6nn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:300 CHAPTER 6 The Definite Integral
- 17/1dx
16. / (+1/2+2√x) dx
14.
18. /(277-Vx) dx
20.
· fex dx
•1222, dx
24. /(-36²" + 2x^²) de
22.
-3e-x -
dx
25.
27.
28.
29.
30.
31.
32.
In Exercises 25-36, find the value of k that makes the antidifferen-
tiation formula true. [Note: You can check your answer without
looking in the answer section. How?]
33.
34.
35.
[5e-2
36.
Se-21 dt = ke-21+ C
2e4x-1 dx = ke4x-1 + C
4
Ta
e³x+1 dx =
Ja
- dx =
(8 - x)4
15.
3. [x√x dx
+C
3
17. [(x-20²¹ + 1) de
dx
k
(8-x)³
2+x
k
3x+1
(5x-7)-2 dx = k(5x-7) + C
2-3x
19.
f3e-2x dx
fedx
· S-26²²² +1
21.
23.
√x + 1 dx = k(x + 1)³/² + C
[(4-
(4x) dx = k ln 14-x + C
[(3x + 2)¹ dx = k(3x + 2)5+ C
・/120
26.
[30/10
+ C
(2x - 1)³ dx = k(2x - 1)4 + C
-dx = k ln |2 + x + C
do khi
dx = k ln |2-3x + C
+ 1)dx
3e¹/10 dt = ke¹/10 + C
Find all functions f(1) that satisfy the given condition.
37. f'(1) = 13/2
4
38. f(1) =
6+1
39. f'(t)=0
40. f'(t)=1²-5t-7
Find all functions f(x) that satisfy the given conditions.
41. f'(x) = .5e-0.2x, f(0) = 0 42. f'(x)=2x-ex, f(0) = 1
43. f'(x)= x, f(0) = 3 44. f'(x) = 8x¹/3, f(1) = 4
45. f'(x)=√x + 1, f(4)=0 46. f'(x) = x² + √x, f(1) = 3
47. Figure 4 shows the graphs of several functions f(x) for which
f'(x)=. Find the expression for the function f(x) whose
graph passes through (1, 2).
(1, 2)
(a)
8
6
6+
Figure 4
48. Figure 5 shows the graphs of several functions f(x) for which
f'(x)=3. Find the expression for the function f(x) whose
graph passes through (6, 3).
+ C
1
(c) • (In x)² + C
Figure 5
49. Which of the following is In x dx?
1
(6,3)
4
Figure 6
2
6
Y
6
-2-
50. Which of the following is fxVx+ 1 dx?
(a) (x + 1)/2-(x + 1)³/² + C
(b) x²(x+1)/2 + C
8
8
2
51. Figure 6 contains the graph of a function F(x). On the same
coordinate system, draw the graph of the function G(x) hav-
ing the properties G(0)= 0 and G'(x) = F'(x) for each .x.
10 12
(b) x lnx-x+ C
A
4
+x
10 12
y = F(x)
6
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