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Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![102 CHAPTER 1 The Derivative
Check Your Understanding 1.6
1. Find the derivative
dx (x).
EXERCISES 1.6
Differentiate.
1. y = 6x³
3. y=3√x
5.ya_2
7. f(x)=x+x³+x
9. y = (2x+4)³
11. y=(x³ + x² +1)
13. y=-=
15. y=3√2x²+1
17. y = 2x + (x + 2)³
19. y = 1
= 5x5
1
21. y ³+1
1
23. y=x+x+1
25. f(x)=5√3x³ + x
27. y= 3x + 7³
29. y= V1+x+x²
2
1-5x
31. y T
45
1 + x + √x
35. y=x+1+√x+1
37. f(x) =
33. y=-
22. y =
24. y = 2√x+1
1
x³ + x + 1
V1+x²
1
2x+5
7
VI+x
34. y=(1+x+x²)
26. y
28. y=
30, y =
32. y=-
2. y = 3x4
4. y=_1_
3.x²
6. f(x) = 12 + 7/3
8. y = 4x³ - 2x²+x+1
10. y=(x²-1)³
12. y = (x² + x)-2
14. y - 4(x²-6)-3
16. y = 2√x+1
18. y (x-1)³ + (x+2)*
20. y(x²+ 1)² + 3(x² - 1)²
im
36. y = ²x
8. y = (x - 1) ¹
y=
39. f(x)=3x² - 2x + 1, (1, 2)
40. f(x) = x10 + I + VI − x, (0, 2)
1/(a+h)+ g(a+h)]-[f(a)+ g(a)
h
[f(a+h)-f(a) g(a+h)-g(a)
h
h
f(a+h)-f(a)
h
-f(a)+g'(a).
In Chapter 3, the general power rule will be proved as a special case of the chain rule.
-lim
In Exercises 39 and 40, find the slope of the graph of y=f(x) at the
designated point.
-lim
g(a+h)-g(a)
h
+lim
Solutions can be found following the section exercises.
x + (x³ + 1) ¹0
2. Differentiate the function y
Limit of a sum.
41. Find the slope of the tangent line to the curve y = x³ + 3x - 8
at (2,6).
42. Write the equation of the tangent line to the curve
y=x+3x-8 at (2,6).
43. Find the slope of the tangent line to the curve y(x²-15)6
at x4. Then write the equation of this tangent line.
44. Find the equation of the
angent line to the curve
y" x²+x+2
at x = 2.
45. Differentiate the function f(x)-(3x²+x-2)² in two
ways.
(a) Use the general power rule.
(b) Multiply 3x²+x-2 by itself and then differentiate the
resulting polynomial.
46. Using the sum rule and the constant-multiple rule, show that
for any functions f(x) and g(x)
8 mua
vest
y6r+1
f(x)-8(x)=f(x)-8(x)
dx
47. Figure 2 contains the curves y = f(x) and y= g(x) and the
tangent line to y=f(x) at x = 1, with g(x)=3-f(x). Find
g(1) and g'(1).
y = f(x)
d
y = g(x)
Figure 2 Graphs of f(x) and g(x) = f(x).
x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F270a7167-3c73-48e2-a405-df008c65eadb%2F3d126863-b941-4182-be76-81516220e859%2Fl26modn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:102 CHAPTER 1 The Derivative
Check Your Understanding 1.6
1. Find the derivative
dx (x).
EXERCISES 1.6
Differentiate.
1. y = 6x³
3. y=3√x
5.ya_2
7. f(x)=x+x³+x
9. y = (2x+4)³
11. y=(x³ + x² +1)
13. y=-=
15. y=3√2x²+1
17. y = 2x + (x + 2)³
19. y = 1
= 5x5
1
21. y ³+1
1
23. y=x+x+1
25. f(x)=5√3x³ + x
27. y= 3x + 7³
29. y= V1+x+x²
2
1-5x
31. y T
45
1 + x + √x
35. y=x+1+√x+1
37. f(x) =
33. y=-
22. y =
24. y = 2√x+1
1
x³ + x + 1
V1+x²
1
2x+5
7
VI+x
34. y=(1+x+x²)
26. y
28. y=
30, y =
32. y=-
2. y = 3x4
4. y=_1_
3.x²
6. f(x) = 12 + 7/3
8. y = 4x³ - 2x²+x+1
10. y=(x²-1)³
12. y = (x² + x)-2
14. y - 4(x²-6)-3
16. y = 2√x+1
18. y (x-1)³ + (x+2)*
20. y(x²+ 1)² + 3(x² - 1)²
im
36. y = ²x
8. y = (x - 1) ¹
y=
39. f(x)=3x² - 2x + 1, (1, 2)
40. f(x) = x10 + I + VI − x, (0, 2)
1/(a+h)+ g(a+h)]-[f(a)+ g(a)
h
[f(a+h)-f(a) g(a+h)-g(a)
h
h
f(a+h)-f(a)
h
-f(a)+g'(a).
In Chapter 3, the general power rule will be proved as a special case of the chain rule.
-lim
In Exercises 39 and 40, find the slope of the graph of y=f(x) at the
designated point.
-lim
g(a+h)-g(a)
h
+lim
Solutions can be found following the section exercises.
x + (x³ + 1) ¹0
2. Differentiate the function y
Limit of a sum.
41. Find the slope of the tangent line to the curve y = x³ + 3x - 8
at (2,6).
42. Write the equation of the tangent line to the curve
y=x+3x-8 at (2,6).
43. Find the slope of the tangent line to the curve y(x²-15)6
at x4. Then write the equation of this tangent line.
44. Find the equation of the
angent line to the curve
y" x²+x+2
at x = 2.
45. Differentiate the function f(x)-(3x²+x-2)² in two
ways.
(a) Use the general power rule.
(b) Multiply 3x²+x-2 by itself and then differentiate the
resulting polynomial.
46. Using the sum rule and the constant-multiple rule, show that
for any functions f(x) and g(x)
8 mua
vest
y6r+1
f(x)-8(x)=f(x)-8(x)
dx
47. Figure 2 contains the curves y = f(x) and y= g(x) and the
tangent line to y=f(x) at x = 1, with g(x)=3-f(x). Find
g(1) and g'(1).
y = f(x)
d
y = g(x)
Figure 2 Graphs of f(x) and g(x) = f(x).
x
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