Advanced Engineering Mathematics
6th Edition
ISBN: 9781284105902
Author: Dennis G. Zill
Publisher: Jones & Bartlett Learning
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Chapter 9.4, Problem 13E
To determine
To calculate: the first partial derivatives of the given function
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In Problems 11–20, for the given functions f and g. find:
(a) (f° g)(4)
(b) (g•f)(2)
(c) (fof)(1)
(d) (g ° g)(0)
\ 11. f(x) = 2x; g(x) = 3x² + 1
12. f(x) = 3x + 2; g(x) = 2x² – 1
1
13. f(x) = 4x² – 3; g(x) = 3
14. f(x) = 2x²; g(x) = 1 – 3x²
15. f(x) = Vx; 8(x) = 2x
16. f(x) = Vx + 1; g(x) = 3x
%3D
1.
17. f(x) = |x|; g(x) =
18. f(x) = |x – 2|: g(x)
x² + 2
2
x + 1
x² + 1
19. f(x) =
3
8(x) = Vĩ
20. f(x) = x³/2; g(x) =
X + 1'
In Problems 2–4, for the given functions fand g find:
(a) (f° g) (2)
(b) (g • f)(-2)
(c) (fo f) (4)
(d) (g ° 8) (-1)
2. f(x) = 3x – 5; g(x) = 1 – 2r
3. f(x) = Vx + 2: g(x) = 2x² + 1
4. f(x) = e"; g(x) = 3x – 2
In Problems 39–46, show that (f ° g) (x) = (g° f) (x) = x.
%3D
39. f(x) = 2x; g(x) =
40. f(x)
= 4x; g(x) = i*
41. f(x) = x; g(x)
%3!
%3D
43. f(x) = 2x – 6; 8(x) = ; (x + 6)
46. fl+) = s(*) =
42. f(x) = x + 5; g(x) = x - 5
44. f(x) = 4 – 3x; g(x) = (4 - x)
%3D
45. f(x) = ax + b; g(x) = - (x - b) a + 0
%3D
a
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- In Problems 27–36, verify that the functions f and g are inverses of each other by showing that f(g(x)) = x and g(f(x)) any values of x that need to be excluded. = x. Give 27. f(x) = 3x + 4; g(x) = (x- 4) 28. f(x) = 3 – 2x; g(x) = -(x – 3) 29. f(x) = 4x – 8; 8(x) = + 2 30. f(x) = 2x + 6; 8(x) = ;x - 3 31. f(x) = x' - 8; g(x)· Vx + 8 32. f(x) = (x – 2)², 2; g(x) = Vĩ + 2 33. f(x) = ; 8(x) = 34. f(x) = x; g(x) x - 5 2x + 3' 2x + 3 4x - 3 3x + 5 35. f(x) *: 8(x) = 8(x) 36. f(x) = 1- 2x x + 4 2 - x 1.7 82 CHAPTER 1 Graphs and Functions In Problems 37-42, the graph of a one-to-one function f is given. Draw the graph of the inverse function f"1. For convenience (and as a hint), the graph of y = x is also given. 37. y= X 38. 39. y =X 3 (1, 2), (0, 1) (-1,0) (2. ) (2, 1) (1, 0) 3 X (0, -1) -3 (-1, -1) 3 X -3 (-2, -2) (-2, -2) -하 -하 -하 40. 41. y = x 42. y = X (-2, 1). -3 3 X (1, -1)arrow_forward4. Suppose the following functions are a general solution of: y(4) + a3y" +a2y" + a1y' + a0y = 0arrow_forward2. Find if y=x +3x-7 and x 21+1. dtarrow_forward
- For each dif erential equation in Problems 1–21, find the general solutionby finding the homogeneous solution and a particular solution. Please DO NOT YOU THE PI method where 1/f(r) * x. Dont do that. Instead do this, assume for yp = to something, do the 1 and 2 derivative of it and then plug it in the equation to find the answer.arrow_forwardProblem 4. Find the total derivative of the following function: df af of dy + dr Dy dr F(x,y) = (x²-3y)-(x+y³) if y = 2x² + 3xarrow_forwardIn Problems 33–44, determine algebraically whether each function is even, odd, or neither. 34. f(x) = 2x* –x? 38. G(x) = Vĩ 33. f(x) = 4x 37. F(x) = V 35. g(x) = -3x² – 5 39. f(x) = x + |x| 36. h (х) — Зx3 + 5 40. f(x) = V2r²+ 1 x² + 3 -x 42. h(x) =- 1 2x 44. F(x) 41. g(x) 43. h(x) x2 - 1 3x2 - 9arrow_forward
- 2. Let P(t) represent the population of Los Angeles t years after 1900. (a) Interpret P(10) = 319, 198 in words. P(10) - Р(0) (b) Given that P(0) = 102, 479 and P(10) = 319, 198, calculate and interpret 10 – 0 in words. (c) of Los Angeles reached 200,000. Set up an equation that could be used to find how many years after 1900 the populationarrow_forward23. What is the domain of the function f(x) = Vx² – 16? %3D In Problems 25–32, use the given functions f and g. (a) Solve f(x) = 0. (e) Solve g(x) s 0. (b) Solve g(x) = 0. (f) Solve f(x) >g(x).arrow_forwardProblem 5: Find all first partial derivatives for the function f(x, y) = х — хуarrow_forward
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