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mpute(g(x)), f(x)= x³, g(x) = 6x - 1 30. f(x)=√x, g(x)=x² + 1 31. f(x) = g(x)=1-x² where f(x) and g(x) are the following: 1 1 + √²² 8(x) = 1/ X 33. f(x)=x²-x², g(x)=x²-4 32. f(x)=- 34. f(x) = =+ x², g(x) = 1 - x* x4 X 35. f(x) = (x³ + 1)², g(x)=x² + 5 36. f(x)= x(x-2), g(x) = x³ Compute using the chain rule in formula (1). State your answer dx in terms of x only. 37. y=u³/2, u = 4x + 1

mpute(g(x)), f(x)= x³, g(x) = 6x - 1 30. f(x)=√x, g(x)=x² + 1 31. f(x) = g(x)=1-x² where f(x) and g(x) are the following: 1 1 + √²² 8(x) = 1/ X 33. f(x)=x²-x², g(x)=x²-4 32. f(x)=- 34. f(x) = =+ x², g(x) = 1 - x* x4 X 35. f(x) = (x³ + 1)², g(x)=x² + 5 36. f(x)= x(x-2), g(x) = x³ Compute using the chain rule in formula (1). State your answer dx in terms of x only. 37. y=u³/2, u = 4x + 1

Advanced Engineering Mathematics
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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Q31,33,35&Q37 These are easy questions solve all please needed to be solved correctly in 1 hour and get the thumbs up please show neat and clean work for it by hand solution needed
In Exercises 21-26, a function h(x) is defined in terms of a differen- tiable f(x). Find an expression for h'(x). 21. h(x) = f(x²) 22. h(x)=2f(2x + 1) 23. h(x) = -f(-x) 24. h(x)=f(f(x)) f(x²) 26. h(x)=√f(x²) x 27. Sketch the graph of y = 4x/(x + 1)², x>-1. 28. Sketch the graph of y = 2/(1+x²). 25. h(x) = d mpute f(g(x)), where f(x) and g(x) are the following: dx f(x)= x³, g(x) = 6x - 1 30. f(x)=√x, g(x)=x² + 1 () == √, g(x)=1-x² 31. f(x)= = 1 + ²√²² 8(x) = ² / 2 33. f(x)=x²-x², g(x)=x²-4 34. f(x)= =+ x², g(x) = 1 - x4 35. f(x) = (x³ + 1)², g(x)=x² + 5 36. f(x)= x(x-2), g(x)= x³ 32. f(x) = dy Compute dx in terms of x only. 37. y=u³/2, u = 4x + 1 38. y Vu+1, u = 2x² 2 39. y=+,u=x-x² 2 u u²+2u u+1 40. y= u Compute 43. y= using the chain rule in formula (1). State your answer dy 41. y=x²-3x, x= 1² +3, to=0 42. y = (x² - 2x + 4)², x=- u= x(x + 1) x + 1 x-1' 1+1 40=1 40=3 44. y = √x + 1, x = √t + 1, to = 0 45. Find the equation of the line tangent to the graph of y = 2x(x-4) at the point (5, 10). 46. Find the equation of the line tangent to the graph of y = at the point (1, 1). √2-x² X 47. Find the x-coordinates of all points on the curve y=(-x² + 4x - 3)³ with a horizontal tangent line. 48. The function f(x)=√x² - 6x + 10 has one relative mini- mum point for x ≥ 0. Find it. The length, x, of the edge of a cube is increasing. (a) Write the chain rule for dV dt' volume of the cube. , the time rate of change of the dV (b) For what value of x is equal to 12 times the rate of dt increase of x? 50. Allometric Equation Many relations in biology are expressed by power functions, known as allometric equations, of the form y = kx", where k and a are constants. For example, the weight of a male hognose snake is approximately 446x³ grams, where x is its length in meters. If a snake has length .4 meters and is growing at the rate of .2 meters per year, at what rate is the snake gaining weight? (Source: Museum of Natural History.) 51. Suppose that P, y, and t are variables, where P is a function of y and y is a function of t. (a) Write the derivative symbols for the following quantities: (i) the rate of change of y with respect to t; (ii) the rate of change of P with respect to y; (iii) the rate of change of P with respect to t. Select your answers from the following: dP dy dy dP dt dy' dP' dt' dt' dP' dP dt (b) Write the chain rule for 3.2 The Chain Rule 207 dy dx dQ dx dQ dx dy' dx' do dy' (b) Write the chain rule for 52. Suppose that Q, x, and y are variables, where Q is a function of x and x is a function of y. (Read this carefully.) (a) Write the derivative symbols for the following quantities: (i) the rate of change of x with respect to y; (ii) the rate of change of Q with respect to y; (iii) the rate of change of Q with respect to x. Select your answers from the following: dQ dy P = (a) Find the marginal cost, and dP dx 53. Marginal Profit and Time Rate of Change When a company produces and sells x thousand units per week, its total weekly profit is P thousand dollars, where dt dy and 200x 100+ x² The production level at t weeks from the present is x = 4 + 2r. (a) Find the marginal profit, dP (b) Find the time rate of change of profit, dt (c) How fast (with respect to time) are profits changing when t = 8? dC dx dy do 54. Marginal Cost and Time Rate of Change The cost of manufac- turing x cases of cereal is C dollars, where C= 3x + 4√x + 2. Weekly production at t weeks from the present is estimated to be x = 6200 + 100r cases. dC (b) Find the time rate of change of cost, dt (c) How fast (with respect to time) are costs rising when / = 2? 55. A Model for Carbon Monoxide Levels Ecologists estimate that, when the population of a certain city is x thousand persons, the average level L of carbon monoxide in the air above the city will be L ppm (parts per million), where L = 10 + 4x + .0001x². The population of the city is
Transcribed Image Text:In Exercises 21-26, a function h(x) is defined in terms of a differen- tiable f(x). Find an expression for h'(x). 21. h(x) = f(x²) 22. h(x)=2f(2x + 1) 23. h(x) = -f(-x) 24. h(x)=f(f(x)) f(x²) 26. h(x)=√f(x²) x 27. Sketch the graph of y = 4x/(x + 1)², x>-1. 28. Sketch the graph of y = 2/(1+x²). 25. h(x) = d mpute f(g(x)), where f(x) and g(x) are the following: dx f(x)= x³, g(x) = 6x - 1 30. f(x)=√x, g(x)=x² + 1 () == √, g(x)=1-x² 31. f(x)= = 1 + ²√²² 8(x) = ² / 2 33. f(x)=x²-x², g(x)=x²-4 34. f(x)= =+ x², g(x) = 1 - x4 35. f(x) = (x³ + 1)², g(x)=x² + 5 36. f(x)= x(x-2), g(x)= x³ 32. f(x) = dy Compute dx in terms of x only. 37. y=u³/2, u = 4x + 1 38. y Vu+1, u = 2x² 2 39. y=+,u=x-x² 2 u u²+2u u+1 40. y= u Compute 43. y= using the chain rule in formula (1). State your answer dy 41. y=x²-3x, x= 1² +3, to=0 42. y = (x² - 2x + 4)², x=- u= x(x + 1) x + 1 x-1' 1+1 40=1 40=3 44. y = √x + 1, x = √t + 1, to = 0 45. Find the equation of the line tangent to the graph of y = 2x(x-4) at the point (5, 10). 46. Find the equation of the line tangent to the graph of y = at the point (1, 1). √2-x² X 47. Find the x-coordinates of all points on the curve y=(-x² + 4x - 3)³ with a horizontal tangent line. 48. The function f(x)=√x² - 6x + 10 has one relative mini- mum point for x ≥ 0. Find it. The length, x, of the edge of a cube is increasing. (a) Write the chain rule for dV dt' volume of the cube. , the time rate of change of the dV (b) For what value of x is equal to 12 times the rate of dt increase of x? 50. Allometric Equation Many relations in biology are expressed by power functions, known as allometric equations, of the form y = kx", where k and a are constants. For example, the weight of a male hognose snake is approximately 446x³ grams, where x is its length in meters. If a snake has length .4 meters and is growing at the rate of .2 meters per year, at what rate is the snake gaining weight? (Source: Museum of Natural History.) 51. Suppose that P, y, and t are variables, where P is a function of y and y is a function of t. (a) Write the derivative symbols for the following quantities: (i) the rate of change of y with respect to t; (ii) the rate of change of P with respect to y; (iii) the rate of change of P with respect to t. Select your answers from the following: dP dy dy dP dt dy' dP' dt' dt' dP' dP dt (b) Write the chain rule for 3.2 The Chain Rule 207 dy dx dQ dx dQ dx dy' dx' do dy' (b) Write the chain rule for 52. Suppose that Q, x, and y are variables, where Q is a function of x and x is a function of y. (Read this carefully.) (a) Write the derivative symbols for the following quantities: (i) the rate of change of x with respect to y; (ii) the rate of change of Q with respect to y; (iii) the rate of change of Q with respect to x. Select your answers from the following: dQ dy P = (a) Find the marginal cost, and dP dx 53. Marginal Profit and Time Rate of Change When a company produces and sells x thousand units per week, its total weekly profit is P thousand dollars, where dt dy and 200x 100+ x² The production level at t weeks from the present is x = 4 + 2r. (a) Find the marginal profit, dP (b) Find the time rate of change of profit, dt (c) How fast (with respect to time) are profits changing when t = 8? dC dx dy do 54. Marginal Cost and Time Rate of Change The cost of manufac- turing x cases of cereal is C dollars, where C= 3x + 4√x + 2. Weekly production at t weeks from the present is estimated to be x = 6200 + 100r cases. dC (b) Find the time rate of change of cost, dt (c) How fast (with respect to time) are costs rising when / = 2? 55. A Model for Carbon Monoxide Levels Ecologists estimate that, when the population of a certain city is x thousand persons, the average level L of carbon monoxide in the air above the city will be L ppm (parts per million), where L = 10 + 4x + .0001x². The population of the city is
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