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In Exercises 21-26, a function /(x) is defined in terms of a differen- tiable f(x). Find an expression for h'(x). 21. h(x)=f(x²) 23. h(x) = -f(-x) f(x²) 25. h(x)=1 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). mpute f(g(x)), where f(x) and g(x) are the following: A f(x) = x³, g(x) = 6x - 1 30. f(x)=√x, g(x) = x² + 1 31. f(x) = g(x)=1-x² 32. f(x) = =, 8(x) = 1/2 X 33. f(x)=x²-x², g(x)=x²-4 34. f(x)==+x2², g(x) = 1 - x4 f(x) = 4 x 35. f(x) = (x³ + 1)², g(x) = x² + 5 36. f(x)= x(x-2), g(x) = x³ 39. y= dy Compute using the chain rule in formula (1). State your answer dx in terms of x only. 37. y=u³/2, u = 4x + 1 38. y = Vu+1, u = 2x² 2 +-,u=x-x² 40. y = 1+Vx u Compute 22. h(x) = 2f(2x + 1) 24. h(x)=f(f(x)) u ²+2u u²+ u+1 -, u = x(x + 1) (b) For what value of x is equal to 12 times the rate of 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 1. (a) Write the derivative symbols for the following quantities: (i) the rate of change of y with respect to r; (ii) the rate of change of P with respect to y; (iii) the rate of change of P with respect to 1. Select your answers from the following: dP dy dy dP dt dy dp di dt' dP' (b) Write the chain rule for- dP di dy dx do dx do dx' dy dx' do' dy' 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: (b) Write the chain rule for and do dy dt dy and dy do 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 P= 200x 100+ x² The production level at / weeks from the present is x = 4 + 21.

In Exercises 21-26, a function /(x) is defined in terms of a differen- tiable f(x). Find an expression for h'(x). 21. h(x)=f(x²) 23. h(x) = -f(-x) f(x²) 25. h(x)=1 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). mpute f(g(x)), where f(x) and g(x) are the following: A f(x) = x³, g(x) = 6x - 1 30. f(x)=√x, g(x) = x² + 1 31. f(x) = g(x)=1-x² 32. f(x) = =, 8(x) = 1/2 X 33. f(x)=x²-x², g(x)=x²-4 34. f(x)==+x2², g(x) = 1 - x4 f(x) = 4 x 35. f(x) = (x³ + 1)², g(x) = x² + 5 36. f(x)= x(x-2), g(x) = x³ 39. y= dy Compute using the chain rule in formula (1). State your answer dx in terms of x only. 37. y=u³/2, u = 4x + 1 38. y = Vu+1, u = 2x² 2 +-,u=x-x² 40. y = 1+Vx u Compute 22. h(x) = 2f(2x + 1) 24. h(x)=f(f(x)) u ²+2u u²+ u+1 -, u = x(x + 1) (b) For what value of x is equal to 12 times the rate of 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 1. (a) Write the derivative symbols for the following quantities: (i) the rate of change of y with respect to r; (ii) the rate of change of P with respect to y; (iii) the rate of change of P with respect to 1. Select your answers from the following: dP dy dy dP dt dy dp di dt' dP' (b) Write the chain rule for- dP di dy dx do dx do dx' dy dx' do' dy' 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: (b) Write the chain rule for and do dy dt dy and dy do 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 P= 200x 100+ x² The production level at / weeks from the present is x = 4 + 21.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter4: Calculating The Derivative
Section4.CR: Chapter 4 Review
Problem 5CR: Determine whether each of the following statements is true or false, and explain why. The chain rule...
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Ex 3.2 Q31, Q33,Q35&Q37 needed These are easy questions please solve all mentioned above Needed to be solved correctly in 30 minutes and get the thumbs up please show me 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) 23. h(x) = -f(-x) S(x²) 25. h(x)=1 26. h(x)=√(x²) 27. Sketch the graph of y= 4x/(x + 1)², x>-1. 28. Sketch the graph of y = 2/(1+x²). 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 f(x) = g(x)=1-x² 31. 1 1 + √² 8(x) = ! x 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=u²³/2, u = 4x + 1 38. y Vu+ I, u = 2x² +=₁u=x-x² 39. y= 40. y = Compute u 2 2 43. using the chain rule in formula (1). State your answer 22. h(x) = 2f(2x + 1) 24. h(x)=f(f(x)) u ²+2u u+1 -to u= x(x + 1) 41. y=x²-3x, x= 1² +3, to=0 42. y = (x² - 2x + 4)², x=- v = x + 1₁ x = — -₁ 10 = x-1 1 to 1 1+1' 44. y = √x + 1, x = V1 + 1, to = 0 45. Find the equation of the line tangent to the graph of y = 2x(x-4)6 at the point (5, 10). 46. Find the equation of the line tangent to the graph of at the point (1, 1). y = V2-x² 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 1. Select your answers from the following: 3.2 The Chain Rule 207 dP dy dy dP dt dy dP' dp di dt dP dt (b) Write the chain rule for dy dx do dx do 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: do dy and dt dy and do 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 P = dC (a) Find the marginal cost, dx 200x 100+x² The production level at t weeks from the present is x = 4 + 2t. dP dx (a) Find the marginal profit, dP (b) Find the time rate of change of profit, t (c) How fast 1 = 8? respect to time) are profits changing when 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 t = 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) 23. h(x) = -f(-x) S(x²) 25. h(x)=1 26. h(x)=√(x²) 27. Sketch the graph of y= 4x/(x + 1)², x>-1. 28. Sketch the graph of y = 2/(1+x²). 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 f(x) = g(x)=1-x² 31. 1 1 + √² 8(x) = ! x 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=u²³/2, u = 4x + 1 38. y Vu+ I, u = 2x² +=₁u=x-x² 39. y= 40. y = Compute u 2 2 43. using the chain rule in formula (1). State your answer 22. h(x) = 2f(2x + 1) 24. h(x)=f(f(x)) u ²+2u u+1 -to u= x(x + 1) 41. y=x²-3x, x= 1² +3, to=0 42. y = (x² - 2x + 4)², x=- v = x + 1₁ x = — -₁ 10 = x-1 1 to 1 1+1' 44. y = √x + 1, x = V1 + 1, to = 0 45. Find the equation of the line tangent to the graph of y = 2x(x-4)6 at the point (5, 10). 46. Find the equation of the line tangent to the graph of at the point (1, 1). y = V2-x² 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 1. Select your answers from the following: 3.2 The Chain Rule 207 dP dy dy dP dt dy dP' dp di dt dP dt (b) Write the chain rule for dy dx do dx do 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: do dy and dt dy and do 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 P = dC (a) Find the marginal cost, dx 200x 100+x² The production level at t weeks from the present is x = 4 + 2t. dP dx (a) Find the marginal profit, dP (b) Find the time rate of change of profit, t (c) How fast 1 = 8? respect to time) are profits changing when 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 t = 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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