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216 CHAPTER 3 13. x³y²-4x²=1 15. x³+³=x³3 17. x²y + y²x = 3 Techniques of Differentiation Use implicit differentiation of the equations in Exercises 19-24 to determine the slope of the graph at the given point. 19. 4³x²=-5; x-3, y = 1 20. y² = x³+1; x=2, y = -3 21. xy³=2; x = -1, y = -2 22. √x + √y=7; x = 9, y = 16 23. xy + y³ = 14; x = 3, y = 2 24. y²=3xy-5; x = 2, y = 1 25. Find the equation of the tangent line to the graph of x²4=1 at the point (4,) and at the point (4,-). 14. (x + 1)(y-1)² = 1 16. x² + 4xy + 4y = 1 18. x³y + xy = 4 26. Find the equation of the tangent line to the graph of xy² = 144 at the point (2, 3) and at the point (2, -3). 27. Slope of the Lemniscate The graph of x4 + 2x²y² + y² = 4x²-4y² is the lemniscate in Fig. 6. dy (a) Find - by implicit differentiation. dx (b) Find the slope of the tangent line to the lemniscate at (√6/2, V2/2). Figure 6 A lemniscate. 28. The graph of x² + 2x²y² + y² = 9x² - 9y² is a lemniscate similar to that in Fig. 6. (2²+ ²)² = 4²-4² (a) Find by implicit differentiation. dx (b) Find the slope of the tangent line to the lemniscate at (√5, -1). 29. Marginal Rate of Substitution Suppose that x and y represent the amounts of two basic inputs for a production process and that the equation 30x¹/32/3= 1080 describes all input amounts where the output of the process is 1080 units. Find (a) Find dy dx (b) What is the marginal rate of substitution of x for y when x = 16 and y = 54? (See Example 4.) dy dx 30. Demand Equation Suppose that x and y represent the amounts of two basic inputs for a production process and 10x¹/21/2=600. when x= 50 and y = 72. In Exercises 31-36, suppose that x and y are both differentiable functions of r and are related by the given equation. Use implicit dy determined in terms of x. y. differentiation with respect to to dx dt and 31. x+y=1 32. 14-x²=1 33. 3xy-3x²-4 34. y²=8+ xy 35. x² + 2xy = y³ 36. x²y² = 2y³ + 1 37. Point on a Curve A point is moving along the graph of x²-4y²=9. When the point is at (5,-2), its x-coordinate is increasing at the rate of 3 units per second. How fast is the y-coordinate changing at that moment? 38. Point on a Curve A point is moving along the graph of x³y²= 200. When the point is at (2, 5), its x-coordinate is changing at the rate of -4 units per minute. How fast is the y-coordinate changing at that moment? 39. Demand Equation Suppose that the price p (in dollars) and the weekly sales x (in thousands of units) of a certain commodity satisfy the demand equation 2p²+x²=4500. Determine the rate at which sales are changing at a time when x = 50, p= 10, and the price is falling at the rate of 5.50 per week. 40. Demand Equation Suppose that the price p (in dollars) and the weekly demand, x (in thousands of units) of a commodity satisfy the demand equation 6p+ x + xp = 94. How fast is the demand changing at a time when x = 4, p=9, and the price is rising at the rate of $2 per week? 41. Advertising Affects Revenue The monthly advertising rev- enue, A, and the monthly circulation, x, of a magazine are related approximately by the equation A=6Vx²-400, x ≥ 20, where A is given in thousands of dollars and x is measured in thousands of copies sold. At what rate is the advertising rev- enue changing if the current circulation is x = 25 thousand copies and the circulation is growing at the rate of 2 thousand copies per month? dA dA dx Hint: Use the chain rule - dt dx dt 42. Rate of Change of Price Suppose that in Boston the whole- sale price, p, of oranges (in dollars per crate) and the daily supply, x (in thousands of crates), are related by the equation px + 7x+8p= 328. If there are 4 thousand crates available today at a price of $25 per crate, and if the supply is changing at the rate of -3 thousand crates per day, at what rate is the price changing? 43. Related Rates Figure 7 shows a 10-foot ladder leaning against a wall. (a) Use the Pythagorean theorem to find an equation relating x and y. (b) If the foot of the ladder is being pulled along the ground at the rate of 3 feet per second, how fast is the top end of the ladder sliding down the wall at the time when the foot of the ladder is 8 feet from the wall? That is, what is at dx dt dt the time when 3 and x = 8?

216 CHAPTER 3 13. x³y²-4x²=1 15. x³+³=x³3 17. x²y + y²x = 3 Techniques of Differentiation Use implicit differentiation of the equations in Exercises 19-24 to determine the slope of the graph at the given point. 19. 4³x²=-5; x-3, y = 1 20. y² = x³+1; x=2, y = -3 21. xy³=2; x = -1, y = -2 22. √x + √y=7; x = 9, y = 16 23. xy + y³ = 14; x = 3, y = 2 24. y²=3xy-5; x = 2, y = 1 25. Find the equation of the tangent line to the graph of x²4=1 at the point (4,) and at the point (4,-). 14. (x + 1)(y-1)² = 1 16. x² + 4xy + 4y = 1 18. x³y + xy = 4 26. Find the equation of the tangent line to the graph of xy² = 144 at the point (2, 3) and at the point (2, -3). 27. Slope of the Lemniscate The graph of x4 + 2x²y² + y² = 4x²-4y² is the lemniscate in Fig. 6. dy (a) Find - by implicit differentiation. dx (b) Find the slope of the tangent line to the lemniscate at (√6/2, V2/2). Figure 6 A lemniscate. 28. The graph of x² + 2x²y² + y² = 9x² - 9y² is a lemniscate similar to that in Fig. 6. (2²+ ²)² = 4²-4² (a) Find by implicit differentiation. dx (b) Find the slope of the tangent line to the lemniscate at (√5, -1). 29. Marginal Rate of Substitution Suppose that x and y represent the amounts of two basic inputs for a production process and that the equation 30x¹/32/3= 1080 describes all input amounts where the output of the process is 1080 units. Find (a) Find dy dx (b) What is the marginal rate of substitution of x for y when x = 16 and y = 54? (See Example 4.) dy dx 30. Demand Equation Suppose that x and y represent the amounts of two basic inputs for a production process and 10x¹/21/2=600. when x= 50 and y = 72. In Exercises 31-36, suppose that x and y are both differentiable functions of r and are related by the given equation. Use implicit dy determined in terms of x. y. differentiation with respect to to dx dt and 31. x+y=1 32. 14-x²=1 33. 3xy-3x²-4 34. y²=8+ xy 35. x² + 2xy = y³ 36. x²y² = 2y³ + 1 37. Point on a Curve A point is moving along the graph of x²-4y²=9. When the point is at (5,-2), its x-coordinate is increasing at the rate of 3 units per second. How fast is the y-coordinate changing at that moment? 38. Point on a Curve A point is moving along the graph of x³y²= 200. When the point is at (2, 5), its x-coordinate is changing at the rate of -4 units per minute. How fast is the y-coordinate changing at that moment? 39. Demand Equation Suppose that the price p (in dollars) and the weekly sales x (in thousands of units) of a certain commodity satisfy the demand equation 2p²+x²=4500. Determine the rate at which sales are changing at a time when x = 50, p= 10, and the price is falling at the rate of 5.50 per week. 40. Demand Equation Suppose that the price p (in dollars) and the weekly demand, x (in thousands of units) of a commodity satisfy the demand equation 6p+ x + xp = 94. How fast is the demand changing at a time when x = 4, p=9, and the price is rising at the rate of $2 per week? 41. Advertising Affects Revenue The monthly advertising rev- enue, A, and the monthly circulation, x, of a magazine are related approximately by the equation A=6Vx²-400, x ≥ 20, where A is given in thousands of dollars and x is measured in thousands of copies sold. At what rate is the advertising rev- enue changing if the current circulation is x = 25 thousand copies and the circulation is growing at the rate of 2 thousand copies per month? dA dA dx Hint: Use the chain rule - dt dx dt 42. Rate of Change of Price Suppose that in Boston the whole- sale price, p, of oranges (in dollars per crate) and the daily supply, x (in thousands of crates), are related by the equation px + 7x+8p= 328. If there are 4 thousand crates available today at a price of $25 per crate, and if the supply is changing at the rate of -3 thousand crates per day, at what rate is the price changing? 43. Related Rates Figure 7 shows a 10-foot ladder leaning against a wall. (a) Use the Pythagorean theorem to find an equation relating x and y. (b) If the foot of the ladder is being pulled along the ground at the rate of 3 feet per second, how fast is the top end of the ladder sliding down the wall at the time when the foot of the ladder is 8 feet from the wall? That is, what is at dx dt dt the time when 3 and x = 8?

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.
Chapter3: The Derivative
Section3.CR: Chapter 3 Review
Problem 8CR
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Ex 3.3 Q13,19,21,23&Q31 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
!!!! <o > O E A₁ X INCORPORATING SOLUTION Assume that p and x are differentiable functions of t, and differentiate the demand equation with respect to t: TECHNOLOGY 1. x²-y²=1 3. 5-3x² = x dx dp dp We want to know at a time when x = 4, p = 6, and -=-.40. (The derivative 18 dt dt dt 3.3 Implicit Differentiation and Related Rates 215 dx negative because the price is decreasing.) We could solve equation (4) for and dt dx dt' then substitute the given values, but since we do not need a general formula for easier to substitute first and then solve: d d d d (p) + (2x) + (xp) = (38) dt dt dt dp dx dp dx dt dt +2 +x +p== 0. dt dt dt Check Your Understanding 3.3 Suppose that x and y are related by the equation 3y2 - 3x² + y = 1. 1. Use implicit differentiation to find a formula for the slope of the graph of the equation. EXERCISES 3.3 In Exercises 1-18, suppose that x and y are related by the given dy equation and use implicit differentiation to determine dx dx dt Thus, sales are rising at the rate of .25 thousand units (or 250 units) per week. 2. x² + y²-6=0 4. x² + (y + 3) = x² dx -40+25 +4(-40) + 6- dt dx 8 = 2 dt dx dt 215 =0 5. Substitute all specified values for the variables and their derivatives. 6. Solve for the derivative that gives the unknown rate. = 25. Suggestions for Solving Related-Rates Problems 1. Draw a picture, if possible. 2. Assign letters to quantities that vary, and identify one variable-say, t-on which the other variables depend. 3. Find an equation that relates the variables to each other. 4. Differentiate the equation with respect to the independent variable t. Use the chain rule whenever appropriate. (4) The graph of an equation in x and y can be easily obtained when y can be expressed as one or more functions of x. For instance, the graph of x² + y² = 4 can be plotted by simultaneously graphing y= V4- x² and y = -√4-x². 5. y* - x^4 = y? - xả 7. 2x³ + y = 2y³ + x 9. xy=5 11. x(y + 2)² = 8 >> Now Try Exercise 39 it is Solutions can be found following the section exercises. 2. Suppose that x and y in the preceding equation are both functions of 1. Differentiate both sides of the equation with dy respect to t, and find a formula for in terms of x, y, and dt dx dt 6. x² + y² = x² + y² 8. x4 + 4y = x - 4y³ 10. xy³ = 2 12. x²y³ = 6
Transcribed Image Text:!!!! <o > O E A₁ X INCORPORATING SOLUTION Assume that p and x are differentiable functions of t, and differentiate the demand equation with respect to t: TECHNOLOGY 1. x²-y²=1 3. 5-3x² = x dx dp dp We want to know at a time when x = 4, p = 6, and -=-.40. (The derivative 18 dt dt dt 3.3 Implicit Differentiation and Related Rates 215 dx negative because the price is decreasing.) We could solve equation (4) for and dt dx dt' then substitute the given values, but since we do not need a general formula for easier to substitute first and then solve: d d d d (p) + (2x) + (xp) = (38) dt dt dt dp dx dp dx dt dt +2 +x +p== 0. dt dt dt Check Your Understanding 3.3 Suppose that x and y are related by the equation 3y2 - 3x² + y = 1. 1. Use implicit differentiation to find a formula for the slope of the graph of the equation. EXERCISES 3.3 In Exercises 1-18, suppose that x and y are related by the given dy equation and use implicit differentiation to determine dx dx dt Thus, sales are rising at the rate of .25 thousand units (or 250 units) per week. 2. x² + y²-6=0 4. x² + (y + 3) = x² dx -40+25 +4(-40) + 6- dt dx 8 = 2 dt dx dt 215 =0 5. Substitute all specified values for the variables and their derivatives. 6. Solve for the derivative that gives the unknown rate. = 25. Suggestions for Solving Related-Rates Problems 1. Draw a picture, if possible. 2. Assign letters to quantities that vary, and identify one variable-say, t-on which the other variables depend. 3. Find an equation that relates the variables to each other. 4. Differentiate the equation with respect to the independent variable t. Use the chain rule whenever appropriate. (4) The graph of an equation in x and y can be easily obtained when y can be expressed as one or more functions of x. For instance, the graph of x² + y² = 4 can be plotted by simultaneously graphing y= V4- x² and y = -√4-x². 5. y* - x^4 = y? - xả 7. 2x³ + y = 2y³ + x 9. xy=5 11. x(y + 2)² = 8 >> Now Try Exercise 39 it is Solutions can be found following the section exercises. 2. Suppose that x and y in the preceding equation are both functions of 1. Differentiate both sides of the equation with dy respect to t, and find a formula for in terms of x, y, and dt dx dt 6. x² + y² = x² + y² 8. x4 + 4y = x - 4y³ 10. xy³ = 2 12. x²y³ = 6
<O> Q = A₁ X 216 CHAPTER 3 Techniques of Differentiation 13. x³y²-4x² = 1 15. x³ + ³ = x³y3 17. x²y + y²x = 3 Use implicit differentiation of the equations in Exercises 19-24 to determine the slope of the graph at the given point. 19. 4³x²=-5; x = 3, y = 1 20. y² = x³ + 1; x = 2, y = -3 21. xy³ = 2; x = -1, y = -2 22. √x + √ỹ= 7; x = 9, y = 16 23. xy + y³ = 14; x = 3, y = 2 24. y² = 3xy-5; x = 2, y = 1 25. Find the equation of the tangent line to the graph of x²y = 1 at the point (4,) and at the point (4, -1). 14. (x + 1)(y-1)² = 1 16. x² + 4xy + 4y = 1 18. x³y + x³ = 4 26. Find the equation of the tangent line to the graph of x4y²= 144 at the point (2, 3) and at the point (2, -3). 27. Slope of the Lemniscate The graph of x² + 2x²y² + 4 = 4x² - 4y² is the lemniscate in Fig. 6. dy (a) Find by implicit differentiation. dx (b) Find the slope of the tangent line to the lemniscate at (√6/2, √2/2). € y Figure 6 A lemniscate. 28. The graph of x4 + 2x²y² + y² = 9x² - 9y² is a lemniscate similar to that in Fig. 6. dy (a) Find by implicit differentiation. dx (b) Find the slope of the tangent line to the lemniscate at (√5, -1). (x² + y²)² = 4x² - 4y² 29. Marginal Rate of Substitution Suppose that x and y represent the amounts of two basic inputs for a production process and that the equation Find 30x¹/32/3= 1080 describes all input amounts where the output of the process is 1080 units. dy (a) Find dx (b) What is the marginal rate of substitution of x for y when x = 16 and y = 54? (See Example 4.) dy dx 30. Demand Equation Suppose that x and y represent the amounts of two basic inputs for a production process and 10x¹/21/2= 600. when x = 50 and y = 72. In Exercises 31-36, suppose that x and y are both differentiable functions of and are related by the given equation. Use implicit dy to determine in terms of x, y, dt differentiation with respect to dx dt and 31. x4+4=1 32. 14-x²=1 33. 3xy-3x²=4 34. y² = 8 + xy 35. x² + 2xy = y3 36. x²y² = 2y³ + 1 37. Point on a Curve A point is moving along the graph of x²-4y²=9. When the point is at (5,-2), its x-coordinate is increasing at the rate of 3 units per second. How fast is the y-coordinate changing at that moment? 38. Point on a Curve A point is moving along the graph of x³y² = 200. When the point is at (2, 5), its x-coordinate is changing at the rate of -4 units per minute. How fast is the y-coordinate changing at that moment? 39. Demand Equation Suppose that the price p (in dollars) and the weekly sales x (in thousands of units) of a certain commodity satisfy the demand equation 2p³ + x²=4500. Determine the rate at which sales are changing at a time when x = 50, p = 10, and the price is falling at the rate of $.50 per week. 40. Demand Equation Suppose that the price p (in dollars) and the weekly demand, x (in thousands of units) of a commodity satisfy the demand equation 6p + x + xp = 94. How fast is the demand changing at a time when x = 4, p=9, and the price is rising at the rate of $2 per week? 41. Advertising Affects Revenue The monthly advertising rev- enue, A, and the monthly circulation, x, of a magazine are related approximately by the equation A = 6V x²-400, x ≥ 20, where A is given in thousands of dollars and x is measured in thousands of copies sold. At what rate is the advertising rev- enue changing if the current circulation is x = 25 thousand copies and the circulation is growing at the rate of 2 thousand copies per month? 216 Hint: Use the chain rule AdA dx dt dx dt 42. Rate of Change of Price Suppose that in Boston the whole- sale price, p, of oranges (in dollars per crate) and the daily supply, x (in thousands of crates), are related by the equation px + 7x + 8p = 328. If there are 4 thousand crates available today at a price of $25 per crate, and if the supply is changing at the rate of -.3 thousand crates per day, at what rate is the price changing? 43. Related Rates Figure 7 shows a 10-foot ladder leaning against a wall. (a) Use the Pythagorean theorem to find an equation relating x and y. (b) If the foot of the ladder is being pulled along the ground at the rate of 3 feet per second, how fast is the top end of the ladder sliding down the wall at the time when the foot dy dt of the ladder is 8 feet from the wall? That is, what is at dx the time when = 3 and x = 8? dt Page 217
Transcribed Image Text:<O> Q = A₁ X 216 CHAPTER 3 Techniques of Differentiation 13. x³y²-4x² = 1 15. x³ + ³ = x³y3 17. x²y + y²x = 3 Use implicit differentiation of the equations in Exercises 19-24 to determine the slope of the graph at the given point. 19. 4³x²=-5; x = 3, y = 1 20. y² = x³ + 1; x = 2, y = -3 21. xy³ = 2; x = -1, y = -2 22. √x + √ỹ= 7; x = 9, y = 16 23. xy + y³ = 14; x = 3, y = 2 24. y² = 3xy-5; x = 2, y = 1 25. Find the equation of the tangent line to the graph of x²y = 1 at the point (4,) and at the point (4, -1). 14. (x + 1)(y-1)² = 1 16. x² + 4xy + 4y = 1 18. x³y + x³ = 4 26. Find the equation of the tangent line to the graph of x4y²= 144 at the point (2, 3) and at the point (2, -3). 27. Slope of the Lemniscate The graph of x² + 2x²y² + 4 = 4x² - 4y² is the lemniscate in Fig. 6. dy (a) Find by implicit differentiation. dx (b) Find the slope of the tangent line to the lemniscate at (√6/2, √2/2). € y Figure 6 A lemniscate. 28. The graph of x4 + 2x²y² + y² = 9x² - 9y² is a lemniscate similar to that in Fig. 6. dy (a) Find by implicit differentiation. dx (b) Find the slope of the tangent line to the lemniscate at (√5, -1). (x² + y²)² = 4x² - 4y² 29. Marginal Rate of Substitution Suppose that x and y represent the amounts of two basic inputs for a production process and that the equation Find 30x¹/32/3= 1080 describes all input amounts where the output of the process is 1080 units. dy (a) Find dx (b) What is the marginal rate of substitution of x for y when x = 16 and y = 54? (See Example 4.) dy dx 30. Demand Equation Suppose that x and y represent the amounts of two basic inputs for a production process and 10x¹/21/2= 600. when x = 50 and y = 72. In Exercises 31-36, suppose that x and y are both differentiable functions of and are related by the given equation. Use implicit dy to determine in terms of x, y, dt differentiation with respect to dx dt and 31. x4+4=1 32. 14-x²=1 33. 3xy-3x²=4 34. y² = 8 + xy 35. x² + 2xy = y3 36. x²y² = 2y³ + 1 37. Point on a Curve A point is moving along the graph of x²-4y²=9. When the point is at (5,-2), its x-coordinate is increasing at the rate of 3 units per second. How fast is the y-coordinate changing at that moment? 38. Point on a Curve A point is moving along the graph of x³y² = 200. When the point is at (2, 5), its x-coordinate is changing at the rate of -4 units per minute. How fast is the y-coordinate changing at that moment? 39. Demand Equation Suppose that the price p (in dollars) and the weekly sales x (in thousands of units) of a certain commodity satisfy the demand equation 2p³ + x²=4500. Determine the rate at which sales are changing at a time when x = 50, p = 10, and the price is falling at the rate of $.50 per week. 40. Demand Equation Suppose that the price p (in dollars) and the weekly demand, x (in thousands of units) of a commodity satisfy the demand equation 6p + x + xp = 94. How fast is the demand changing at a time when x = 4, p=9, and the price is rising at the rate of $2 per week? 41. Advertising Affects Revenue The monthly advertising rev- enue, A, and the monthly circulation, x, of a magazine are related approximately by the equation A = 6V x²-400, x ≥ 20, where A is given in thousands of dollars and x is measured in thousands of copies sold. At what rate is the advertising rev- enue changing if the current circulation is x = 25 thousand copies and the circulation is growing at the rate of 2 thousand copies per month? 216 Hint: Use the chain rule AdA dx dt dx dt 42. Rate of Change of Price Suppose that in Boston the whole- sale price, p, of oranges (in dollars per crate) and the daily supply, x (in thousands of crates), are related by the equation px + 7x + 8p = 328. If there are 4 thousand crates available today at a price of $25 per crate, and if the supply is changing at the rate of -.3 thousand crates per day, at what rate is the price changing? 43. Related Rates Figure 7 shows a 10-foot ladder leaning against a wall. (a) Use the Pythagorean theorem to find an equation relating x and y. (b) If the foot of the ladder is being pulled along the ground at the rate of 3 feet per second, how fast is the top end of the ladder sliding down the wall at the time when the foot dy dt of the ladder is 8 feet from the wall? That is, what is at dx the time when = 3 and x = 8? dt Page 217
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