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?
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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