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25.N-+4, p-u+ow, g-отию, routur ƏN ƏN ƏN when u = 2, v = 3, w = 4 ди душps 20 02

25.N-+4, p-u+ow, g-отию, routur ƏN ƏN ƏN when u = 2, v = 3, w = 4 ди душps 20 02

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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8:34 1 944 P 9 p+r' 9. z = In(3x + 2y), x= s sin t, y = t cos silami novig 25. N = 10. z = √√√xe", x= 1+ st, y=g²p²0 lib ƏN ƏN ON wollot 28 0 du dv dw 11. z = e' cos 0, 12. z =tan(u/v), CHAPTER 14 Partial Derivatives 13. Let p(t)=f(x, y), where f is differentiable, x = g(t), y=h(t), g(2) = 4, g'(2) = -3, h(2) = 5, h'(2) = 6, f. (4, 5) = 2, f,(4, 5) = 8. Find p'(2). 14. Let R(s, t)=G(u(s, t), v(s, t)), where G, u, and v are differen- tiable, u(1, 2) = 5, u,(1, 2) = 4, u,(1, 2) = -3, v(1, 2) = 7, v,(1, 2) = 2, v,(1, 2) = 6, G,(5, 7)-9, G.(5, 7) = -2. Find R,(1, 2) and R,(1, 2). 4. P = 15. Suppose f is a differentiable function of x and y, and g(u, v)=f(e" + sin v, e" + cos v). Use the table of values to calculate g.(0, 0) and g.(0, 0). 22. T= 17. u=f(x, y), 18. w=f(x, y, z), 19. T = F(p, q, r), r=r(x, y, z) 20. R = F(t, u) V 2u + v aT ƏT ƏT ap aq ər rst, 0= √√√5² + 1² u=2s + 3t, v=3s-2t (0, 0) (1, 2) ap ap dx dy f 3 6 nitab 17-20 Use a tree diagram to write out the Chain Rule for the given ( case. Assume all functions are differentiable.pnol a asszontits where x = x(r, s, t), y = y(r, s, t) where x = x(u, v), y = y(u, v), z = 2(u, v) bus where p = p(x, y, z), q = q (x, y, z), mol SVIO: where t = t (w, x, y, z), u = u(w, x, y, z) g u² + v² + w², 6 3 21-26 Use the Chain Rule to find the indicated partial derivatives. 21. z = x + x²y x=s+ 2t-u, y = stu²; əz əz əz when s = 4, t = 2, u = 1 as at du fx 4 2 16. Suppose f is a differentiable function of x and y, and dong(r,s) = f(2r-s, s² - 4r). Use the table of values in thiqal on Exercise 15 to calculate g,(1, 2) and g, (1, 2). 36. fy 8 U= = pq√r, v= p√qr; 5 when p = 2, q=1, r=4 23. w = xy + yz + zx, x= r cos 0, y = r sin 0, z = r0; dw dw when r = 2,0 = π/2 är 20 when x = 0, y = 2 u = xe', v = yes, w=e*y; when u = 2, v = 3, w = 4 26. u = xe, x=a²B, y=B²y, t = y²α; -1, ß = 2, y = 1 ди ди ди da' aß' ay when a = 20 15 10 27-30 Use Equation 6 to find dy/dx. 27. y cos x = x² + y² 29. tan '(x²y) = x + xy² 31-34 Use Equations 7 to find az/ax and əz/ay. 31. x² + 2y² + 3z² = 1 33. e² = xyz p=u+vw, DA 5. q=v + uw, 35. The temperature at a point (x, y) is T(x, y), measured in degrees Celsius. A bug crawls so that its position after t seconds is given by x = √√1+1, y = 2+1, where x and y are mea- sured in centimeters. The temperature function satisfies T. (2, 3) = 4 and T,(2, 3) = 3. How fast is the temperature rising on the bug's path after 3 seconds? 10 Wheat production W in a given year depends on the average temperature I and the annual rainfall R. Scientists estimate that the average temperature is rising at a rate of 0.15°C/year and rainfall is decreasing at a rate of 0.1 cm/year. They also estimate that at current production levels, aw/aT = -2 16 and aW/OR = 8. x6 (a) What is the significance of the signs of these partial derivatives? 20 37. The speed of sound traveling through ocean water with salinity 2635 parts per thousand has been modeled by the equation 30 (b) Estimate the current rate of change of wheat production, dW/dt. SVEIT 40 t 28. cos(xy) = 1 + sin y 30. e' sin x = x + xy (min) 82 C = 1449.2 + 4.67 0.055T² + 0.000297³ +0.016D where C is the speed of sound (in meters per second), T is the temperature (in degrees Celsius), and D is the depth below the ocean surface (in meters). A scuba diver began a leisurely dive into the ocean water; the diver's depth and the surrounding water temperature over time are recorded in the following graphs. Estimate the rate of change (with respect to time) of the speed of sound through the ocean water experienced by the diver 20 minutes into the dive. What are the units? r=w+uv, 32. x² - y² +2²-2z = 4 34. yz + x ln y = z² TA 16 14- 12 10- 8. × 10 20 30 十二 40 (min)
Transcribed Image Text:8:34 1 944 P 9 p+r' 9. z = In(3x + 2y), x= s sin t, y = t cos silami novig 25. N = 10. z = √√√xe", x= 1+ st, y=g²p²0 lib ƏN ƏN ON wollot 28 0 du dv dw 11. z = e' cos 0, 12. z =tan(u/v), CHAPTER 14 Partial Derivatives 13. Let p(t)=f(x, y), where f is differentiable, x = g(t), y=h(t), g(2) = 4, g'(2) = -3, h(2) = 5, h'(2) = 6, f. (4, 5) = 2, f,(4, 5) = 8. Find p'(2). 14. Let R(s, t)=G(u(s, t), v(s, t)), where G, u, and v are differen- tiable, u(1, 2) = 5, u,(1, 2) = 4, u,(1, 2) = -3, v(1, 2) = 7, v,(1, 2) = 2, v,(1, 2) = 6, G,(5, 7)-9, G.(5, 7) = -2. Find R,(1, 2) and R,(1, 2). 4. P = 15. Suppose f is a differentiable function of x and y, and g(u, v)=f(e" + sin v, e" + cos v). Use the table of values to calculate g.(0, 0) and g.(0, 0). 22. T= 17. u=f(x, y), 18. w=f(x, y, z), 19. T = F(p, q, r), r=r(x, y, z) 20. R = F(t, u) V 2u + v aT ƏT ƏT ap aq ər rst, 0= √√√5² + 1² u=2s + 3t, v=3s-2t (0, 0) (1, 2) ap ap dx dy f 3 6 nitab 17-20 Use a tree diagram to write out the Chain Rule for the given ( case. Assume all functions are differentiable.pnol a asszontits where x = x(r, s, t), y = y(r, s, t) where x = x(u, v), y = y(u, v), z = 2(u, v) bus where p = p(x, y, z), q = q (x, y, z), mol SVIO: where t = t (w, x, y, z), u = u(w, x, y, z) g u² + v² + w², 6 3 21-26 Use the Chain Rule to find the indicated partial derivatives. 21. z = x + x²y x=s+ 2t-u, y = stu²; əz əz əz when s = 4, t = 2, u = 1 as at du fx 4 2 16. Suppose f is a differentiable function of x and y, and dong(r,s) = f(2r-s, s² - 4r). Use the table of values in thiqal on Exercise 15 to calculate g,(1, 2) and g, (1, 2). 36. fy 8 U= = pq√r, v= p√qr; 5 when p = 2, q=1, r=4 23. w = xy + yz + zx, x= r cos 0, y = r sin 0, z = r0; dw dw when r = 2,0 = π/2 är 20 when x = 0, y = 2 u = xe', v = yes, w=e*y; when u = 2, v = 3, w = 4 26. u = xe, x=a²B, y=B²y, t = y²α; -1, ß = 2, y = 1 ди ди ди da' aß' ay when a = 20 15 10 27-30 Use Equation 6 to find dy/dx. 27. y cos x = x² + y² 29. tan '(x²y) = x + xy² 31-34 Use Equations 7 to find az/ax and əz/ay. 31. x² + 2y² + 3z² = 1 33. e² = xyz p=u+vw, DA 5. q=v + uw, 35. The temperature at a point (x, y) is T(x, y), measured in degrees Celsius. A bug crawls so that its position after t seconds is given by x = √√1+1, y = 2+1, where x and y are mea- sured in centimeters. The temperature function satisfies T. (2, 3) = 4 and T,(2, 3) = 3. How fast is the temperature rising on the bug's path after 3 seconds? 10 Wheat production W in a given year depends on the average temperature I and the annual rainfall R. Scientists estimate that the average temperature is rising at a rate of 0.15°C/year and rainfall is decreasing at a rate of 0.1 cm/year. They also estimate that at current production levels, aw/aT = -2 16 and aW/OR = 8. x6 (a) What is the significance of the signs of these partial derivatives? 20 37. The speed of sound traveling through ocean water with salinity 2635 parts per thousand has been modeled by the equation 30 (b) Estimate the current rate of change of wheat production, dW/dt. SVEIT 40 t 28. cos(xy) = 1 + sin y 30. e' sin x = x + xy (min) 82 C = 1449.2 + 4.67 0.055T² + 0.000297³ +0.016D where C is the speed of sound (in meters per second), T is the temperature (in degrees Celsius), and D is the depth below the ocean surface (in meters). A scuba diver began a leisurely dive into the ocean water; the diver's depth and the surrounding water temperature over time are recorded in the following graphs. Estimate the rate of change (with respect to time) of the speed of sound through the ocean water experienced by the diver 20 minutes into the dive. What are the units? r=w+uv, 32. x² - y² +2²-2z = 4 34. yz + x ln y = z² TA 16 14- 12 10- 8. × 10 20 30 十二 40 (min)
3 О 19. ди Ət 1 дх : at ат ду ӘТ əz 21. ar 29. 31. SI || || Әх ди дх дr ər + X дх др Әх ат др ӘТ Әт др др ду ӘТ др др дz 1582, 3164, -700 + + ди ду ду д + ay ar' a + ӘТ dq дадх 55 25. 14, -96, 14 27. ат да да ду от да + ат dr Әт + да дz д 23. 2x + + 4..4 - 1 + x+y2 + y2 + x+y x² - 2xy - 2x³y 5..3 2y 37 33. COS X е eº У
Transcribed Image Text:3 О 19. ди Ət 1 дх : at ат ду ӘТ əz 21. ar 29. 31. SI || || Әх ди дх дr ər + X дх др Әх ат др ӘТ Әт др др ду ӘТ др др дz 1582, 3164, -700 + + ди ду ду д + ay ar' a + ӘТ dq дадх 55 25. 14, -96, 14 27. ат да да ду от да + ат dr Әт + да дz д 23. 2x + + 4..4 - 1 + x+y2 + y2 + x+y x² - 2xy - 2x³y 5..3 2y 37 33. COS X е eº У
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