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25. N=P p +9 p+r' ƏN др ƏN ди ry ƏN ƏN ди p=u + vw, g = b +uw, r = w + uv. when u=2, v = 3, w = 4 y = B² - wZB f - 2.2.

25. N=P p +9 p+r' ƏN др ƏN ди ry ƏN ƏN ди p=u + vw, g = b +uw, r = w + uv. when u=2, v = 3, w = 4 y = B² - wZB f - 2.2.

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
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Coc J 944 CHAPTER 14 Partial Derivatives 9. z In(3x + 2y), x= s sin t, 10. z = √xe, 11. z = e' cos 0, 12. z tan(u/v), 22. T = 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). x= 1+ st, y = $²² 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). (0, 0) (1, 2) r=st, 0 = √5² + 1² u=2s + 3t, v=3s-2t 2u+v' ат ат ат ap aq ar f 3 y = t cos similami novig x 25. N = P + q p+r' alib ƏN ƏN ƏN of 2n 0-du dv dw 26. u = xe", ди ди ди θα aß dy 6 9 1. P = √√√u² + v² + w², ap ap ах ду 6 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 Exercise 15 to calculate g,(1, 2) and g, (1, 2). 3 17-20 Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable. ionul a 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) fx fy 4 8 2 5 21-26 Use the Chain Rule to find the indicated partial derivatives. 21. z = x² + x²y, x= s + 2t - u, y = stu²; dz dz Əz when s = 4,1 = 2, u = 1 at du when x = 0, y = 2 u=pa√r, v=p√qr; when p = 2, q=1, r=4 23. w = xy + yz + zx, x = r cos 0, y = r sin 0, z = 0; dw dw when r = 2,0 = π/2 ar ae u = xe', v=ye", w = ey; 8 aliqual on 36. Wheat production W in a given year depends on the average temperature T and the annual rainfall R. Scientists estimate szanitob 0(that the average temperature is rising at a rate of 0.15°C/year where x = x(r, s, t), y = y(r, s, t) insq diw oldsins 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 W/R = 8. where x = x(u, v), y = y(u, v), z = z(u, v) bus where p = p(x, y, z), q = g(x, y, z), mormoT 1 - O x6 (a) What is the significance of the signs of these partial derivatives? where t = t (w, x, y, z), u = u(w, x, y, z) 08 27-30 Use Equation 6 to find dy/dx. 27. y cos x = x² + y² 29. tan '(x²y) = x + xy² 20 15 10 5 p=u+vw, when u = 2, v = 3, w = 4 x=a²ß, y = B²y, t = y²α; when a = -1, ß = 2, y = 1 DA 31-34 Use Equations 7 to find az/ax and əz/ay. 31. x² + 2y² + 3z² = 1 33. e² = xyz Kium 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 q=v+uw, r=w+4v; 37. The speed of sound traveling through ocean water with salinity 35 parts per thousand has been modeled by the equation 20 (b) Estimate the current rate of change of wheat production, dW/dt. SVEIT 30 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? 28. cos(xy) = 1 + sin y 30. e' sinx= x + xy 40 1 (min) 32. x² - y² + 2²-2z = 4 34. yz + x ln y = z² TA 16 14 12 10 8. 10 20 30 40 / (min)
Transcribed Image Text:Coc J 944 CHAPTER 14 Partial Derivatives 9. z In(3x + 2y), x= s sin t, 10. z = √xe, 11. z = e' cos 0, 12. z tan(u/v), 22. T = 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). x= 1+ st, y = $²² 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). (0, 0) (1, 2) r=st, 0 = √5² + 1² u=2s + 3t, v=3s-2t 2u+v' ат ат ат ap aq ar f 3 y = t cos similami novig x 25. N = P + q p+r' alib ƏN ƏN ƏN of 2n 0-du dv dw 26. u = xe", ди ди ди θα aß dy 6 9 1. P = √√√u² + v² + w², ap ap ах ду 6 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 Exercise 15 to calculate g,(1, 2) and g, (1, 2). 3 17-20 Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable. ionul a 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) fx fy 4 8 2 5 21-26 Use the Chain Rule to find the indicated partial derivatives. 21. z = x² + x²y, x= s + 2t - u, y = stu²; dz dz Əz when s = 4,1 = 2, u = 1 at du when x = 0, y = 2 u=pa√r, v=p√qr; when p = 2, q=1, r=4 23. w = xy + yz + zx, x = r cos 0, y = r sin 0, z = 0; dw dw when r = 2,0 = π/2 ar ae u = xe', v=ye", w = ey; 8 aliqual on 36. Wheat production W in a given year depends on the average temperature T and the annual rainfall R. Scientists estimate szanitob 0(that the average temperature is rising at a rate of 0.15°C/year where x = x(r, s, t), y = y(r, s, t) insq diw oldsins 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 W/R = 8. where x = x(u, v), y = y(u, v), z = z(u, v) bus where p = p(x, y, z), q = g(x, y, z), mormoT 1 - O x6 (a) What is the significance of the signs of these partial derivatives? where t = t (w, x, y, z), u = u(w, x, y, z) 08 27-30 Use Equation 6 to find dy/dx. 27. y cos x = x² + y² 29. tan '(x²y) = x + xy² 20 15 10 5 p=u+vw, when u = 2, v = 3, w = 4 x=a²ß, y = B²y, t = y²α; when a = -1, ß = 2, y = 1 DA 31-34 Use Equations 7 to find az/ax and əz/ay. 31. x² + 2y² + 3z² = 1 33. e² = xyz Kium 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 q=v+uw, r=w+4v; 37. The speed of sound traveling through ocean water with salinity 35 parts per thousand has been modeled by the equation 20 (b) Estimate the current rate of change of wheat production, dW/dt. SVEIT 30 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? 28. cos(xy) = 1 + sin y 30. e' sinx= x + xy 40 1 (min) 32. x² - y² + 2²-2z = 4 34. yz + x ln y = z² TA 16 14 12 10 8. 10 20 30 40 / (min)
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