Fx(a, b, c)(x 15.5 EXERCISES Preliminary Questions 1. Which of the following is a possible value of the gradient Vf of a 4. function f(x. y) of two variables? po W (a) 5 (b) (3,4) (c) (3, 4. 5) 2. True or false? A differentiable function increases at the rate |IVfell in the direction of Vfp. 3. Describe the two main geometric properties of the gradient Vf. Exercises 1. Let f(x, y) = xy2 and r(1) = ( 2,13). (a) Calculate Vf and r(t). (b) Use the Chain Rule for Paths to evaluate - f(r(1) at t = 1 and 1 = -1. 2. Let f(x, y) = exy and r(t) = (13, 1 + 1). (a) Calculate Vf and r(t). (b) Use the Chain Rule for Paths to calculate - f(r(t)). (c) Write out the composite f(r(t)) as a function of t and differentiate. Check that the result agrees with part (b). 3. Figure 14 shows the level curves of a function f(x, y) and a path r(t), traversed in the direction indicated. State whether the derivative - f(r(t)) is positive, negative, or zero at points A-D. CX -20 -10 B 0 A 10 20 30 -4 8 FIGURE 14 4. Let f(x, y) = x2 + y2 and r(t) = (cost, sint). dt (a) Find - f(r(t)) without making any calculations. Explain. (b) Verify your answer to (a) using the Chain Rule.ES nts in the direction of fastest rate of decrease, and that rate hogonal to the level curve (or surface) through P. tangent plane to the level surface F(x, y, z) = k at P = (a, b, c): VFp . ( x - a, y - b, z - c) = 0 a, b, c)(x - a) + Fy(a, b, c) (y - b) + Fz(a, b, c)(z - c) = 0 a 4. You are standing at a point where the temperature gradient vector is pointing in the northeast (NE) direction. In which direction(s) should you walk to avoid a change in temperature? (b) NW (c) SE (d) SW in (a) NE 5. What is the rate of change of f(x, y) at (0, 0) in the direction making an angle of 45 with the x-axis if Vf (0, 0) = (2, 4)? In Exercises 5-8, calculate the gradient. X 5 . f ( x , y ) = cos (x2 + y ) 6. 8(x, y) = * 2 + 1 2 d 7. h (x, y, z) = xyz-3 8. r(x, y, Z, w) = xzeyw In Exercises 9-20, use the Chain Rule to calculate - f(r(t)) at the value of t given. e. 9. f(x, y) = 3x - 7y, r(t) = (cost, sint), t =0 10. f (x, y) = 2x + 3y, r(t) = (+3, 12), 1 =-2 11. f(x, y) = x2 - 3xy, r(t) = (cost, sint), t =0 12. f(x, y) = x2 - 3xy, r(t) = (cost, sint), t = ? 13. f(x, y) = sin(xy), r(t) = (e2t, e3t), t = 0 14. f(x, y) = cos(y - x), r(t) = (et, e2t), t = In 3 15. f (x, y) = x - xy, r(t) = (12, +2 - 4t), 1=4 16. f(x, y) = 3xe-y, r(t) = (212, +2 - 21), 1= 0 17. f(x, y) = Inx + Iny, r(t) = (cost, 12), 1 = " 18. g(x, y, z) = xyez, r(t) = (12, 13, t - 1), 1 = 1 19. g(x, y, z) = xyz-1, r(t) = (et, t, 12), t = 1 20. g (x, y, z, w) = x + 2y + 3z+5w, r(t) = (12, 13, 1, 1 - 2), 1 =1 In Exercises 21-30, calculate the directional derivative in the direction of v at the given point. Remember to use a unit vector in your directional derivative computation. 21. f (x, y) = x2 + y3, v= (4, 3), P = (1, 2) 22. f(x, y) = xy3 - x2, v=i-j, P = (2, -1) 23. f(x, y) = x2y3, v=itj, P = (6,3)
