6. [12 Points] DETAILS LARCALCET7 13.6.035. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Find the gradient of the function at the given point. Function Point fIX. y, 2): VX2+y2+z2 (3,9, 3) Vf(8, 9, 3) = Find the maximum value of the directional derivative at the given point. Need Help? _ 7. [/4 Points] DETAILS LARCALCET7 13.6.046. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Consider the following. f(x, y) = 9x2 + 4y2 C = 85, P(3, 1) (a) Find the gradient of the function at P. Vf(3,1)= (b) Find a unit vector that is normal to the level curve f(x, y) = C at P. (c) Find the tangent line to the level curve f(x, y) = c at P. 1. [-/4 Points] DETAILS LARCALC10 13.5.014. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Find aw/as and aw/at using the appropriate Chain Rule. Function Values w = y' - 3xzy s = -1, t = 2 x = es, y = et aw as aw at Evaluate each partial derivative at the given values of s and t. as aw at 2. [-/3 Points] DETAILS LARCALC10 13.5.034.MI. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Differentiate implicitly to find the first partial derivatives of w. x2 + 12 + 22 - 7yw + 4w2 = 3 ax dw ay Ow az Need Help? Master It 3. [-/2 Points] DETAILS LARCALCET7 13.5.012. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Consider the following w = xy2 + x2z+ yz2, x = t2, y = 5t, z=5 (a) Find dw/dt using the appropriate Chain Rule. dw dt (b) Find dw/dt by converting w to a function of t before differentiating. dw dt Need Help? Read It1. [-l1 Points] DETAILS LARCALC10 13.6.002. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Use Theorem 13.9 to find the directional derivative of the function at P in the direction of the unit vector u = cos 6i + sin Bj. 7T f(x, y) = , P(2, 0), 9 = _ E 2. [-/1 Points] DETAILS LARCALC10 13.6.006. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Find the directional derivative of the function at P in the direction of v. f x. y) = x3 y3. P(8, 3), v = _2 (i + j) 2 3. [-11 Points] DETAILS LARCALC10 13.6.013. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Find the gradient of the function at the given point. r(x, y) = 4x + 2y2 + 3, (4, 5) Vf(4, 5) = 4. [-l1 Points] DETAILS LARCALC10 13.6.022. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Find the directional derivative of the function at P in the direction of v. f(X, y, z) =xy+ yz+xz, P(8, 9, 8), v = 16i +j - k I: 5. [l1 Points] DETAILS LARCALCET7 13.6.026.MI. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER . Use the gradient to find the directional derivative of the function at P in the direction of PQ. f(x, y) = 3x2 - y2 + 4, P(1, 1), Q(6, 8) I: Need Help? _'_' 8. [-l1 Points] DETAILS LARCALCET7 13.6.053.MI. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER The surface of a mountain is modeled by the equation h(x, y) = 7000 0.001x2 0.004y2. A mountain climber is at the point (700, 700, 4550). In what direction should the climber move in order to ascend at the greatest rate? Need Help? _@ 9. [-/2 Points] DETAILS LARCALCET7 13.6.058. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER The temperature in degrees Celsius on the surface of a metal plate is given by T(x, y), where X and y are measured in centimeters. Find the direction from point P where the temperature increases most rapidly. T(X, y) = 45 x2 5y2, P(3, 1) =ind the rate of increase. (Round your answer to two decimal places.) per centimeter Need Help? _