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- Let r(t) = (sin(t), cos(t), 6 sin(t) + 9 cos(2t)). Find the projection of r(t) onto the xz-plane for -1 ≤ x ≤ 1. (Use symbolic notation and fractions where needed.) z(x) =Find F. dr for given F and C. (a) (2,4). F = -y sin(x)i + cos(x)] and C is the parabola y = x² between (0,0) and (b) F = 2yi – (sin y)] counterclockwise around the unit circle C starting at the point (1,0).Let f (x, y) = e3x-4y). Find the equation for the tangent plane to the graph of ƒ at the point (1,2). (Use symbolic notation and fractions where needed.) z =
- (8) If f(z) is analytic function expressed in polar form, show that: of (sin+icos) of Ər r ᎧᎾ f'(z)=(cos-isin). =Suppose that the temperature T at point (x, y, z) is given by T(x, y, z) = sin(xyz) and that an insect currently at the point (5, 0, 8) moves slightly in the direction parallel to the positive y-axis. As a result, the bug will feel the temperature... decrease stay the same increaseWrite the function f(x,y) = xcosh(x)cos(y) - ysinh(x)sin(y) + i(xsinh(x)sin(y) + ycosh(x)sin(y)) in terms of z
- 4. Below is a contour map for a function f(x, y) that describes the topography of a park. (a) Estimate the values of f, f and fy at the point (r, y) = (0.6, 0.4). (b) What is the equation for the tangent plane to the graph of f at the point corresponding to (x, y) = (0.6,0.4)? (c) Estimate fr, fay, fyæ and fyy at the point (x, y) = (0.6, 0.4). -187- 158 0.6 30 -28 -100 -86 0.4 -71 -57 -42 16 -13 45 So 59 74 88 30 -28 16 -13 -13 16 -13 -1 0.2 16 30 16 У о 59 74 45 -28 30 -0.2 16 -16 -13 30 16 -13 16 -0.4 -28 30 -0.6 今 -1 -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 132 117 -13 -13 - L-28 -42 -13 - -91- -13 172- 45Evaluate √(2² + yz sin(xyz))dx+(y²+xz sin(xyz))dy+(x+xysin(xyz))dz where C is the curve following the outline for the triangle from (1,0,0) to (0,1,0) to (0, 0, 1) and back to (1,0,0).6) The contour map of a function C(x, y) is shown below. Use it to answer the following questions. -6 37 -2/ 2 O i O P -2 0 -2- -1 3 d S J a. Estimate the value of C(-1,4). (Your answer does not need to be exact.) b. Is Cx (-4,2) positive, negative, or zero? c. In which direction is VC at the point (0,5)? d. C has a critical point the point labelled P, near (-1.3,0). Is this point a local minimum, local maximum, or saddle point?