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- EXERCISE 2 Consider f(x, y) = 3x² – 4y² and - U = cos T î + sin 7 ĵ. Use the Theorem above to find the directional derivative of f in the direction of the Ū. Note that Djf(x, y) = (cos 0 î + sin 6 j) · [fa(x, y) î + fy(x, y) )- 5arrow_forwardPlease provide with proper handwritingarrow_forwardaf of f(x, y) = tan (x²y³+ 2xy)? ду What is thearrow_forward
- Definition 3.27 Let P(x, y) be a point in R² and 0 be the radian measure of the angle formed by OP (O is the origin) and the positive side of the x-axis. Suppose f is a function of the two variables x and y and U cos 0 î + sin 0 ĵ. The directional derivative of ƒ in the direction of U, denoted by D7#f, is given by f(x + h cos 0, y + h sin 0) – f(x, y) DGf(r, y) = lim h→0 h if this limit exists. Note that the directional derivative is a scalar (a real number). The partial derivatives fr and fy are special cases of the directional derivative. 1 Remark (a) If U = î, then cos 0 = 1 and sin 0 = 0. Thus, f(r + h, y) – f(x,y) D;f(x, y) = lim h-0 h which is the partial derivative of f with respect to x. (6) If U = î, then cos 0 = 0 and sin 0 = 1. Thus, f(x, y + h) – f(r, y) D;f(x, y) = lim h→0 h which is the partial derivative of f with respect to y.arrow_forwardUse the definition of directional derivative to solve Duf(xo, Yo, zo) at point P(xo, Yozo) where f(x, y, z) = x² + y² + z² + 2xz and u = = (1,0, 1) at P(1, 1, 1).arrow_forwardLet u(t) = 2t'i+ (P -7)j-8k and v(t) = e'i+3 ej- ek. Compute the derivative of the following function. 3t L u(t) • v(t) Select the correct choice below and fill in the answer box(es) to complete your choice. O A. The derivative is the scalar function 訓 這 O B. The derivative is the vector-valued function (i+ (Dj+ ( k.arrow_forward
- [4ху + 2х 2x2 + 1 5) Let F(x, y) = Is F conservative? If so, give a potential function for F. 6) Let C be the curve parametrized by (1 + sin(t),t³), where 0arrow_forwardSex. (x + y)ds where C is the straight-line segment x = 3t, y = (3 - 3t), z = 0 from C (0,3,0) to (3,0,0). Evaluate [(x+y)ds = C (Type an exact answer.)arrow_forward5. Find the directional derivative of the scalar function of (x, y, z) = xyz in the direction of the outer 27 normal to the surface z = xy at the point (3, 1, 3). Ans.arrow_forwardarrow_back_iosarrow_forward_ios
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