(a) Describe the shape of the curve C parameterized by r(t) [0, 10π]. = (t cost, t sint, t) where tЄ (b) Find the gradient Vf of the scalar function f(x, y, z) = xex+y+z (c) Find the value of the vector line integral of Vf, the gradient field from part (b), along the curve C from part (a).

Determine the directional derivative of f(x, y, z) = tan-1 (2²) U = (-1,2,2) at P(0, 1, 1). Determine also the maximum and minimum directional derivative of f along the unit vector. where

Find the directional derivative of ø = x² + y² + z² at point (1, 2, 1) for a direction determined by dx = 2dy = -2dz.

Calculate the gradient of f(x, y) = sin(x² - y) Vf=

Let vector r(t) = <5sin(t), 5cos(t), 2t^(3/2)> be the position function of an object.a. Find vector v(t) and vector a(t), the velocity and acceleration of the object.b. Find the speed of the object.c. Find the distance the object travels on the interval [0, 1]. Also find the average speed of theobject on this interval.

Let 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.

f(x,y,z) = e-2xyz-2   ; P(ln2, ln4, ln3) ; <-2/3, 2/3, -1/3> compute the gradient at point P; Find unit vector of the gradient (U); compute directional derrivitive of the funciton at point P in the Direction of U.