Consider again the vector field F(x, y) = (18y² x, 18yz² + 4e¹"). Find a parameterisation of a curve C such that the circulation of F along C is equal to e¹ +8. A curve for this circulation has parameterisation r(t) = Use square brackets for vectors (be careful NOT to use an interval from the palette). for 0 ≤t≤1

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.

linearise R = AT + BT2

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.

Given the vector-valued function below, solve for the following:a. Domain of Rb. Show that R is continuous at t = 2.c. If f(t) = t^2, evaluate (R◦f)'(-1)

Please help with steps!

Q8..

Suppose that the directional derivative of a function f at a point Pin the direction of the vector ū = (-1,0) is - Relate this 4 value to the behaviour of f. A. From the point P, the function f is increasing at a rate of - in the positive x-direction. B. From the point P, the function f is increasing at a rate of - in the positive y-direction. C. From the point P, the function f is decreasing at a rate of - in the negative y-direction.

Suppose that f(x, y) = x³y². Find the directional derivative of f(x, y) in the direction of (-3, 1) at the point (-2,2). Note that the direction is not a unit vector.

Please do the graph