Question

Let f(x,y)=xy² and r(t)=( 1/2t²,t³ ).
(a) Calculate ∇f and r′(t).

Answer 1

The gradient vector ∇f of f(x,y) = xy² is (y², 2xy), and the derivative of the vector function r(t)=(1/2t²,t³) is r'(t) = (t, 3t²).

Step-by-step explanation:

The question involves calculus and vector analysis used to find the gradient vector (denoted as ∇f) of the function f(x,y) and the derivative of the vector-valued function r(t). To calculate these, we must use partial derivatives for ∇f and regular derivatives for r'(t).

To find the gradient of f(x,y) = xy², we take the partial derivatives with respect to x and y:

• ∇f = (∂f/∂x, ∂f/∂y) = (y², 2xy)

The derivative of r(t) is calculated by taking the derivative of each component:

• r'(t) = (d/dt (1/2t²), d/dt (t³)) = (t, 3t²)

Remember to perform the derivative operations according to the respective rules of differentiation.