Exercises

01

Compute gradient

For each of the following functions, compute the gradient:

• (a) $f(x,y)=x^2-3xy$
• (b) $f(x,y)=\cos (y-x)$
• (c) $g(x,y,z,w)=xz{e}^{yw}$.

02

Derivative using gradient

Let $g(x,y,z)=z^2-x{y}^2$. Find $D_{\mathbf{v}}(f)$ at the point $(2,1,3)$, where $\mathbf{v}=(-1,2,2)$, by first computing the gradient and evaluating a dot product.

03

Direction with given slope

You are hiking on a mountain with altitude in meters given by the function:

$$f(x,y)=2500+100(x+y^2)e^{-0.3y^2}.$$

You are at the point $(-1,-1)$. Which two directions could you head (as an angle $\theta$ from the $+x$-axis) in order to ascend at a 20% grade?