# MTH 101 Chapter 7: dis 7

Calculus II Discussion Problems (Week 7) Spring 2021

1. Let f(x, y) = 2xy −3y2. Evaluate its directional derivative at the point (1,2) in the direction of

the unit vector u=3

5i+4

5jin two diﬀerent ways:

(a) Using the deﬁnition of directional derivative.

(b) Using the gradient formula for directional derivative.

2. Let f(x, y) be a diﬀerentiable function whose graph looks like this:

Can you guess the value of its directional derivative at (0,0) in the direction of 2i+j?

3. Let f(x, y, z) = x2+y2+zand X0= (1,2,4).

(a) In which direction does fincrease most rapidly at X0?

(b) Find three directions at X0in which fhas a zero rate of change.

(c) Compute the directional derivative of fat X0in the direction of i+j−k.

(d) Consider the level surface of fpassing through the point X0. Find an equation of the tangent

plane to this surface at X0.

4. Let f(x, y) be a diﬀerentiable function on R2, and consider its tangent plane at a point (a, b, f(a, b)).

We can view this tangent plane as the graph of some function g(x, y). How are the directional

derivatives of fand gat (a, b) diﬀer from each other?

5. Let F(x, y, z) = xy +yz +zx and

x(u, v) = u+v, y(u, v) = u−v, z(u, v) = uv.

Express ∂F

∂u and ∂F

∂v in terms of uand v.

END

## Document Summary

Spring 2021: let f (x, y) = 2xy 3y2. Evaluate its directional derivative at the point (1, 2) in the direction of the unit vector u = 3. 5 j in two di erent ways: (a) using the de nition of directional derivative. (b) using the gradient formula for directional derivative: let f (x, y) be a di erentiable function whose graph looks like this: Find an equation of the tangent plane to this surface at x0: let f (x, y) be a di erentiable function on r2, and consider its tangent plane at a point (a, b, f (a, b)). We can view this tangent plane as the graph of some function g(x, y).