Linear Approximation, Gradient, and Directional Derivatives Summary

Potential Test Questions from Sections 14.4 and 14.5

1. Write the linear approximation (aka, the tangent plane) for the given function at the givenpoint:

(a) f(x, y) =x2

y2 + 1at the point (4, 1)

(b) g(x, y) = ex2+y at the point (0, 0)

2. Use your previous answers to give pencil-and-paper approximations for f(0.01,−0.02) andg(4.01, 0.98). Use a calculator to compare each answer to the actual value.

3. Use the Chain Rule to evaluated

dtf(c(t)) in each case below:

(a) f(x, y) = x2 − 3xy and c(t) = 〈cos t, sin t〉

(b) g(x, y, z) = xyz−1 and c(t) = 〈et, t, t2〉

4. In each of the following cases, calculate the derivative of the given function with respect to vand the directional derivative of f in the direction of v at the given point P .

(a) f(x, y) = x2y3, v = 〈1, 1〉, and P = (16, 3).

(b) g(x, y, z) = z2 − xy2, v = 〈−1, 2, 2〉, and P = (2, 1, 3)

5. Find the equation of the plane tangent to the surfce x2 + 3y2 + 4z2 = 20 at the point P =(2, 2, 1).