- Suppose f : U → ℝ
^{q}and let x ∈ U and v be a unit vector. Show that D_{v}f= Dfv. Recall that - Let f=. Find where f is differentiable and compute the derivative at all these points.
- Let
Show that f is continuous at

and that the partial derivatives exist atbut the function is not differentiable at. - Let
Find Df

. - Let
Find Df

. - Let
Find Df

. - Let
Show that all directional derivatives of f exist at

, and are all equal to zero but the function is not even continuous at. Therefore, it is not differentiable. Why? - In the example of Problem 7 show that the partial derivatives exist but are not continuous.
- A certain building is shaped like the top half of the ellipsoid, ++= 1 determined by letting z ≥ 0. Here dimensions are measured in feet. The building needs to be painted. The paint, when applied is about .005 feet thick. About how many cubic feet of paint will be needed. Hint: This is going to replace the numbers, 900 and 400 with slightly larger numbers when the ellipsoid is fattened slightly by the paint. The volume of the top half of the ellipsoid, x
^{2}∕a^{2}+ y^{2}∕b^{2}+ z^{2}∕c^{2}≤ 1,z ≥ 0 isπabc. - Suppose r
_{1}=,r_{2}=, and r_{3}=. Find the rate of change with respect to t of the volume of the parallelepiped determined by these three vectors when t = 1. - A trash compactor is compacting a rectangular block of trash. The width is changing at the rate of −1 inches per second, the length is changing at the rate of −2 inches per second and the height is changing at the rate of −3 inches per second. How fast is the volume changing when the length is 20, the height is 10, and the width is 10?
- A trash compactor is compacting a rectangular block of trash. The width is changing at the rate of −2 inches per second, the length is changing at the rate of −1 inches per second and the height is changing at the rate of −4 inches per second. How fast is the surface area changing when the length is 20, the height is 10, and the width is 10?
- The ideal gas law is PV = kT where k is a constant which depends on the number of moles and on the gas being considered. If V is changing at the rate of 2 cubic cm. per second and T is changing at the rate of 3 degrees Kelvin per second, how fast is the pressure changing when T = 300 and V equals 400 cubic cm.?
- Let S denote a level surface of the form f= C. Show that any smooth curve in the level surface is perpendicular to the gradient.
- Suppose f is a C
^{1}function which maps U, an open subset of ℝ^{n}one to one and onto V , an open set in ℝ^{m}such that the inverse map, f^{−1}is also C^{1}. What must be true of m and n? Why? Hint: Consider Example 21.7.6 on Page 1399. Also you can use the fact that if A is an m × n matrix which maps ℝ^{n}onto ℝ^{m}, then m ≤ n. - Finish Example 21.7.5 by finding f
_{y}in terms of θ,r. Show that f_{y}= sinf_{r}+f_{θ}. ^{∗}Think of ∂_{x}as a differential operator which takes functions and differentiates them with respect to x. Thus ∂_{x}f ≡ f_{x}. In the context of Example 21.7.5, which is on polar coordinates, and Problem 16, explain how_{xx}+ u_{yy}. Use the above observation to give a formula Δu in terms of r and θ. You should get u_{rr}+u_{r}+u_{θθ}. This is the formula for the Laplacian in polar coordinates.

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