Thomas’ Calculus Early Transcendentals 13ed

Homework Assignment for Chapter 14

1. Section 14.2: Problems 41, 43, 48, 49, 51, 57, 61, 63, 67.

Hint for problems 61, 63: Read ‘Changing to Polar Coordinates’ on page 781. Note that in equation (1), page 781, “|r| < δ” should be “0 < |r| < δ” instead.

Hint for problem 49: Find two paths that pass through (1, 1).

2. Section 14.3: Problems 19, 21, 53, 60, 65, 67, 69, 81, 91.

3. Show that if g(x, y) = ε_{1} · (x − x_{0}) + ε_{2} · (y − y_{0}) as (x, y) → (x_{0}, y_{0}) then g(x, y) = ε · \sqrt{(x − x_{0})^{2} + (y − y_{0})^{2}} and vice versa (the converse). Here lim_{(x,y)→(x_{0},y_{0})}(ε_{1}, ε_{2}, ε) = (0, 0, 0).

Remark: A brief version of the above statement reads:

Show that o(1)·Δx+o(1)·Δy = o(1)·\sqrt{Δx^{2} + Δy^{2}} where the o(1)’s refer to 2D limits as (Δx, Δy) → (0, 0).

Hint: \sqrt{Δx^{2} + Δy^{2}} = \frac{Δx}{\sqrt{Δx^{2}+Δy^{2}}} Δx + \frac{Δy}{\sqrt{Δx^{2}+Δy^{2}}} Δy

4. Section 14.4: Problems 1, 7, 10, 21, 24, 29, 31, 43, 51.

5. Suppose that F (x, y, z) = 0 can implicitly define x = f(y, z), or y = g(z, x), or z = h(x, y) near some point (x_{0}, y_{0}, z_{0}) with F (x_{0}, y_{0}, z_{0}) = 0. (for example, F (x, y, z) = x + 2y + 3z − 4 can). Show that, for any such point (x_{0}, y_{0}, z_{0}), we have

\frac{∂f}{∂y} \frac{∂g}{∂z} \frac{∂h}{∂x} = \frac{∂f}{∂z} \frac{∂g}{∂x} \frac{∂h}{∂y} = −1

6. Section 14.5: Problems 9, 15, 19, 25, 27, 29, 35, 36, 40 (See page 807).

7. Let f(x, y) = x^{2}y/(x^{2} + y^{2}) for (x, y) ≠ (0, 0) and f(0, 0) = 0.

(a) Is f continuous at (0, 0)?

(b) Do f_{x} and f_{y} exist at (0, 0)?

(c) Evaluate the directional derivative of f at (x_{0}, y_{0}) = (0, 0) in the direction (cos θ, sin θ), i.e. df/ds_{(0,0),(cos θ,sin θ)}, if it exists.

(d) Is f differentiable at (0, 0)?

8. Section 14.6: Problems 1, 9, 13, 19, 25, 33, 39, 45, 53, 55, 58, 57.

9. Section 14.7: Problems 1, 19, 31, 35, 39, 43, 44, 49, 51, 55, 61.

Hint for problem 44: Do this problem using the gradient analysis. That is, plot ∇f near the critical point to determine whether the critical point is a local minimum, local maximum or neither.

10. Section 14.8: Problems 1, 23, 27, 37, 39.

11. Section 14.9: Problems 7, 9, 11.

12. Taylor’s formula for functions of 2 variables can be summarized as

f(x_{0} + Δx, y_{0} + Δy) = f(x_{0}, y_{0}) + ∑_{k=1}^{n} \frac{1}{k!}(Δx∂_{x} + Δy∂_{y})^{k}f(x_{0}, y_{0})

+\frac{1}{(n + 1)!}(Δx∂_{x} + Δy∂_{y})^{n+1}f(x_{0} + cΔx, y_{0} + cΔy), 0 < c < 1.

Derive a similar formula for functions of 3 variables.

13. Section 14.10: Problems 3, 7, 9, 12.

14. Continue on problem 12: Give a formula for (\frac{∂u}{∂x})_{y} where u = u(x, y, z, w) and f(x, y, z, w) = 0, g(x, y, z, w) = 0.

15. Follow up on example 5 of section 14.8: Let f(x, y, z) = x^{2} + y^{2} + z^{2} and compute \frac{df}{dx} at P_{1} = (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}, 1 − \sqrt{2}), subject to the two constraints g_{1}(x, y, z) = x^{2} + y^{2} − 1 = 0 and g_{2}(x, y, z) = x + y + z − 1 = 0. That is, evaluate \frac{d}{dx}f(x, y(x), z(x)) at x = \frac{\sqrt{2}}{2}, where y = y(x) and z = z(x) are implicitly given by the constraints g_{1} = 0 and g_{2} = 0. Can you explain why you get \frac{df}{dx} = 0 there?