Calculus II, Spring 2019 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 11

1. Section 14.7: Problems 31, 35, 43, 44.

Hint: You can use the method of gradient analysis introduced in 190430’s class. That is, plot the gradient vector ∇f near a critical point to determine whether the critical point is a local minimum, local maximum or neither.

2. Section 14.8: Problems 1, 23, 27, 33, 35.

3. Section 14.9: Problems 7, 9, 10, 11.

Hint: An alternative method for problem 9, 10: try Taylor’s formula for \frac{1}{1−z} with appropriate choice of z. For problem 11: read example 1 in page 840.

4. Section 14.9:

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.

(Note: the textbook uses the notations a, b, h, k in place of x_{0}, y_{0}, Δx, Δy). Derive a similar formula for functions of 3 variables. Then use it to derive the error estimate |E| < ⋯ in page 816 (section 14.6).

5. Section 14.9:

Use Taylor’s formula for 2 variables to derive the formula Δf = Δ_{1} + Δ_{2} + Δ_{3} on page 5 of mse19s 190425.pdf.