Calculus I, Fall 2012 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 08

1. Section 4.6: Problems 13, 23(b), 25, 26, 28, 29, 33.

2. Read and compare the difference between Intermediate Value Theorem and Mean Value Theorem.

3. Section 5.2: Problems 34, 36, 38, 39, 46.

4. (s5.2-extra1) Evaluate

lim_{n→∞} ∑_{k=1}^{n} \frac{1}{n}\sqrt{1 − (\frac{k}{n})^{2}}

Hint: What is the area of a circle? what does it have to do with this problem?

5. (s5.2-extra2) Use the binomial expansion for (k + 1)^{ℓ} to show by induction that

lim_{n→∞} ∑_{k=1}^{n} \frac{1}{n}(\frac{k}{n})^{ℓ} = \frac{1}{ℓ + 1}

where ℓ is a positive ingteger.

Hint: (k + 1)^{ℓ+1} = k^{ℓ+1} + (ℓ + 1)k^{ℓ} + · · ·.