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

Homework Assignment for Chap 05

1. Section 5.2: Problems 33, 35, 39, 45.

2. Section 5.3: Problems 1, 31, 35, 36, 39, 40.

3. Section 5.4: Problems 7, 11, 13, 19, 23, 41, 43, 45, 51, 69, 73.

4. Section 5.5: Problems 11, 23, 36, 37, 45, 49, 55.

5. Section 5.6: Problems 53, 55, 56, 58, 67, 71, 73.

6. (s5.2-extra1) Evaluate

lim_{n→∞} ∑_{k=1}^{\frac{n}{2}} \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?

7. (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 integer.

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

8. (s5.2-extra1) Evaluate

lim_{n→∞} ∑_{k=1}^{\frac{n}{2}} \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?

9. (s5.5-extra1) Evaluate

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