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

Homework Assignment for Chap 02

Last update Sep 19, 2012.

1. Section 2.2: problems 25, 55, 72(b).

2. Section 2.3: problems 23, 33, 40, 46.

3. State (need not prove) the ‘x → c^{+}’ and ‘x → ∞’ versions of the Sandwich Theorem.

Part of the assumption in the standard Sandwich Theorem reads

⋯ for all x ≠ c in some open interval about c ⋯

How would you change this sentence in the ‘x → c^{+}’ and ‘x → ∞’ versions, respectively?

4. Section 2.4: problems 54, 55, 56, 57.

5. Chap 2: problems 41, 42, 53, 69, 70.

6. For those of you who are really not confident about your high school mathematics, pick some among section 2.3 problems 1-22, 35-40, section 2.4 problems 37-48 and practice yourself. Normally you don’t need this.

7. Evaluate lim_{θ→π/6} \frac{sin θ − 1/2}{θ − π/6}

8. (Challenge problem, optional)

Let f : (0, 1) ⟶ R be defined as

f(x) = \{ \begin{matrix}1/p & if x = q/p, p, q ∈ N, (p, q) = 1 \\ 0 & otherwise \end{matrix}

For what values of c ∈ (0, 1) is f continuous at c?

9. Section 2.5: problems 30, 32, 35, 38, 41.

10. Chap 2: problems 74, 75.

11. How would you define the following limits formally using ϵ and δ? (Need not prove anything, just define them) a.

lim_{x→c^{+}} f(x) = L

b.

lim_{x→c} f(x) = ∞

c.

lim_{x→−∞} f(x) = L

Hint: The formal definition of lim_{x→c} f(x) = L is a translation of

f(x) can be arbitrarily close to L as long as x ≠ c is close enough to c The ‘f(x) → ∞’ part, in plain words can be like ‘f(x) be arbitrarily large’ while the ‘as x → ∞’ part can be ‘whenever x is large enough’. The latter, in mathematical language, would be ‘there is an M such that for all x > M, ⋯’

12. Use the ϵ − δ argument to show that, if f(x) and g(x) are both continuous at x = c, then so is f(x) + g(x) and f(x) − g(x).

13. Use the ϵ − δ argument to show that if lim_{x→c}f(x) = L and g(y) is continuous at y = L,

then lim_{x→c}g(f(x)) = g(L).

14. Use the ϵ − δ argument to show that if f(x) is continuous at x = c, then so is 3f(x).