Calculus I, Fall 2015 (http://www.math.nthu.edu.tw/˜ wangwc/)

Brief answer to selected problems in HW03

1. Section 3.2: Problem 48.

a. {x| − 3 ≤ x ≤ 3, and x ≠ 2, ≠ −2}
b. {2, −2}
c. ϕ(empty set)

2. Section 3.2: Problem 54. Assume tangent line intersect with the curve at (x_{0}, y_{0}). Then solve the equations for x_{0} ≥ 0

\{ \begin{matrix}y_{0} = \frac{1}{2\sqrt{x_{0}}} (x_{0} + 1) \\ y_{0} = \sqrt{x_{0}} \end{matrix}

3. Section 3.2: Problem 57. Observing following limits for lim_{t→0} \frac{g(t)}{h(t)}

(1) \{ \begin{matrix}g(t) = t^{2} \\ h(t) = t \end{matrix} (2) \{ \begin{matrix}g(t) = t \\ h(t) = 2t \end{matrix} (3) \{ \begin{matrix}g(t) = t \\ h(t) = t^{2} \end{matrix}

4. Section 3.3: Problem 70.

continuity: −a + b = b − 3
differentiable: a = −2b

5.

\frac{d^{n}}{dx^{n}} (f(x)g(x)) = ∑_{k=0}^{n} C_{k}^{n}(\frac{d^{k}}{dx^{k}} f)(\frac{d^{n−k}}{dx^{n−k}} g)