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

Brief answer to selected problems in HW04

1. Section 3.5: Problem 58. For continuity, we can find b = 1, and g′(x) is not differentiable at x = 0 since

lim_{x→0^{+}} g′(x) ≠ lim_{x→0^{−}} g′(x)

2. Hw04: Problem 3

\frac{d}{dx}f(g(2)) = f′(g(2))g′(2) = 4 × 0.1 = 0.4

3. Section 3.7: Problem 42. Using implicit differentiation to get

\frac{dy}{dx} = \frac{y + 2}{1 − x}

And normal parallel to the line 2x + y = 0 lead us to solve the equations

\{ \begin{matrix}\frac{y+2}{1−x} = \frac{1}{2} \\ xy + 2x − y = 0 \end{matrix}

4. Section 3.7: Problem 48. Implicit differentiation gives us

qy^{q−1} \frac{dy}{dx} = px^{p−1}

By using y = x^{\frac{p}{q}} to express \frac{dy}{dx} leads to the result.