Problem 1 a) [D\frac{3t}{ln~t} = \frac{3ln~t - 3t(1/t)}{(ln~t)^2} = \frac{3(ln~t - 1)}{(ln~t)^2}] b) [\frac{d}{dx}sin^{-1}x = \frac{1}{\sqrt{1-x^2}}] [\frac{d}{dx}tan^{-1}x = \frac{1}{1+x^2}] [\frac{d}{dx}sec^{-1}x = \frac{1}{|x|\sqrt{x^2-1}}] c) [D^3sin~kx = k^3(-cos~kx)] [D^3cos~kx = -k^3sin~kx] d) [D(at~sin^2t) = a + k~sin^2t + 2akt~sin~t~cos~t] [= \frac{2}{3}\frac{k~sin~t~cos~t}{(a + k~sin^2t)^{2/3}}] Problem 2 a) [\lim_{x \to 0} \frac{sin~x - x + x^3/6}{x^5} = \frac{0}{0}] Apply L'Hopital's rule: [\lim_{x \to 0} \frac{cos~x - 1 + x^2/2}{5x^4} = \frac{0}{0}] Apply L'Hopital's rule again: [\lim_{x \to 0} \frac{-sin~x + x}{20x^3} = \frac{0}{0}] Apply L'Hopital's rule again: [\lim_{x \to 0} \frac{-cos~x + 1}{60x^2} = \frac{0}{0}] Apply L'Hopital's rule again: [\lim_{x \to 0} \frac{sin~x}{120x} = \frac{0}{0}] Apply L'Hopital's rule again: [\lim_{x \to 0} \frac{cos~x}{120} = \frac{1}{120}] b) [f(x) = cos~x] [f'(x) = -sin~x] Zeros of f'(x): x = 0, ±π, ±2π, ±3π, ... f(0) = 1, f(±π) = -1, f(±2π) = 1, ... So, no inflection points at x = 0 Problem 3 a) [A = A_0e^{-rt}] [A/A_0 = 1/4 = e^{-rt}] [-ln~4 = -rt] [t = ln~4/r] Problem 4 ... (omitted due to image quality)

Practice Exam 1 Solutions 1

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