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Math 400 Calculus I 1 Homework 6 - Select Problems Homework 6 - Assigned Problems Section 3.4 3.4.11 Use Theorem 3.11 to evaluate the following limits. lim x→0 tan(7x) sin x Solution. tan(7x) sin(7x) 1 = lim x→0 sin x x→0 cos(7x) sin x sin(7x) 1 1 = lim x→0 1 cos(7x) sin x sin(7x) 7x 1 1 = lim · · x→0 1 7x cos(7x) sin x 1 7x sin(7x) · = lim x→0 7x cos(7x) sin x sin(7x) 1 x = lim · lim · 7 lim x→0 x→0 cos(7x) x→0 sin x 7x =1·1·7 =7 lim 3.4.19 Find dy dx for the following functions. y = sin x cos x Solution. We have to use the product rule. y 0 = cos x · cos x + sin x(− sin x) = cos2 (x) − sin2 (x) = cos (2x) By an identity. 1 ARC Math 400 Calculus I Homework 6 - Select Problems ARC Section 3.6 3.6.41 Calculate the derivative of the following function. √ tan e 3x Solution. We have four “layers” of applying the chain rule for this problem. √ √ 1 0 2 y = sec e 3x · e 3x · (3x)−1/2 · 3 2 For the above, the function between each dot is a separate derivative of an “inner” function. 2