MATH 180 Study Guide - Final Guide: Scilab, Quotient Rule, Intermediate Value Theorem
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Problem 1 solution: di erentiate each function with respect to x. Leave your answer in unsimpli ed form. (a) ex x3 1. Solution: (a) use the quotient rule. (cid:18) ex x3 1. 2x (cid:19) (2x)(ex x3 1) (ex x3 1)(2x) (2x)2 (2x)(ex 3x2) (ex x3 1)(2) (2x)2 (b) use the chain rule. ( sin(x)) (c) use the exponential and power rules. (cid:0)3x + x3(cid:1) . Solution: (a) we nd dy dx using implicit di erentiation. d dx (x) + x + 16 = y2x d d dx (cid:0)y2x(cid:1) (16) = dx. 1 + 0 = y2 d dx (x) + x d dx (cid:0)y2(cid:1) dx(cid:19) dy. 2xy (b) the value of dy dx at (2, 3) is the slope of the tangent line. An equation for the tangent line at (2, 3) is then: 3 dy dx(cid:12)(cid:12)(cid:12)(cid:12)(2,3) y 3 = . Problem 3 solution: calculate each limit and indicate the method used: (a) lim x 1 x2 5x + 4.
