Transcription of Week 1: Calculus I Practice Problem Solutions
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Week 1: Calculus I Practice Problem Solutions
Christian ParkinsonGRE Prep: Calculus I Practice Problem Solutions1 Week 1: Calculus IPractice Problem SolutionsProblem is the tangent line to the graph ofy=x+exatx= 0? tangent line is given by`(x) =y(0) +y (0)(x 0) = 1 + limx 0(1 +x) 1xfor l H opital s rule, we seelimx 0(1 +x) 1x= limx 0 (1 +x) 11= .This gives a first order approximation (1 +x) 1 + xwhenx limx 0cos( x) 1x2for l H opital s rule, we seelimx 0cos( x) 1x2= limx 0 sin( x)2x= limx 0 2cos( x)2= >0. Find the minimum value off(x) =ex cxamongx the derivative to zero shows that extreme points occur whenex c= 0 x= log(c).The second derivative offis always positive so any extreme point is a minimum.
Practice Problem Solutions Problem 1. What is the tangent line to the graph of y= x+ ex at x= 0? Solution. The tangent line is given by ‘(x) = y(0) + y0(0)(x 0) = 1 + 2x. Problem 2. Evaluate lim x!0 (1 + x) 1 x for 2R Solution. Using l’H^opital’s rule, we see lim x!0 (1 + x) 1 x = lim x!0 (1 + x) 1 1 = : This gives a rst order ...
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