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.
Christian Parkinson GRE Prep: Calculus I Practice Problem Solutions 3 so fis constant. Problem 11. Let f(x) = x2+sin(x) for x>0. Find f0(x). Solution. The temptation here is to use the power rule or the exponential rule but in the current form, neither apply since both the base and the exponent depend on x. To x this, we write f(x) = e(2+sin( x ...
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