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# SINGLE VARIABLE CALCULUS I MATH 101

Rice University

GPA 3.86

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This 3 page Class Notes was uploaded by Jayde Lang on Monday October 19, 2015. The Class Notes belongs to MATH 101 at Rice University taught by Staff in Fall. Since its upload, it has received 17 views. For similar materials see /class/224934/math-101-rice-university in Mathematics (M) at Rice University.

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Date Created: 10/19/15

Calculus 102 Lecture 1 Jeff Dajiang Liu June 7 2005 This course is an introduction to Calculus This course could be taken as a continuation of Calculus 101 but the material from 101 is not a necessary prerequisite In this First lecture we would like to give an review of the funda mental things from math 1011 The rst thing we would like to recall here is of course the derivativei Here is the de nition for a given function fa and a given point 10 the derivative is de ned by limgc cono W geometrically this is the numerical value of the slope of of the tangent line of a curve It is the limiting case of the secant liner for the straight lines of course the tangent line and the secant line coincide but for the curved lines this is the limiting case 2i The absolute value of the the derivative of a function measures the growth rate of fa with respect to I Here are some examples from which you can see what the rate of change is example 1 what is the slope of the line fa 31 example 2how about fa 12 hint use the de nition to show this sometimes people would like to use another equivalent way to de ne deriva tives This way is useful here to compute the derivative of 121 De nition Given and x0 fl limh ow if you have learnt triagonometric functions we would like see the example of fa sinz and fa cosz let us see a real world problem using derivatives For a moving object on the xaxis the position of the object 1t 372 3t 1 please nd out the speed and the rate of acceleration at t 1 2 3 Actually Calculus was created by Newton and Leibniz when they studied the mechanics and astronomy They wanted to study the relations between objects 1see blackboard for the picture 2limiting case when certain parameter changed just a little bit When it was rst invented it is not strict at alli I heard Leibniz even made the mistake like uv MW in his book which I think is very interesting OK we had better stop now and start with some more sophisticated rules please make sure these are clear in your mind before we can proceed 1i Constant do you know what is derivative of a constant pay attention that the derivative measure the the rate of change the constant does not change anything so i 2 The derivative of the linear combination afzbgz af zbgz 3 then you are supposed to know the derivative of a certain polynomial function ie 312 4x 5 4 The product rule given 2 functions u and v uv uv uv 5 the reciprocal rule 7 1632 Use the product rule and reciprocal rule to deduce 6 The power rulezfor the function fa I then n 7 lz 1 Pay attention here n E R F1 The exponential rule 66 emf Amazingll The derivative of the func tion equals itself Acutally this is the only nontrivial case for the relation f Could you tell me the trival case Hint It appears previ ously 00 i The logrithmic rule log z i 2 1n class Exercise Compute am 0 The chain rule This is the most frequently used rules in calculus espe cially when you learn some multivariable calculus later The rule is as follows fgzgz sometimes when you see some abstract functions the rule is usefu Example fe fe e Example 2 Example 312 31 52 Well I have omitted several important conditions in the previous rulesi Could you nd themi Diffentiability Here I would like to give several other interesting applications of the deriva tives 1 nd the equation of the tangent line of 12 at point 12 2 Show the rate of the volume growth of a ball With respect to its radius is equal to its surface area so far we summarized the rules for nding derivatives There Will be several self practicing problems given at the end of the class You DO NOT have to turn them in If you want I can give grade for them Which Will be counted as the extra bonusi sometimes people use derivatives to nd the local maxima or minima of given functions example please nd the local maxima of the function sinz z E T if Please remind yourself how the local extrema is defined3 Finally people Will sometimes see the case When the derivative does not ex isti One special case of this is the derivative is ooiWehaveapictuTeofthisi 3picture

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