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Week 1 Notes

by: Olga Lukashina

Week 1 Notes Math 21A

Olga Lukashina
Michael Kapovich

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About this Document

Thorough Lecture Notes of Functions, Average Rate of Change, Instantaneous Rate of Change, Intuitive Definition of Limit, Limit Laws and Theorems, and Rigorous Definition of Limit
Michael Kapovich
Class Notes
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This 11 page Class Notes was uploaded by Olga Lukashina on Friday October 2, 2015. The Class Notes belongs to Math 21A at University of California - Davis taught by Michael Kapovich in Fall 2015. Since its upload, it has received 44 views. For similar materials see Calculus in Mathematics (M) at University of California - Davis.

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Date Created: 10/02/15
Lecture 1 92515 Calculus is the study of functions 0 Everything else limits derivatives integralsetc are all TOOLS that help us study functions What are functions 0 A functions fof one real variable is a rule which assigns a real number fX the output to a real number X the input A function assigns a unique single element fX to each element X 0 X is the independent variable 0 y is the dependent variable A function can be represented by an equation a graph a numerical table or a verbal description h i 7 z r 39 7 7 y Input Glllpul alm u i r1 m age Below are some examples offunctions EXAMPLE 1 fXX2 Input real number X and the machine outputs the real number X squared EXAMPLE 2 f R gtR EXAMPLE 3 Phase by phase analysis Piece Wise functions are described in pieces by using different formulas on different parts of the domain Most common example is the Absolute Value Formula EXAMPLE 4 Some functions are even described in words fX is number y yA3 X fXy fXXAl3 The number fX is the value of the function f at X A function does not need a formula It simply needs an input and an output 0 All possible input values are called the domain All possible output values is called the range 0 In calculus domain and range are a set of real numbers When we de ne domain and range in calculus we always assume that it is the largest possible set of real numbers Many functions are given a single formula but many are not For a function given by single formula the natural domain is all the values of x for which the operations used by the functions make sense Functions can have the same value at two different input elements in the domain but each input element is assigned a single output value D domain set F Effl mm tuiniug Not every curve is a function Since each input has a single unique output not vertical line can intersect the graph of a function more than onceThis is called the Vertical Line Test A way to manipulate functions is composition if a ellix E f lgm Composition is very sensitive to which order you compose the function To have an inverse every input has to have a di erent output g gt39i f ful ll A function that has distinct values at distinct elements in the domain is called onetoone These functions take on any one value in the range exactly once A function yfX is one to one if and only if its graph intersects each horizontal line at most onceThis is called the Horizontal Line Test A function that undergoes or inverts the effects of a function f is called the inverse of f not all functions have inverses Inverses can only be taken of ONLY of onetoone functions Composing a function and its inverse has the same e ect as doing nothing How to find the inverse function 1 Solve the equation yfX for X This gives you formula xf inversey Where X is expressed as a function of y 2 Interchange y and X Lecture 2 92815 Rates of Change and Tangents are an introduction to derivatives They explain how to think about derivatives informally and explains the origins of derivatives Average rate of change of yfX with respect to X over the interval X1 X2 is fix fill ail h x I In Average Rate of Change is basically the slope of the line through points X1 fXl and X2 fX2 o In geometry a line joining two points of a curve is a secant of the curve Average rate of change thus is a slope of a straight line The formula says nothing about what is happening in between two points 1 an V 11 iii A El 39r E Secumbr H H PM 39 N r 39 a l mug I A 3 Jr 3 gig Below are some examples of Average Rate of Change EXAMPLE 1 Average Velocity vtvelocity of an object at time t t is from 0 to 10 Suppose v065 mph v 1 070mph Rate of changeAYAXfl0 f0l 007065l 00510 Average Rate of changel2 miles per hour2 EXAMPLE 2 Economics A factory is producing refrigerators at the rate 100 per day at day 0 and in 10 days it produces 90 refrigerators What is the average rate of change in production ARCAYAXfl0f010090 lOOlOO l refrigerator per day Calculus however is about instantaneous rate of change Instantaneous rate of change is the value the average rate approaches as the length of h approaches 0 AYAX at AX0 is impossible so we need to nd a way to do it But how do we measure instantaneous rate of change 0 We need to find the slope of the tangent line which can be achieved by zooming in on the graph Seeants l EXAMPLE fXXA2 fXhfXh fXhA2XA2h lh2 1A2h l 2hhA2 lh 2hhA2h 2h This is the Average Rate of Change 2 This is the Instantaneous Rate of Change Lecture 3 93015 Mathematical OR 030 True statement 132 True statement 0 It is inclusiveboth are not required RE VIE W How to de ne Instantaneous Rate of Change AYAXfxh fXh AX0 This is impossible It took 150 years to come up with a solution to this puzzle the key is the concept of LIMIT llim I L E Which is read the limit of fX as X approaches c is L Informal De nition of Limit Values of fX are close to L whenever X is close to c T he limit value of a function does not depend on how the function is de ned at the point that is being approached EXAMPLE 1 fXL then limL Xa i5 10 5 a 5 10 i5 Indeed if X is approximately a then fXL and L is approximately L At any point in the domain the limit will be L t EXAMPLE 2 E lim XA3 lXl Xl then lim X l XAZ l39X l l X l X 1 then lim XA2X1 X 1 then 1113 Sometimes the limit does not exist Below is a graphical representation of What it means for the limit to not exist I It i 31 I i 5 a If EH3 LIMIT LAWS 1 Tautological Rule lim xa X 1 If x approaches a then x approaches a 2 Sum Rule Suppose limit as X approaches a of fX and limit as X approaches a of gX both exist then limfxgx lim fxlim gx Xa Xa Xa 3 Product Rule Both limits have to eXist limfxgx lim fx lim gX Xa Xa Xa 4 Ratio Rule limfXgx lim fXlim gx Xa Xa Xa Denominator cannot be 0 so lim gx cannot be 0 5 Power Rule lim fxnlim fxn Xa Xa 6 Root Rule lim fxAlnlim fxln Xa Xa n has to be greater than 0 DO NOT take even roots of a negative number 7 Inequality Rule Limits preserve non strict inequalities 8 Sandwhich Rule suppose fX is less than gX y L suppose gX is less than hX if lim fXlim hXL then Lim gxL THlEg EM 1 11m1 Laws if L1 11 and l are real numbera and Jim x L L 311m Rah391 5113me R1131 1 Con rm Multiig ile Rule 41 Pmdm Hale Q f f m Hale F war Hula T Ram Rule and i111 11 1 M1 than Jr 4391 llmi il 511121 L I quotquot 12121 L M I lmt i fix I L I gsle 1 M rst L a limi ail L 111 a oaitive integer IH if 5391 L39f n a positing integer IF 11 is 13113111 we 1155111111 that lim x L 3 U 11 Intuitive de nition of limit lirrngivI I Lecture 4 10215 which means that if X is approximately a but not equal to a then fx is approximately equal to L Cauchy tried to gure out how to de ne the limit precisely but could not gure it out What exactly does approximately mean Then this 30 years after that mathematicians came up with a rigorous de nition for a limit It includes the notion of Deltas and Epsilons Delta and Epsilon are both errors How big of an error in the calculation of a limit will you allow yourself to make They are degrees of approximation When calculating the limit the objective is to verify it You can do that by nding delta when given epsilon and both values are positive De nition of Rigorous Limit Suppose yfx lim fXL Xa For arbitrary Epsilon that is greater than 0 we can nd a Delta that is greater than 0 Epsilon is the change in either direction from L Delta is the change in either direction from a Objective Figure out the accuracy of the output by specifying the accuracy of the input F Li L 55 Te Satisfy r liee L 39 7 39h E g 1 HM in here Eager nquot all I i e 39 TwirlEl in here 113 thie 1 L 5 39 a E e e I E Haw in Find l gebraicaly a fur 3 Given L m and E Era1 I The pf if nding 21 5 If guah thank Fm 311x iLE Elii lfLElI Ll iiE can lb HEC l pllli h l in twn STEPS L SHEW The i equ ifiy lf x E tn Find an p i n rval at b cantain ing 12 SIM whigh Eh Ein qmality h ld f tquot allll x 2 Find a mine f 13 that pl th 0pm minimal 1 s 3 cantata am a insidlethe interval 15 b Th3 inaquality l x LI 1 E will hmld far all A r in this 3int rv k


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