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# Math 221 Midterm 1 Study Guide MATH 221 001

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This 5 page Study Guide was uploaded by y-chen9 on Sunday October 4, 2015. The Study Guide belongs to MATH 221 001 at University of Wisconsin - Madison taught by Tonghai Yang in Fall 2015. Since its upload, it has received 356 views. For similar materials see Basic Concepts of Elementary Mathematics I in Applied Mathematics at University of Wisconsin - Madison.

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Date Created: 10/04/15

Math 221 First Semester Calculus Midterm I Review By Yang Chen Professor Tonghai Yang TA Shenyuan Huang De nition of Derivative The derivative measures the instantaeous rate of change of the function for all values of X and is de ned as the limit of the average rate of change in the function The derivative at a point is the instantaneous rate of change at XC The derivative at a point is the slope of the tangent line at that x value THE DEFINITION a3 Hm fxh fx fix fret h Concept of Derivative Geometrically a gt I ii39uuizicm i iiff iquot 11 dcn39t mlfi w 113 Numerically YXA2 X Y Y O O O 1 1 2 5 25 10 10 100 20 Analytically Simply nding the average rate of change over increasingly small intervals provides a close approximation but the two are not the same That is where limits come in Basic Derivative Rules 5 Pemrnlle itali a e4 m1 it ett a Preth rule tat1ait 1im em eh e er 393939 Quiz eet rule 2 en rt r e 1 1 E Reelpreeelnlle 2 em rt 1 ti r BF r 7 BF 5 39EI ChejILr39ule x i quotij1Iquotj1 3 mu 1 It L H Derivatives of Trigonometric Functions equot emx 2 ewquot E if li i tit tit E V at 1 teem E e1llft eeet eeettenx tr tit if 2 t 2 tit li j eee 1 eett S 1 it tit Trick to Finding LimitDerivatives L Hospitals Rule Take the derivative again and againand again if needed when your answer continues to be an indeterminate form OO in nityin nity Oin nity or in nityO L H nepitl e Rule Siqu f and g are dh 39erentiehlie ent 13 ii it an an epen interwl I that eemtejne a pihljr at Suppeee that Hm x ii and timely I er int Hm ten and Hm im ht ether Iillirirtrtilei we have an indietem nate them Of type E e 13 Then Jim W x H3 e113 thtihe limitva the right eide er iiE w er E However there are cases where L Hospital s rule does not work Such as Lim X in nity Xquot3 equotx O in nity If this happens change the function into a fraction so you can do L H s rule Lim X in nity equotxxquot3 in nityin nity Note exponentpower expgtpower Derive again equotX6x in nityin nity Derive AGAIN equotX6 in nity6 THE LIMIT DOES NOT EXIST AYSYMPTOTES LIMITS ob INFINITY PRODUCE ASYMPTOTES Vertical Asymptotes Occur at a value of X that makes the denominator 0 Horizontal Aysmptotes Occur when there is a limit to the Y value of the func on There are 3 situations 1 The degree in the numerator is higher than the degree of the denominator a No HA 2 The degree of the numerator is less than the degree of denominator a HA is zero 3 The degrees are the same a The coef cient of the highest degree of numerator divided by highest degree in denominator is the symptote 9 1o it2the 1i39ertioal 3 fr asymptote quot3 y4 the e horizontal asymptote I 2 I WI I e 4 2 4 e 2 l I 4 Slant Asymptotes Occurs when the polynomial in the numerator is a higher degree than the polynomial in the denominator To nd the slant asymptote you must divide the numerator by the denominator using either long division or synthetic division I g CONTINUITY A function f is said to be continuous on an interval I iff is continuous at each point x in l 1 The sum of continuous functions is continuous 2 The difference of continuous functions is continuous 3 The product of continuous functions is continuous 4 The quotient of continuous function is continuous except when the denominator O 5 Any polynomial is continuous for all points of X 6 equotx sinx and cosX are continuous for all values of x Basic Properties of LimitsContinuity A limit is the value a sequencefunction approaches as the imputXvalue approaches some value In other words a limit is a yvalue that fx can be kept arbirarily close to just by keeping X close enough to C but not equal to C Limits involving in nity are related to vertical and horizontal asymptotes A limit at C exisits if the limit approaching the xvalue from the left and right are the same In this example the lim XD 5 fx does not exist because the left and right limits are different Relationship Between Differentiability amp Continuity All differentiable functions are continuous but not all continuous functions are differentiable Coninuity requirements 1 lim xa fx must exist 2 lime a fx fa for all points a Differentiability requirements 1 A function is differentiable anywhere its dervative is de ned It is undifferentiable at discontinuities The derivative of a vertical line is also unde ned RemovableNonRemovable Discontinuities A holegap in a graph That is a removable discontinuity can be repaired by lling in a single point In other words a removable discontinuity is a point at which a graph is not connected but can be by lling in a single point You cannot do this in a nonremovable discontinuity such as the common form a jump or step discontinuity Discontinuity Examples 1 and 3 are removable while 2 and 4 are not 42 2 4i 2 2 f X I X i 4 2 Ni 2 L 4 2 X that

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