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This 13 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 1431 at University of Houston taught by Jiwen He in Fall. Since its upload, it has received 11 views. For similar materials see /class/208373/math-1431-university-of-houston in Mathmatics at University of Houston.
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Date Created: 09/19/15
Lecture 5Section 31 The Derivative Jiwen He 1 Review 11 Info Homework and Quizzes Homework 2 amp 3 o Homework 2 is due today in lab 0 Homework 3 is due September 16th in lab Online Quizzes o Quizzes 1 and 2 have expired 0 Quiz 3 is posted and due on this Friday before 1159 PM 0 Quiz 4 is posted and due 919 Weekly Written Quizzes in Lab 0 Quizzes will be given every week on Thursday in lab beginning THIS WEEK 0 The weekly written quizzes form is posted on the course homepage You must print out this form and BRING it to class every Thursday Daily Grades 0 Daily grades start Today 0 The daily grades form is posted on the course homepage You must print out this form and BRING it to class every day Quiz 1 Quiz 1 902 1 hrn x gt1 13 1 a 1 b 2 C 1 d None of these Quiz 2 Quiz 2 Classify the discontinuity at 36 1 for 902 1 a Jump b In nite C Removable d None of these 12 The Pinching Theorem The Pinching Theorem J x h Theorem 1 Let p gt 0 Suppose that for all 36 such that 0 lt 30 0 lt p M90 S ft S WE If lirn L and lirn 930 2 L than lirn L lihe Pincihing Theorem Continuity of Sine and Cosine eorem lim sinx 0 lim cosx 1 r gt0 r gt0 lim sinx sin 0 lim cosx cos c LC N ZU N 0 lt lsinxl lt lxl le ISin xl he Pinclrgng Theorem I rigonometric Limits eorem sinx 1 cosx 11m 1 11m r gt0 L r gt0 IL O 1 U tanx an X 03 0777 7 77 COS X gt 1 1 Proof Use Geometric argument to get sinz cosz lt 7 lt 17 1 then apply the pinching theorem mampr iggno metric Limits 71 sinmszorxnearo 270 Sinz Theorem 5 1 7 1 1 lim 7 gt cosz z 1 7 7m2 for z near 0 0270 x 2 2 Theorem 6 For my number 04 7 O sin om 1 7 cos om 11m717 11m 07 0270 04m 0270 04 om 1 7 cos 04 1 hmAi17 11111 277 170 Sin om x70 04 2 Quiz 3 Quiz 3 What day is today a Monday b Wednesday C Friday d None of these 13 The IntermediateValue Theorem The IntermediateValue Theorem Theorem 7 If f is continuous on ab and K is any number between fa and fb then there is at least one number 0 in the interual ab such that f0 K yll O K fa fO Q 393 A our Q U gt4 The IntermediateValue Theorem Roots of Equation Theorem 8 ff is continuous on a b and falt0ltfba 07 fblt0ltfa then the equation fac 0 has at least a root in a b The IntermediateValue Theorem Solution of Inequality Solve the inequality x3 x2 6xgt0 Solution 2 0 U 3 00 Solution of Inequality Solve the inequality 30 332x 1x 42 S 0 i Z x Solution 312 14 The ExtremeValue Theorem The ExtremeValue Theorem Theorem 9 A function f continuous on a bounded closed cub takes on both a mascimum ualue M and a minimum ualue m Q Q L1 393 w 2 Section 31 The Derivative 2 1 The Derivative Secant Lines vs Tangent Lines gt6IV r X De nition 10 The slope of the graph at the point e is given by hmf M f h gtO h 7 provided the limit exists Derivative and Differentiation J A fc P l x C x De nition 11 A function f is di ei entiable at e if hmf M f exists h gtO h If this limit exists it is called the derivative of f at e and is denoted by f 22 Derivative as Function Derivative as Function De nition 12 The derivative of a function f is the function f With value at v given by h jw hmf fa h gtO h 7 provided the limit exists To di ei entiate a function f is to nd its derivative Examples 13 E pb ggvmas Function More Examples 3 1 1035 V57 fx 1 2 39 x Woe amp 9er 1S x V35 square root function xquot Examples 15 Line Functions For a line function fx mx b the derivative f x m Examples 16 NO rv f 1 y 2 2 Examples 17 i 2 ng7 i 0 xlt07 f x2 xgt0 f 1 xgt0 l x 23 Nondifferentiability Nondi erentiability Discontinuity A function f is not di erentiable at C if hmf M f h h gtO does not exist Jump discontinuity at 0 as v gt 0 f gt 00 as v gt 0 f gt 0 y 10 In nite discontinuity at 0 as a gt 0 f x gt 00 as a gt 0 f 1 gt 00 y xv slope i x2 tangent 1 y Nondifferentiability Corner Points A function f is not differentiable at e if hm fc h fc h h gtO does not exist Corner point at 0 as a gt 0 f x gt 1 as a gt 0 f x gt 1 VI 2 2 2 2 Corner point at 1 as a gt 1 f 1 gt 2 as 1 gt 17 f 1gt 11 l x 1 x no derivative at 1 Nondifferentiability Vertical tangents A function f is not di e39r entz able at c if hm fc h fc h does not exist h gtO vertical tangent at 0 as x gt 0 f x gt 00 y fx x13 vertical tangent at 2 as x gt 2 f gt oo ylk 12 Nondifferentiability Vertical Cusps A function f is not di erentiable at c if hm fc h fc h gtO 2 does not exist vertical cusp at O as 6 gt 0 f 6 gt 00 as 6 gt 0 f 6 gt 00 YA RV fx 9623 vertical cusp at 1 as 6 gt 1 f 6 gt 00 as 6 gt 1 f 6 gt 00 ylk 1 2 39 gtltv gm2 a 1w5 13