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# Class Note for MATH 1431 with Professor He at UH

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This 13 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 22 views.

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Date Created: 02/06/15

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 2 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 ZU N LC N 0 lt lsinxl lt lxl l d 3w i 7 Isin xl m L 1 B r J he Pinclrgng Theorem I rigonometric Limits eorem sinx 1 cosx 11m 1 11m r gt0 L r gt0 IL O 1 tan x Proof Use Geometric argument to get sinz coszlt lt1 7 then apply the pinching theorem MiBErgkiggnometric Limits lim 1 i sinm z m for m near 0 070 Sinz Theorem 5 1 7 cosz 1 1 11m 7 7 gt cosz z 1 7 7m2 for z near 0 170 2 2 2 Theorem 6 For my number 04 7 O sin om 1 7 cos om hm 17 11m 07 9270 04m 9270 04 om 1 7 cos 04 1 hm 17 11m 2 7 170 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 M M fb fb J K K l x 39a t i 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 0 000 O l39 0 II gtlt 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 2 Section 31 The Derivative 2 1 The Derivative Secant Lines vs Tangent Lines RV r C X C 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 3 A fc P C gtltv 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 gtltv E pb ggvmas Function More Examples 1 fxE7 square root function Examples 15 tangent HIi t Line Functions For a line function fx mx b the derivative f x m Examples 16 1 06 7 f 1 y 2 2 Examples 17 23 Nondifferentiability Nondi erentiability Discontinuity A function f is not di erentiable at C if flt0h flt0gt h lim does not exist h gtO 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 slope i x2 tangent HID t Nondifferentiability Corner Points A function f is not differentiable at e if fC h fC h lim does not exist h gtO Corner point at 0 as a gt 0 f x gt 1 as a gt 0 f x gt 1 ylk 2 2 2 2 Corner point at 1 as a gt 1 f 1 gt 2 as 1 gt 17 f 1gt 11 y 1 gtltv 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 R fx x13 vertical tangent at 2 as x gt 2 f gt 00 311 k gx 2 x15 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 J A RV fx 923 vertical cusp at 1 as 6 gt 1 f 6 gt 00 as 6 gt 1 f 6 gt 00 J A 1 2 39 Xv gm2 a 1w5 13

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