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Calculus I

by: Edgar Jacobi

Calculus I MATH 115

Edgar Jacobi
GPA 3.6


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This 3 page Class Notes was uploaded by Edgar Jacobi on Monday September 7, 2015. The Class Notes belongs to MATH 115 at Kansas taught by Staff in Fall. Since its upload, it has received 15 views. For similar materials see /class/182397/math-115-kansas in Mathematics (M) at Kansas.


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Date Created: 09/07/15
Review of Chapter 4 A l quot quot of the Derivative Keywords applications 1 Formulas 2 Concepts7 theorems7 and techniques 21 Graph of a function 0 Critical points nd points where 0 or is discontinuous or not de ned 0 Intervals where is increasingdecreasing intervals where gt Ofz lt 0 Key is to nd the critical points 0 Concavity of Concave upward is increasing or f z gt 0 Concave downward is decreasing or f z lt 0 o lntervals where is concave upwarddownward key is to nd the points where f z 0 or f z is discontinuous or not de ned 0 ln ection points points where the concavity of changes 22 Optimization 0 Find the relative or local extrema 1 First derivative test Find the critical points and check the sign change of f around these points Let I c is a critical point 7 If is increasing around I c is a relative minimum 7 If is decreasing around I c is a relative maximum 2 The second derivative test I c is a critical point and fc 0 7 If f c gt 07 then is a relative minimum 7 If f c lt 07 then is a relative maximum 7 If f c 07 cannot be determined 0 Find the absolute or global extrema Compares the values of at the critical points7 the endpoints of the domain7 and possibly limgch00 andor limgchar00 0 Optimization application problems 1 Set up mathematical model or formulas 2 Eliminate one variable 3 Find the absolute maximum andor minimum of the function 23 Curve sketching 0 Vertical asymptotes z a lingHaJr ioo andor limxnai ioo 0 Horizontal asymptotes y b limgch00 12 andor limgch00 b o lntercepts with zaxis and y axis Review of Chapter 2 Functions Limits and the Derivative Keywords function limit continuity derivative 1 Formulas MW di erence quotient slope of a secant line average rate of change lim W derivative 0 slope of the tangent line rate of change 2 Concepts theorems and techniques 21 Functions 0 function domain range from expression and graph 0 independent and dependent variables 0 graph of a function or an equation sketch ordered pairs 0 verticalline test 0 composite function g o f 7 f o g o particular functions polynomial functions constant linear quadratic cubic rational functions and poWer functions 0 application to economics demand function supply function market equilibrium equilibrium quantity and price 22 Limit 0 limit and its calculation by using table graph and calculator W o indeterminate form 2 tricks eg 7 4 x72 7 an 0 limit at in nity limx w x limx uw limac gt0 L for n gt O 0 one sided limits limw a limx ai 23 Continuity o continuity and discontinuity of a function at a point 0 relation between graph and continuity 0 properties of continuous functions 0 The intermediate value theorem 0 The theorem for existence of zeros of a continuous function 0 The bisection method 24 Derivative o secant line and its equation 0 tangent line and its equation to the graph of o derivative rate of change and slope of tangent line 0 relation between di erentiability and continuity o calculation of derivative 0 Notation fx x yI 3 Review Problems All odd numbered problems Review of Chapter 5 Exp and Log functions Keywords Exp and Log 1 Exponential functions 0 Exponential function with base I y by 0 The laws of exponents 0 Basic properties of Exp functions domain range special values continuity and monotonicity o e limr h00 1 27 and function y e o Differentiation em e 0 Chain rule e ef fz 2 Logarithmic functions Logarithmic function with base I y log I if and only if by z o The laws of logarithms 0 Basic properties of Log functions domain range special values continuity and monotonicity 0 Common logarithmic function y log 1 Natural logarithmic function y lnz o Differentiation lnlzl 0 Chain rule ln bfz o Logarithmic differentiation 3 Relations between Exp functs and Log functs 0 ln 6 z 4 Applications 7 Compound interest 0 Exponential growth and decay 0 Learning curves 0 Logistic growth functions


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