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by: logeybearrr

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# MATH 3A: Final Study Guide

Marketplace > University of California Santa Barbara > Math > MATH 3A Final Study Guide
logeybearrr
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Differentiation Rules Common Derivatives Implicit Differentiation Related Rates Logarithmic Differentiation Rolle's Theorem Mean Value Theorem Closed Interval Method First and Second Deriv...
COURSE
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Kabbabe
TYPE
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PAGES
3
WORDS
CONCEPTS
Math, Calculus, 3A
KARMA
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## Popular in Math

This 3 page Bundle was uploaded by logeybearrr on Saturday June 7, 2014. The Bundle belongs to a course at University of California Santa Barbara taught by a professor in Fall. Since its upload, it has received 86 views.

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Date Created: 06/07/14
Cheat Sheet Final Exam Differentiation rules f g fl g f g fl gl cf cf f939 f g f9 39 f 9 f9w39 f39993939rE c 0 1 m 1 e e am amlna ln 3 loga 3 x m sin 13 cosa cos 3 sing tan 3 sec2a sec 3 secxtana csc 3 csca cot 1 cot 3 csc2a arcsin 3 11m2 arccos 3 1962 arctan 3 1 sec 1 3 Wm cot 1 13 1 csc 1 13 10 Two important limits sinac 1 1 cosac 0 0 Implicit Differentiation When the relation between the dependent variable y and the independent variable 3 is given by an equation we can nd y by taking derivatives of both sides of the equation with respect to 3 treating y as a function of 3 In other words y should be treated as the variable u in the u substitution scheme Related Rates Sometimes We Wish to nd the rate of change in time of a certain quan tity say A but We have no Way of doing it directly Usually we can nd the rate of a change of a related quantity B In order to nd the rate of change of A it is necessary rst to nd an equation relating A and B and then proceed to differentiate said equation with re spect to time treating both A and B as functions of time like in the u substitution scheme Logarithmic Differentiation In order to nd the derivative of a function of the form y f x9 take natural logarithms in both sides of the previous equation and then pro ceed to differentiate using implicit differentiation Rolle s Theorem If f is a continuous function in a b differentiable in 0 19 such that fa fb then there exists a number 0 in a 19 such that fC 0 Mean Value Theorem If f is a continuous function in a b and differentiable in a b then there exists a number 0 in a 19 such that fC Relations between a function and its derivatives A function f is increasing at each point 3 Where fl gt 0 On the other hand it is decreasing at each point 3 where f13 lt 0 It is concave up at 3 if f33 gt 0 and concave down at as if fquot lt 0 Important De nitions Let f be a function and c a point of its domain D 1 We say that c is an absolute maximum of f if f 3 f c for all 3 in D On the other hand we say that c is an absolute minimum of f if f 2 f c for all 1 in D 2We say that c is a local maximum of f if R 3 fc for the points 3 near c and that c is a local minimum of f if f 2 f c for the points 3 near c 3We say that c is a critical point of f if fC 0 or if fc doesn t exist 4We say that c is an in ection point of f if f changes concavity at c The Closed Interval Method Given a continuous function f in a closed interval a b We can nd its absolute maximum and its absolute minimum value by following these steps 1Find the critical points of f in a b 2Evaluate f at the critical points 3Evaluate f at the endpoints of the interval 4Oompare values and decide First Derivative Test Let f be a function and let c be a critical point of f If fl changes sign from negative to positive at c then c is a local minimum of f On the other hand if fl changes sign from positive to negative at c then c is a local maximum of f If fl doesn t change signs at c then c isn t either a local maximum nor a local minimum of f Second Derivative Test Let f be a function and let c be a point such that fc 0 If fC lt 0 then c is a local maximum of f If fC gt 0 then c is a local minimum of f De nitions Whenever we Wish to compute a limit of the form m 010133 9013 and lima f 0 limag1 0 We say that our original limit is an indeterminate form of type On the other hand if limma f oo and lima g1 oo we refer to our original limit as an indeterminate form of type L Hospital s Rule If the limit frv iliitm is an indeterminate form of type or of type g then lim E im fl o H g1 H g Slant Asymptotes We say that the line y ma b is a slant asymptote of the function f if lim ma b 0 ac gtoo This usually occurs in rational functions whenever the degree of the numerator is 1 higher than the degree of the denominator Guidelines for sketching a curve Domain Intercepts Symmetry Asymptotes Vertical Horizontal Slant Intervals of increase and decrease Local maxima and local minima Concavity and points of in ection Sketch the curve mQquot1ti1UQwDgt Good luck on the Final

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