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UCR - MATH 009A - Study Guide for Final - Study Guide

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UCR - MATH 009A - Study Guide for Final - Study Guide

School: University of California Riverside
Department: Applied Mathematics
Course: First-Year Calculus
Professor: Steven Gindi
Term: Fall 2018
Tags: Limits, Derivatives, L'Hopital, Implicit_Differentiation, Local Maximum and Minimums, concavity, and Inverse_Derivative
Name: Study Guide for Final
Description: These are some questions that from the lessons we have learned in Class. THEY ARE HARD...well some, but they are meant to be a bit difficult so that you can practice. Answers for this will be uploaded later on. Hope this is helpful.
Uploaded: 12/13/2018
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background image   Things to know  This is a study guide for the final. I know that stuff in here might be a bight too                                       complicated, but if we want to prepare for the exam and have practice, we need to challenge                                   ourselves, that way when it is test day, the material is easier than usual. Please contact me if there                                       is a question. Be aware that this is NOT to represent a test but, a practice. Test yourself, and good                                         luck! 
Power Rule :  (x)
  (x)     f x n f · x 1   Derivative using limits(first principle) : (x) f = lim h→0 h f(x+hf(x)   Product Rule :  (x) (x) (x) (x) (x) (x) (x) (x)   h f · g h f · g g · f   Quotient Rule :   (x) (x) h = f(x) g(x) h = g  (x) 2 (x)g(x(x)f(x)   Chain Rule : (x) (g(x)) (x) (g(x)) (x)   h f h f · g   Implicit Differentiation : (y ) (x  y  ) x  y  x  y  d dx = d dx n n dx dy + 1 = n 1 n n n 1 dx dy dx dy = (1 nx  y  ) n n 1 (nx  y  1) 1 n   Inverse Derivative :  example   ; ; (y)     f y 2   (x) f = √x (y)   (x) dx df  = 1 (y) f   L’hopital’s Rule : 
Notes on infinity in terms of L'hopital's: 
; ; ;   1 = 0   0 1 = ∞   c = ∞ c =   − ∞   RULE 1:( If  ,   ):  0 0 im im l f(x) g(x) l (x) (x)   RULE 2: ( If   ):   Apply Rule 1 now! im f(x) (x)   l · g = 0 · ∞ im or lim  l 1 f(x) g(x) 1 g(x) f(x)   RULE 3: ( If   ):    ,   ,     0 0 0 1 (x   f g(x)   STEP 1 im f(x     l g(x) e L   STEP 2 et y (x   l f g(x)   STEP 3 n y (x)ln f(x)   l g   STEP 4 im ln y im(g(x)ln f(x))   l l   STEP 5 im ln y   l   STEP 6 im y   ; for L is the final result of step 5.   l e L     Do the limit of  using RULE 2. After applying rule 2, do what Rule 2 im(g(x)ln f(x))   l   says (Do rule 1) 
RULE 4 ( If 
 ) im f(x) (x)  or    l ± g = ∞ − ∞ − ∞ + ∞   Try to combine  f(x) and g(x) and use L’hopital’s  First Derivative test (Increasing/Decreasing) :  (xis increasing on an interval if (x)  for ALL x in interval   f > 0   (xis decreasing on an interval if (x)  for ALL x in interval   f < 0   
background image  
Second Derivative Test (Concavity) : 
f (x)  in the intervalthen it is concave down   I ′′ < 0     f (x)  in the intervalthen it is concave up   I ′′ > 0     List of Derivatives :  (x) in x (x) os   f s f c (x) os x (x) in   f c f =   − s   (x) an x (x) ec     f t f s 2 (x)   (x)     f e x f e x    (x) n x (x)   f l f = x 1 (x) ec x (x) ec x an   f s f s · t   (x) sc x (x) sc x ot   f c f =− c · c (x) ot x (x) sc    f c f =   − c 2   (x)   (x)   (ln a)   f a x f a x (x) og  x (x)   f l a f = 1 xlna   (x) rcsin x (x)   f a f = 1 x  2 (x) rccos x (x)   f a f =   − 1 x  2   (x) rctan x (x)   f a f = 1 1+x  2    
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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School: University of California Riverside
Department: Applied Mathematics
Course: First-Year Calculus
Professor: Steven Gindi
Term: Fall 2018
Tags: Limits, Derivatives, L'Hopital, Implicit_Differentiation, Local Maximum and Minimums, concavity, and Inverse_Derivative
Name: Study Guide for Final
Description: These are some questions that from the lessons we have learned in Class. THEY ARE HARD...well some, but they are meant to be a bit difficult so that you can practice. Answers for this will be uploaded later on. Hope this is helpful.
Uploaded: 12/13/2018
3 Pages 116 Views 92 Unlocks
  • Better Grades Guarantee
  • 24/7 Homework help
  • Notes, Study Guides, Flashcards + More!
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