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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

##### Uploaded: 12/13/2018
This preview shows pages 1 - 2 of a 3 page document. to view the rest of the content 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 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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##### Uploaded: 12/13/2018
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