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Math 103, week 2 notes

by: Cambria Revsine

Math 103, week 2 notes MATH 103 001

Cambria Revsine

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About this Document

These notes cover sections 1.5-2.2 from Thomas' Calculus
Intermediate Algebra Part III
William Simmons
Class Notes
Math, Calculus
25 ?




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This 4 page Class Notes was uploaded by Cambria Revsine on Sunday February 7, 2016. The Class Notes belongs to MATH 103 001 at University of Pennsylvania taught by William Simmons in Spring 2016. Since its upload, it has received 20 views. For similar materials see Intermediate Algebra Part III in Mathematics (M) at University of Pennsylvania.

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Date Created: 02/07/16
Math 103—Week 2 Notes—1.5­2.2 1.5: Exponential Rules: If a > 0 and b > 0… a • a  = a x+y x a x­y y  = a a (a )  = (a )  = axy a  • b  = (ab)x x a a x x  = ( ) b b Exponential Growth/ Decay: kx y = y 0   exponential growth if k > 0         exponential decay if k < 0         *y  is a constant  0 y = Pe   continuously compounded interest model       P is initial monetary investment      r is interest rate (decimal form)       t is time (in units consistent with r) 1.6: One­to­one function: when each range value (y) has one distinct domain value (x)  Passes the horizontal line test Inverse Functions: ­1 Notation: f (x) f (b) =a if f(a) =b  The domain of f  is the range of f and the range of f is the domain of f **To find and/or graph an inverse function, switch x and y values in the original function  Ex:  1 Find the inverse of y =  x + 1 2 1 x =  y + 1 2 2x = y + 2 y = 2x – 2 Logarithmic Functions: y = log a is the inverse of y = a  (a > 0, a ≠ 1) y = ln x is the inverse of y = e x log 10= logx  log e = lnx ln e = 1 Algebraic Properties: lnbx = lnb + lnx  Product rule b ln  = lnb – ln x  Quotient rule x 1 ln  = ­lnx  Reciprocal rule  x lnx  = rlnx  Power rule Inverse Properties: alog  = x (a > 0, a ≠ 1, x > 0) a x log a  = x (a > 0, a ≠ 1, x > 0) e  = x (x > 0)  x lne  = x (x > 0) a  = e (lna = exlna lnx log a =   (a > 0, a ≠ 1)   lna Inverse Trig Functions:     −π π y = sin x is the number in [ ,  ] for which sin y = x 2 2 y = cos x is the number in [0,  π ] for which cos y = x 2.1: Average Rate of Change: d/t; distance travelled over time elapsed y 2y 1 f( 2− f (x 1  =  f(x1+h ) f (x 1 (h ≠ 0) x −x = x −x h 2 1 2 1  aka secant slope between the two points  Instantaneous Rate of Change: Rate at a given time  find the average rates of change (secant lines) from a point closer and  closer to the given point  this rate is the slope of the tangent line, which cuts through the given point 2.2: Limits: The y­value a function approaches as the function approaches a given x­value from both sides lim f x )=L x→ c **c= x­value, L= y­value ** L does not necessarily equal f(x) at c Limit Laws: If  lim f x =L  and  lim g(x =M x→ c x→ c lim ( (x)+g (x))L+M x→ c   Sum Rule lim ( (x)−g (x )=L−M x→ c   Difference Rule x→ c( • f(x)=k• L   Constant Multiple Rule x→ c( (x)•g x ))L•M   Product Rule lim f (x)= L  (M ≠ 0)  Quotient Rule x→ cg(x) M lim [ (x)] =L n x→ c  (n is a positive integer)  Power Rule n n x→ c√f (x)= √  (n is a positive integer)  Root Rule Sandwich Theorem: If  g(x ≤ f (x)≤h(x)  in an interval containing c (except possibly at x=c itself) and  If  lim g(x)=lim h (x =L   x→ c x→ c Then  lim f (x)=L x→ c lim sinø=0 ø→0 lim cosø=1 ø→0


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