Math 103, week 2 notes
Math 103, week 2 notes MATH 103 001
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MATH 103 001
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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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