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Math 103, week 1 notes

by: Cambria Revsine

Math 103, week 1 notes MATH 103 001

Cambria Revsine
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About this Document

These notes cover sections 1.1-1.3 from Thomas' Calculus
Intermediate Algebra Part III
William Simmons
Class Notes
Math, Calculus




Popular in Intermediate Algebra Part III

Popular in Mathematics (M)

This 5 page Class Notes was uploaded by Cambria Revsine on Friday February 5, 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 27 views. For similar materials see Intermediate Algebra Part III in Mathematics (M) at University of Pennsylvania.


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Date Created: 02/05/16
Math 103—Week 1 Notes—1.1­1.3 1.1: Domain and Range of Functions: Domain= all input values of a given function What x­values give y a real value and which do not? Range= all output values of a given function Plug in domain values to find the range Ex:  2 Find the domain and range of  f(x)= √96−x 196−x ≥0   x ≤196                                                         x≤14 ,  x≥−14 Domain= [­14,14] 2 √196− (±14 ) =0 √196−0 =¿ 14 Range= [0,14] Graphing Piecewise Functions: Graph normally for each set of x­values Ex:  −x,x<0 f(x)= 2    {x ,0≤ x≤4 4,x>4 Properties of Graphs: Increasing when x1  x2 and f(1 ) < f2x )  Going SE Decreasing when x < x  and f(x ) > f(x )  Going NE 1  2 1 2 Even if f(­x) = f(x)  symmetrical across the y­axis Odd if f(­x) = ­f(x)  symmetrical about the origin 1.2: Composite Functions: (f ° g)(x) = f(g(x))  plug in the g(x) function into the “x” of f(x) (g ° f)(x) = g(f(x))  plug in the f(x) function into the “x” of g(x) Ex: If  f ( ) √ and  g (x=x+1 , find (g ° f)(x) (g ° f)(x) = g(f(x)) = (f(x) + 1) = √x +1 Shifting Functions: Vertical shift: y= f(x) + k  Shifts the graph vertically k units Horizontal shift: y= f(x + h)  Shifts the graph left h units if h is positive            Shifts the graph right h units if h is negative Scaling Functions: For c > 1… y = c f(x)  Stretches the graph vertically by a factor of c 1 y =  c  f(x)  Compresses the graph vertically by a factor of c y= f(x/c)  Stretches the graph horizontally by a factor of c y = f(cx)  Compresses the graph horizontally by a factor of c Reflecting Functions: y = ­f(x)  Reflects the graph across the x­axis y= f(­x)  Reflects the graph across the y­axis   1.3: Unit Circle:  sin ∅ =  opp csc ∅ hyp hyp 1 =  opp    sin     adj cos ∅ =  sec ∅ hyp hyp 1 =  adj    cos   opp sin tan ∅ =    tan ∅ =  adj cos adj 1 cot ∅ =  opp    tan Trig Functions Positive/ Negative:     S A sin positive all positive T C tan positive cos positive **All Students Take Calculus Trig Graphs: Trig Identities: cos2 ∅  + sin∅  = 1 2 2 1 + tan ∅  = sec∅ 1 + cot ∅  = csc∅ cos(A + B) = cosAcosB – sinAsinB sin(A + B) = sinAcosB + cosAsinB Double­Angle Formulas: 2 2 cos2 ∅  = cos ∅  ­ sin∅ sin2 ∅  = 2sin∅ cos ∅ Half­Angle Formulas: 2 1+cos2∅ cos ∅  =  2 sin ∅  =  1−cos2∅ 2 y = af(b(x + c)) + d a= vertical stretch/ compression; refection across y = d if negative b= horizontal stretch or compression; reflection across x = ­c if negative  c= horizontal shift d= vertical shift applied to the sine function  General sine function: 2π f(x) = A sin ( B (x – C)) + D |A|  = amplitude  vertical distance from center of sine curve |B|  = period  smallest cycle of the function    C = horizontal shift D = vertical shift **Whatever value B is in the function, divideπ2  by it to find the new period


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