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# Math 103, midterm 1 study guide MATH 103 001

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This 11 page Study Guide was uploaded by Cambria Revsine on Tuesday January 26, 2016. The Study Guide belongs to MATH 103 001 at University of Pennsylvania taught by William Simmons in Spring 2016. Since its upload, it has received 84 views. For similar materials see Intermediate Algebra Part III in Mathematics (M) at University of Pennsylvania.

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Date Created: 01/26/16

Math 103-001 1.1: Domain and Range of Functions: Domain= all input values of a given function What x-values 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: ▯ Find the domain and range of ???? ???? = 196 − ???? 196 − ???? ≥ 0 ???? ≤ 196 ???? ≤ 14, ???? ≥ −14 Domain= [-14,14] ▯ 196 − ±14 = 0 196 − 0 =14 Range= [0,14] Graphing Piecewise Functions: Graph normally for each set of x-values Ex: −????, ???? < 0 f ???? = ???? , 0 ≤ ???? ≤ 4 4, ???? > 4 Properties of Graphs: Increasing when x < 1 and2f(x ) < 1(x ) à2Going SE Decreasing when x < x1and 2(x ) > f1x ) à 2oing NE Even if f(-x) = f(x) à symmetrical across the y-axis Odd if f(-x) = -f(x) à symmetrical about the origin 1.2: Composite Functions: (f ° g)(x) = f(g(x)) à plug in the g(x) function into the “x” of f(x) (g ° f)(x) = g(f(x)) à plug in the f(x) function into the “x” of g(x) Ex: If ???? ???? = ???? and ???? ???? = ???? + 1, find (g ° f)(x) (g ° f)(x) = g(f(x)) = (f(x) + 1) = √???? +1 Shifting Functions: Vertical shift: y= f(x) + k à Shifts the graph vertically k units Horizontal shift: y= f(x + h) à Shifts the graph left h units if h is positive à Shifts the graph right h units if h is negative Scaling Functions: For c > 1… y = c f(x) à Stretches the graph vertically by a factor of c ▯ y = ▯(x) à Compresses the graph vertically by a factor of c y= f(x/c) à Stretches the graph horizontally by a factor of c y = f(cx) à Compresses the graph horizontally by a factor of c Reflecting Functions: y = -f(x) à Reflects the graph across the x-axis y= f(-x) à Reflects the graph across the y-axis 1.3: Unit Circle: ▯▯▯ ▯▯▯ ▯ sin∅= csc∅= à ▯▯▯▯ ▯▯▯ ▯▯▯ cos∅= sec∅= à ▯▯▯ ▯▯▯ ▯▯▯ tan∅= ▯▯▯ à tan∅= ▯▯▯ cot∅= ▯▯▯ à ▯ ▯▯▯ ▯▯▯ ▯▯▯ ▯▯▯ Trig Functions Positive/ Negative: S A sin positive all positive T C tan positive cos positive **All Students Take Calculus Trig Graphs: Trig Identities: 2 2 cos ∅ + sin ∅ = 1 2 2 1 + tan ∅ = sec ∅ 1 + cot ∅ = csc ∅ cos(A + B) = cosAcosB – sinAsinB sin(A + B) = sinAcosB + cosAsinB Double-Angle Formulas: 2 2 cos2∅ = cos ∅ - sin ∅ sin2∅ = 2sin∅cos∅ Half-Angle Formulas: 2 ▯▯▯▯▯▯∅ cos ∅ = ▯▯ ▯▯▯▯∅ sin ∅ = ▯ 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 à General sine function: ▯▯ f(x) = A sin ( ▯x – C)) + D ???? = amplitude à vertical distance from center of sine curve ???? = period à smallest cycle of the function C = horizontal shift D = vertical shift **Whatever value B is in the function, divide 2???? by it to find the new period 1.5: Exponential Rules: If a > 0 and b > 0… x y x+y a▯• a = a ▯ = ax-y ▯▯ (a ) = (a ) = a xy x x x a▯• b = (ab) ▯ = ( )x ▯▯ ▯ Exponential Growth/ Decay: y = y e à exponential growth if k > 0 0 exponential decay if k < 0 *y 0s a constant rt y = Pe à 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: One-to-one function: when each range value (y) has one distinct domain value (x) à Passes the horizontal line test Inverse Functions: -1 Notation: f (x) f (b) =a if f(a) =b -1 -1 à 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: ▯ Find the inverse of y = x + 1 ▯ ▯ x = y + 1 ▯ 2x = y + 2 y = 2x – 2 Logarithmic Functions: y = log a is the inverse of y = a (a > 0, a ≠ 1) x y = ln x is the inverse of y = e log x = logx 10 log e = lnx ln e = 1 Algebraic Properties: lnbx = lnb + lnx à Product rule ▯ ln = lnb – ln x à Quotient rule ▯ ln = -lnx à Reciprocal rule ▯ r lnx = rlnx à Power rule Inverse Properties: alogax= x (a > 0, a ≠ 1, x > 0) x log a = x (a > 0, a ≠ 1, x > 0) elnx= x (x > 0) lne = x (x > 0) a = e (lna)x= exlna log x = ▯▯▯ (a > 0, a ≠ 1) a ▯▯▯ Inverse Trig Functions: -1 ▯▯ ▯ y = sin x is the number in ▯ ,▯] for which sin y = x -1 y = cos x is the number in [0, ????] for which cos y = x 2.1: Average Rate of Change: d/t; distance travelled over time elapsed ▯▯▯▯ ▯ = ▯ ▯ ▯▯▯(▯ ▯ = ▯ ▯▯▯▯ ▯▯(▯ ▯ (h ≠ 0) ▯▯▯▯ ▯ ▯▯▯▯ ▯ ▯ à aka secant slope between the two points Instantaneous Rate of Change: Rate at a given time à find the average rates of change (secant lines) from a point closer and closer to the given point à this rate is the slope of the tangent line, which cuts through the given point 2.2: Limits: The y-value a function approaches as the function approaches a given x-value from both sides lim???? ???? = ???? ▯→▯ **c= x-value, L= y-value ** L does not necessarily equal f(x) at c Limit Laws: If lim???? ???? = ???? and lim???? ???? = ???? ▯→▯ ▯→▯ lim ???? ???? + ???? ???? = ???? + ???? à Sum Rule ▯→▯ lim ???? ???? − ???? ???? = ???? − ???? à Difference Rule ▯→▯ lim ???? • ???? ???? = ???? • ???? à Constant Multiple Rule ▯→▯ lim ???? ???? • ???? ???? = ???? • ???? à Product Rule ▯→▯ lim ▯(▯)= ▯ (M ≠ 0) à Quotient Rule ▯→▯ ▯(▯) ▯ lim[???? ???? ] = ???? (n is a positive integer) à Power Rule ▯→▯ lim ▯ ????(????) = ▯ ???? (n is a positive integer) à Root Rule ▯→▯ Sandwich Theorem: If ???? ???? ≤ ???? ???? ≤ ℎ(????) in an interval containing c (except possibly at x=c itself) and If lim???? ???? = limℎ ???? = ???? ▯→▯ ▯→▯ Then lim???? ???? = ???? ▯→▯ lim????????????ø = 0 ø→▯ lim????????????ø = 1 ø→▯ 2.4: One-Sided Limits: Limits as x approaches c from the right or left ▯→▯▯ ???? ???? = ???? à Right-hand limit lim ???? ???? = ???? à Left-hand limit ▯→▯▯ **A function has a limit as x approaches c if it has left- and right-handed limits and if those two limits are equal ????????????ø lim = 1 ø→▯ ø ????????????ø − 1 lim = 0 ø→▯ ø 2.5: Continuity: A function is continuous if its outputs vary consistently with its inputs **Can be drawn without lifting the pencil f(x) is continuous at point x=c if… 1. f(x) exists 2. lim????(????) exists ▯→▯ 3. lim????(????) = ????(????) ▯→▯ lim▯????(????) = ????(????) à function is right-continuous ▯→▯ lim▯????(????) = ????(????) à function is left-continuous ▯→▯ Properties of Continuous Functions: If functions f and g are continuous at x = c, then the following are continuous at x = c f + g à Sums f – g à Differences k • f à Constant multiples f • g à Products f/g (g(c) ≠0) à Quotients n f (n is a positive integer) à Powers ▯ ???? (n is a positive integer) à Roots Composite of Continuous Functions: If f is continuous at c and g is continuous at f(c), then the composite g º f is continuous at c Limits of Continuous Functions: If g is continuous at the point b and lim???? ???? = ????, ▯→▯ Then lim????(???? ???? ) = ???? ???? = ????(lim???? ???? ) ▯→▯ ▯→▯ Ex: lim???????????? ▯▯ 1 − ???? = ???????????? (lim 1 − ???? ) ▯→▯ 1 − ???? ▯ ▯→▯ ▯ =???????????? (lim ) ▯→▯ ▯▯ ▯ ▯ =???????????? ▯ = ▯ Intermediate Value Theorem: If f is continuous on a closed interval [a,b], and if y0is any value between f(a) and f(b), then y = 0 f(c) for some c in [a,b] Ex: Show that there is a solution to x – x – 1 = 0 between 1 and 2 3 f(x) = x – x – 1 f(1) = 1 – 1 – 1 = -1 f(2) = 2 – 2 – 1 = 5 Since -1 < 0 and 5 > 0, there must be a solution in between 1 and 2 **Try values in between 1 and 2 to get a more accurate solution 2.6: Limits Involving Infinity: lim ????(????) ▯→ ±▯ **Plug in “a really big number” into x to represent infinity Limits at Infinity of Rational Functions: ±∞ (depending on signs of coefs) à if highest degree of num > highest degree of denom ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯à if highest degree of num = highest degree of denom ▯▯▯▯▯▯▯ ▯▯▯▯▯▯▯▯▯▯▯ ▯▯ ▯▯▯▯▯ 0 à if highest degree of num < highest degree of denom Horizontal Asymptotes: y = b of function y=f(x) if lim ????(????) ▯→ ±▯ **H.A. = the limit Oblique/Slant Asymptotes: In a rational function, if the highest degree of the numerator is one more than the highest degree of the denominator, the function has an oblique asymptote **Divide the numerator by the denominator to find the equation of the asymptote Vertical Asymptotes: x = a of function y=f(x) if ▯→▯m▯???? ???? = ±∞ or ▯→▯ ▯ ???? = ±∞ **V.A. of a rational function is where the limit does not exist (denominator = 0) Dominance: For limits, log < polynomials < exponents Ex: ???????????????? ▯→▯ ???????????? = 0 ???????????? ▯→▯ ????▯ = 0 3.1: Tangent of a Function: Slope of a curve y=f(x) at the point P0(x ,0f(x )): ???? ????▯+ ℎ − ????(???? ) ▯ ???? = lim ▯→▯ ℎ à tangent line at P has this slope (recap of 2.1) **Plug in point P and m into y=mx + b to find b to complete the tangent equation Derivatives: Same as this slope of the tangent line ???? ???? + ℎ − ????(???? ) ???? ???? = lim ▯ ▯ ▯ ▯→▯ ℎ à aka derivative of function f at poin0 x

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