( ) ( ) ( ) ( ) MATHEMATICS Precalculus Martin Huard Fall 2007 Semester Review. 1. Simplify each expression. 4a b c. x y. 18x. x 2x.

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1 MATHEMATICS Precalculus Martin Huard Fall 007. Simplif each epression. a) 8 8 g) ( ) ( j) m) a b c a b n f) t t ) h) k) n) ( ) i) + n z + l) 8 o) ( + ) ( + ) 9. Rationalize the denominator. a) Factor completel. a) ( + ) ( ) ( ) ( + ) + ( ) ( + ) ( ) ( ) ( ) ( ) f) g) Find all real solutions of each equation. a) + = = + 0 = 0 + = 0 = g) 8 = 0 h) = + 0 i) = 0 j) = 9 f) ( ) ( ) = 0. A rectangular garden is to be twice as long as it is wide. What should be its dimension if it is to have a total area of 80 m?

2 . Solve each inequalit. Give our answer in interval notation. a) < A car radiator contains 0 liters of a 0% antifreeze solution. How man liters will have to be replaced with pure antifreeze if the resulting solution is to be 0% antifreeze. 8. Let A(-) and B() be points in the plane. a) Find the length and midpoint of the segment AB. Find an equation for the line passing through the points A and B. Find an equation for the line perpendicular to the segment AB and passing through the midpoint of the segment AB. Find the equation for the circle having A and B as the endpoints of a diameter. 9. Find the radius and center for the circle = and sketch the graph. 0. Find the domain of the following functions and determine whether the are even or odd. + a) f ( ) = f ( ) = 0 f ( ) = 9 f ( ) = + ( ). Let f = +. Find. Let f ( ) f t+ ) ( ) a) ( + =. Find + a) ( f + f ) ( ) f + ( + ) ( ) f h f h ( + ) ( ) f h f h. Find the and intercepts of the given function and sketch the graph. f = + f ( ) = + 8 f ( ) = + a) ( ) f ( ) = + f ( ) if = if > f) f ( ). For the given function i) find the verte; ii) find the intercepts; iii) find maimum or minimum value that f takes; iv) find the domain and range; v) sketch the graph. f = + f ( ) = + + a) ( ) 9 if < = + if Fall 007 Martin Huard

3 . A sports center has skidoos to rent. The owner finds that if he charges $0 per da all of his skidoos will be rented. However for each $ increase in the price he will rent one skidoo less. What price should he charge to maimize his revenue? ( ) = + g ( ) = +. Let f and. Find the following function and state the domain of each. a) f + g f g f g g f f f f) g g 7. For the given function i) find the inverse f of f; ii) find the domain and range of f and f ; iii) verif that ( f f )( ) = and ( f f )( ) = iv) sketch the graph of f and f. ; a) f ( ) = + ( ) ( ) 8. Assuming that f is invertible find f ( ) if ( ) f =. f = + 9. Divide the following. + a) Factor the following polnomials. a) p ( ) = + + ( ) ( ) = + + ( ) p p = + 0 p = +. Solve the following equations. (Hint: find all rational zeros.) a) + 0=0 + = = = 0. Find the domain range intercepts and asmptotes (if an) for the following functions and sketch the graph. + a) f = f ( ) = e ( ) 9 + f ( ) = ( ) + f ( ) ( ) f ( ) = ln ( ) + f) f ( ) ( ). Write as a single logarithm. a) log log = log + = log + log log ( + ) + Fall 007 Martin Huard

4 . Write the epression as a sum difference and/or multiple of logarithms. a) log ln. Find the solution of the equation. a) + e = = 7 e 7e + 0 = 0 ln = log ( + ) = 7 f) log + log + = log ( ). A culture contains bacteria initiall. After an hour the bacteria count is 000. a) Find the population after hours. How long will it take for the population to double? 7. A man invests $000 in a mutual fund which pas 8% per ear compounded monthl. a) How much mone will the man have in ears? How long would it take for the amount to triple? 8. Find the central angle θ in a circle of radius m that subtends an arc of length m. Give the answer in degrees and in radians. 9. Find the area of a sector with central angle 0 o in a circle of radius m. 0. Find the eact value of the following epressions. a) cos sec cot tan sin f) sin cos g) cos 8 sin sin arccos 8 7 h) ( ) 7 i) sec( arctan ) j) csc( arcsin 7 ) k) arccos( cos ) l) arctan ( tan ) m) sin ( arcsin + arccos ) n) cos( arctan ). If cotθ = and θ is in quadrant II find a) sinθ cosθ tanθ secθ sin θ f) cos θ g) sin θ h) cos θ i) tan θ j) csc θ θ k) secθ l) cot. If sec θ = and tan θ < 0 find a) sinθ cosθ cotθ cscθ sin θ f) cos θ g) sin θ h) cos θ i) tan θ j) csc θ θ k) secθ l) cot. If sinα = and tan β = where α is in quadrant III and β is in quadrant II find the eact value of the trigonometric function. a) sin ( α + β ) sin ( α β ) cos( α + β ) cos( α β ) tan ( α + β ) f) cs c( α + β ) g) sec( α β ) h) cot ( α β ) Fall 007 Martin Huard

5 . Write the first epression in terms of the second if the terminal point determined b θ is in the given quadrant. a) cosθ in terms of tanθ if θ in quadrant III. sinθ in terms of secθ if θ is in quadrant II.. Find the amplitude period and phase shift and sketch the graph. f = cos ) f ( ) sin( ) a) ( ) (. Find the period and phase shift and sketch the graph. a) f ( ) csc( ) f sec f = tan + ) f ( ) tan ( ) = + f ( ) = sin( ) = ( ) = ( ) ( ) f = sec ( + ) ( ) ( = f) f ( ) = cot ( + ) 7. Verif each identit. cscθ + secθ tan θ + a) = cotθ + tanθ = cscθ sinθ + cosθ secθ tanθ sinθ + cot θ = cos θ + cos θ = cot θ sinθ cscθ sin θ + sin θ cscθ csc θ = f) cos θ sin θ = cos θ cosθ ( cosθ ) θ cosθ g) sec = h) tanθ = sin θ sin θ csc Acsc B i) sec( A B) = + cot AcotB j) A + B sin Asin B cot = cosb cosa 8. a) Write cossin as a sum of trigonometric functions. Write cos + cos as a product of trigonometric functions. 9. Rewrite the epression as an algebraic function of. a) sin ( arctan ) cot arccos ( ) 0. Solve each equation on the interval [ 0 ). a) tanθ = sin θ = θ sec = cos θ cosθ = 0 tan θ = sinθ f) sin θ + sin θ = 0 g) tan θ = secθ. Solve the following triangles. a) A = 0 a = b = 8. A =. C =. c =. a = b = c =7 A = b = c = 0. From a point A on the ground the angle of elevation to the top of a skscraper is 9. o. From a point B 00m from point A the angle of elevation is. o. Find the height of the skscraper. Fall 007 Martin Huard

6 ANSWERS. a) 0 b 0 a c 8 8 z g) 9 +9 h) ( ) l) m) ( + ) ( ) ( ) n). a) + + ( + + ) i) + + o) ( ) f) + n t j) ( + )( ) k). a)( ) ( + ) + 9 ( )( ) ( ) ( + ) ( )( + ) f) ( + )( ) ( + ) g). a) 8 g) h) ± ± i) 0 ( + ) + ( ) ( ) ( + ) 0 ± - 8 f) 9 ± j) meters ( ) ( ] [ ). a) ( ) liters [ ) ( ) 8. a) M ( 9 ) = + ( 9. Radius: Center: ( ) 0. a) /{ } odd [ 0) ( 0 ) ( ] [ ) even [ ] neither = + ) ( ) neither. a) t + t+ ( + h+ + ). a) + +. a) -int: = -int: = + 7 ( ) -int: = -int: = ( + + h) + + = -int: none -int: = Fall 007 Martin Huard

7 -int: = 0 -int: = 0 -int: = ± -int: = f) -int: = -int: =. a) i) ( ) ii) -int: -int: iii) Min : - iv) D: R : [ ) i) ( ) ii) -int: - 7 -int: iii) Ma : iv) D: R : ( ]. $80 per da. a) a) i) ( ) ( ) f = + ii) D. of f: [ ) R. of f : [ ) D. of f : [ ) R. of f : [ ) f) + + i) f ( ) = + ii) D. of f: [ ) R. of f : [ ) D. of f : [ ) R. of f : [ ) 8. f ( ) = + 9. a) a) ( ) ( )( + ) ( + ) ( ) f) ( )( )( + ) ( + )( )( + ) + +. a) - - ± - ½ Fall 007 Martin Huard 7

8 . a) D: 9 R : ( ) -int : -int: -8 H.A. : =9 D: 0 R : ( ) -int : none -int: -e H.A. : = 0 D: R : ( ) -int : - -int: H.A. : = D: ( ) R : -int : - -int: 8 V.A. : = D: ( 0 ) R : -int : e -int: none V.A. : = 0 f) D: ( ) R : -int : 7 -int: V.A. : =. a) log 0 log 0 ( + ) + ( ). a) log log ln. a) log ln ln. a) hours or. minutes 7. a) $79..8 ears 8. rad or 9. 0 m 0. a) ½ ln ln e 9 + g) h) i) 7 j) 7 k) m) 7 + n) a) g) h). a) g) h) f) f) l) 7 f) i) 7 j) k) 7 l) 8 i) 7 j) k) 8 7 f) 7 8 l) Fall 007 Martin Huard 8

9 . a) g) h). a) cosθ = sin + tan θ. a) Amplitude: Period: Phase Shift: sec θ θ = secθ Amplitude: Period: Phase Shift: f) Amplitude: Period: Phase Shift:. a) Period: Phase Shift: Period: Period: Phase Shift: Phase Shift: Period: Phase Shift: Period: f) Period: Phase Shift: Phase Shift: 7. a) cscθ+ secθ + sinθ cosθ cosθ+ sinθ sin + cos sin + cos sin cos sin + cos sin cos LS = = = = θ θ θ θ θ θ θ θ θ θ RS = cotθ + tanθ = + = = = LS cosθ sinθ cos θ+ sin θ sin cos sin cos sin cos θ θ θ θ θ θ tan θ+ sec θ secθ cosθ cosθ sec tan sec tan tan sin θ cos sin sin csc cos θ LS = = = = = = = = RS θ θ θ θ θ θ θ θ θ Fall 007 Martin Huard 9

10 ( cscθ+ )( cscθ) θ θ θ θ θ θ+ θ θ θ θ+ θ θ θ θ RS cot θ csc θ sinθ+ RS = = = = cscθ + = + = = LS csc csc csc sin sin cos cos cos cos LS = sin sin = sin cos = cot = θ LS = csc θ = = = = RS csc sin θ sinθ cosθ cosθ f) LS ( )( ) = cos θ sin θ = cos θ sin θ cos θ + sin θ = cos θ = RS θ g) LS = sec = = cos cos θ + θ = + cosθ ( cosθ) ( cosθ) ( cosθ) RS = = = = = LS sin θ cos θ ( cosθ)( + cosθ) + cosθ ( sin θ ) cosθ sin θ sinθ h) RS = = = = = = LS sin θ sinθ cosθ sinθ cosθ cosθ tanθ sin Asin B sin Asin B csc Acsc B sec cos Acos B sin Asin B cosacosb+ sin AsinB cot Acot B sin Asin B sin Asin B sin Asin B i) ( ) cos( A B) LS = A B = = + = = + = RS cos A+ Bsin AB sin A sin B A+ B j) RS cos Bcos A cot sin A+ Bsin AB = = = = LS 8. a) sin 7 + sin 7 cos cos 9. a) + 0. a) f) 0 g). a) B =. C =.8 c =.7 B =.8 C =. c =. B = 0. a =. b =. A = 9. B =.0 C = 8. B =. C = 08. a = m Fall 007 Martin Huard 0