2. Use your graphing calculator to graph each of the functions below over the interval 2, 2
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1 MSLC Math 0 Final Eam Review Disclaimer: This should NOT be used as your only guide for what to study. if 0. Use the piece-wise defined function f( ) to answer the following: if a. Compute f(0), f(), f(-), and f(). Plot the points you found above and sketch a complete graph of y = f().. Use your graphing calculator to graph each of the functions below over the interval, and approimate any local etrema. Also, determine the intervals where the functions are increasing and decreasing. Round your answers to three decimal places where appropriate. f a. g c. h d. k. Determine the average rate of change of the functions between the given values of. a. h 7 from to f c. g from to t from 0 to a. Write the equation of the function F transformed in the following ways: a. shifted units to the left, and shifted up units reflected about the -ais, then shifted down units c. shifted unit to the right, and vertically stretched by a factor of d. shifted unit to the left, then reflected about the y-ais. Given f and g find: a. f g g f c. f f d. g g 6. Algebraically find the maima and minima of the functions given. a. f g 7
2 7. Find the inverse of each function, state the domain and range of both the original function and its inverse. f a. g c. k d. h 8. Find all real zeros and the multiplicity of those zeros for the given polynomials, then find their y-intercepts and sketch the graph. p a. p c. p 9. Find all zeros (both real and comple) of the following polynomials. a. P 7 P 6 c. P Factor the polynomials into linear and irreducible quadratic factors with real coefficients. P a. Q. Factor the polynomials into linear factors with comple coefficients. P a. Q. Find a polynomial with integer coefficients having the following properties: a. Degree: Zeros: 0, i Degree: Zeros:, i
3 . Find all intercepts and asymptotes, then sketch the graph of each rational function. a. R R 6 c. R. Solve the following inequalities. Write your answer using interval notation. 9 a. 6 c. 0 d. e. 7 f.. Solve the equation. w w a c. ln( r) ln(r ) d. log ( ) e. f. log e g. 6 log 6 h. i. 6. How much needs to be invested now in order to have $000 in years given a.% interest rate that is compounded: a. daily? continuously? 7. Determine (correct to decimal places) how long it will take for $000 to double if it s invested in an account that gives 6% interest compounded: a) semiannually? b) quarterly? c) monthly? d) daily? e) continuously?
4 8. The frog population in a small pond grows eponentially. The current population is 70 frogs, and the relative growth rate is % per year. a. Find a function that models the population after t years. Use your equation to find the projected population after years. Round your answer to the nearest frog. c. Use your equation to find the number of years required for the frog population to reach 0 frogs. Round your answer to two decimal places. 9. In the circle pictured below, r is the radius of the circle, is the central angle of the sector, A is the area of the sector, and s is the length of the arc subtended by the central angle. Find: a. A and s if r inches, and radians r and s if A. square miles, and 6 c. A and if r meters, and s.7 meters d. r and A if s meters, and radians 0. Use the given information to find the eact value of the si trigonometric functions of the angle in the picture. a) a 8, b b) a, c c) sin 7 d) cot e) cos f) csc. For each part, determine the quadrant lies in and the values of the five remaining trigonometric functions. a) csc 6 and cot 0 b) sec and sin 0 c) tan and cos 0. Find the eact value of each of the following. 99 a. tan sin 0 cos cos
5 . Given triangle ABC with the following properties: b, c 7, B and C is an obtuse angle Find the measure of angle C. Round your answer to decimal places.. A team of surveyors have been hired to measure the distance across a canyon. Using a tree at point T on the opposite site of the canyon as a reference point, they established points A, B, and C and found the following distances: AB =. ft BC = 6. ft AC = ft a. Find the measure of angle ABC. Round your answer to decimal places. Find the distance TC across the canyon to the nearest foot.. Establish the identity: a. sin sec tan sin sec csc sin cos tan cot tan v sin v c. tan v sin v sin v cot cot y d. cot( y) cot cot y e. sin sin f. sin sec csc sin 6. Find the eact solution of: a. sin tan(6 ) 7 c. cos 8 d. tan 7. Given cot and sin 0 find: a. sin cos c. tan
6 8. Find all solutions of: a. cos 0 on the interval 0, cos sin cos 0 on the interval 0,8 c. sin 0 9. Convert to polar coordinates: a. i i 0. Use DeMoivre s Theorem to find a. z i z cos isin z in the standard a bi form:. Given u i j and v i j find: a. u v u v c. u v d. u v e. uv f. the angle (in degrees) between u and v. Round your answer to decimal places.. Find the complete solution for each of the following systems of linear equations. y 0 a. y 8 y z 0 y z 8 y z 7. A boat on a river travels downstream for 0 miles in one hour. The return trip back upstream takes. hours. How fast does the boat travel in still water, and how fast is the current?. A fruit stand sells a bo of Strawberries for $7 and a bo of Kiwi fruit for $0. If they sold a total of boes of fruit and had revenue of $0, how many boes of each fruit did they sell?. From the top of a 0 foot tall lighthouse, the angle of depression to a ship on the water is 7. How far is the ship from the lighthouse? Round your answer to two decimal places.
7 6. Find the equation of the parabola with verte at the origin and a directri of Find the equation of an ellipse with eccentricity of 9 and foci at 0, 8. Find the transverse ais, vertices, foci, and the equations of the asymptotes of the hyperbola described by the equation 9 y Write each of the following sums of a sequence using sigma notation. a Find the sum of the following: a. k k 0 k 0 c. n d. n0 j0 ANSWERS. a) 0 j 7 f ; f ; f f ; b). a. Min: 0,0 ; Increasing: 0, ; Decreasing:,0 Mins:,0 &,0 ; Ma:0, ; Increasing:,0, ; Decreasing:, 0, c. Min: (0,0); Increasing: 0, ; Decreasing:,0 d. Min: 0.78, 0.8 ; Ma: 0.78, 0.8 ; Increasing:, , ; Decreasing: 0.78,0.78. a) b) 6t c) a a a
8 . a) y b) y c) y.a) 0 6 f g 6. a) Min.:, b) Ma.: b)g f 7, 6 6 d) y 7. a) f Domain f, ; Range f, Domain f, ; Range f, b) g Domain g, ; Range g, Domain g, ; Range g, c) f f d) g g c) Domain k,, ; Range k,0 0, k d) h Domain k 8. a) Zeros: (multiplicity:); (multiplicity:); (multiplicity:); y-int: 0, Domain k,0 0, ; Range k,,,,,, k,, ; Range k Domain k,, ; Range b) Zeros: (multiplicity:); (multiplicity:); y-int: 0, c) Zeros: (multiplicity:); (multiplicity:); y-int: 0,9
9 9. a) Zeros: 0; i 7 ; i 7 b) Zeros: ; i ; i c) Zeros: ; i; i; + i ; i 0. a) b). a) i i b) i i. a) P ; b). a) no y-intercept, -intercept:,0 ; 6 Horizontal Asymptote: y 0; Vertical Asymptotes: 0; b) y-intercept: 0,0 -intercepts: 0,0 &,0 Horizontal Asymptote: y Vertical Asymptotes: ; c) y-intercept: 0, -intercepts:,0 Slant Asymptote: y ; Vertical Asymptotes: ;. a), b), 9,, f) c),, d),,0 e) 9,. a. w 7 c. r 7 d. g. or h. ln or 0 i. ln 7 e. f. ln ln ln ln ln ln
10 6. a. $80.9 $ a).7 yrs b).69 yrs c).8 yrs d). yrs e). yrs 0.t 8. a. nt 70e n 0 c. t.7 years 9. a. s. inches c. (radians) A.7 in r.676 miles s.68 miles A. m d. r 6. meters A m 8 0. a. sin 7 sin c. sin 7 d. sin 0 e. sin f. sin cos tan csc sec cot cos tan csc sec cot 7 7 cos tan csc sec cot 7 0 cos tan csc 0 sec cot 0 cos tan csc sec cot cos tan csc sec cot. a. is in Quadrant III; is in Quadrant II; sin 6 csc 6 sin cos csc sec 8 c. is in Quadrant IV; sin cos csc sec 8 6 tan 8 6 sec cot cos tan 8 8 cot tan cot. a C.9. a. ABC 0.8 TC feet
11 sin sec csc sec tan sin cos sin tan cot sin sin RHS sin sin cos sin RHS sin cos sin sin. a. RHS cos sin sin sin cos sin sin RHS cos sin cos RHS sin cos sin sin RHS cos sin cos cos cos sin cos cos sin sec sec tan tan (sec tan ) sin cos cos sin sin cos c. tan vsin v tan v sin v sin v sin v RHS sin v sin v sin v RHS sin v sin v sin v RHS sin v sin v sin v RHS sin v RHS cos v sin v RHS sin v sin v sin v d. cot cot y cot( y) cot cot y cos( y) RHS sin( y) cos cos y sin sin y sin cos y cos sin y cos cos y sin sin y sin cos y cos sin y RHS sin ysin sin ysin cos cos y sin sin y cot cot y cos y cos cot cot y sin y sin RHS sin sin sin sec csc sin LHS sin cos sin cos sin RHS sin sin e. LHS cos 0 f. LHS cos RHS sin cos sin cos csc sec sec csc 6. a. 6 c. d.
12 7. a. c. 8. a.,,, k c.,,, 9. a., or,, or, 0 0. a. 8 8i i. a. i j 6ij c. 6 d. 7 e. f a., y 6 7 6, y, z. Boat s speed in still water = mph; Current s speed = 6mph. 80 boes of Strawberries; boes of Kiwi fruit feet y y 0 8. Transverse ais: the -ais; Vertices:,0 ; Foci:,0 ; Asymptotes: y 9. a. 0 k k 999 k k k 0. a c. 99 d. 6
x x x 2. Use your graphing calculator to graph each of the functions below over the interval 2,2
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