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Unformatted text preview: December 14 MATH 1 13 Total Points: 200
2006 FINAL EXAM Follow directions carefully:
0 Write your name section number and instructor’s name on this test and on each of the 11 answer sheets. 0 Number the answer sheets 1 through 11 0 Do each problem on ONE answer sheet as directed on the test. Mark your answers clearly. Show all appropriate work. Write answers in simplified form.
0 You may use graphing calculators, but show work algebraically and give exact answers where indicated. Use both front and back of a sheet if needed. If more than one answer is given, or if the answer cannot
be found, you may lose points. POINTS MAY BE DEDUCTED lF DIRECTIONS ARE NOT FOLLOWED. Answer problem 1 answer sheet # 1 1. Solve the following for x, showing your work algebraically:
3 (a) (x2 +4x+7)5 =8
[6] (b) (x—8)2 —32 = 0
[61 (c) log3(x—2)+log3x=1
[8] Answer problem 2 on'anSWer sheet # 2 2. Show your work algebraically for each of the following:
(a) Simplify the following, writing with only positive exponents: — 52 + rt‘I + sz + 81
[5] (b) Subtract and simplify:
[5] 1
(c) Solvefor a: —+__:__ [5] Answer problem 3 on answer sheet # 3 mix—_ x
x2 +5x—14 x—2 f(x+h)f(x),h¢0 3. (a) f (x) = 8x — 3 Find and simplify the difference quotient h [5] (b) Let f(x) = 3 , g (X): 2 Find: (f/g)(x)
x2 x
[5] (c) (i) A company that manufactures MP3 players has a fixed cost of $7,000. it costs $80 to produce each
player. The total cost for the company is the sum of its fixed cost and variable costs. If x
represents the number of MP3 players produced, write the total cost, C as a function of x. ’ (ii) Find and interpret C0 00)
[5'3] 7.“ ,._~_m—*~m_—m—W.—.WW~ Answer problem 4 on answer sheet # 4 4. (a) If a function is an even function, its graph will be symmetric with respect to ?
[4]
3x + 12
(b) g(x) =
x + 3 State the equation(s) of the vertical asymptote(s) and the horizontal asymptote(s) of the graph of g. [6]
(c) f(x) = 5(x + 2)2 —10
[4] (i) What is the vertex of the graph of f ? (ii) State whether the function f has a maximum value or a minimum value, then state what that
value is.
[4] (iii) What is the range of the function f ?
[4]
M Answer problem 5 on answer sheet # 5 (ii) u(x)=Vx—3+«jx+6 he) = log 27 (3 — x) 5. (i) f(t) = 400e"l (iii) g(x) = x3 — 25x2 (iv) (a) Specify the domain of each of the above functions. [8] (b) If the value of k in part (i) above is negative, what does that indicate about the function f ?
[3] (0) Determine the zeros of the function gin (iii) above, and specify whether the multiplicity of each zero is
even or odd.
[4] (d) Determine the exact value of the yintercept of the function h in (iv) above. [4] Answer problem 6 on answer sheet # 6
6. (a) Simplify: e'"(2'+3) +1ne"12
[4] (b) Write as a single log expression: 21nx4 — %1n(x — 9)
[4] (c) A radioactive isotope is decaying according to the function y = 400ek’, where y is the amount in grams of
the isotope that remains after t years. The 1/2 life of this isotope is 30 years. (i) How many grams will be left after 30 years?
[4] (ii) Find the value of k.
[6] »‘._——v————.._———_._——_———.——~—_————.—_—p————~ Answer problem 7 on answer sheet # 7
7. (a) Is the line x — 6y = 24 parallel, perpendicular, or neither parallel nor perpendicular to the line that passes through the points (0,1) and (1, 5) ?
[5] (b) Determine the equation of the vertical line that passes through the point (5,  12), and describe its
slope. [4] (c) The linear function f (x) = —0.34x +1.9 models the percentage of US taxpayers who were audited by the IRS, f (x) , x years after 1998. Find the slope of the line and describe what this means with respect
to the percentage of taxpayers audited and the number of years after 1998. [5] y (d) The graph to the right represents a function, h. (i) As x—>oo, h(x)——> ?
(ii) As x—)2+, h(x)——> 7
[5] Answer problem 8 on answer sheet # 8 8. (a) Draw a graph on your answer sheet that represents a system of equations, one quadratic and one
linear, that has two solutions.
[4]
2x — y = —2
(b) Solve the following system of equations. 2 + 2 Show your work algebraically.
y = x [6] (0) Determine whether or not the point (3, O, 2) a solution of the following system of equations. Justify your
answer: [5]
x+3y—52=—13 —2x+y—z=4
4x—5y+z=—10 Answer problem 9 on answer sheet # 9 9. (a) A pool measuring 30ft. by 60 ft. is surrounded by a path of uniform width, x (see figure). if the area of the
pool and the path combined is 3496 sq. ft., what is the width of the path? Write a mathematical model
and show your work clearly. [8]
(b) Fireworks are launched into the air. The function s(t) = ~16t2 + 320t + 3 models the height, s(t), in feet, t seconds after they are launched. When should the fireworks explode so that they go off at the
greatest height (i.e., after how many seconds is the maximum height reached)? [6] Answer problem 10 on answer sheet # 10 10. (a) Evaluate the product AB, if possible: = [6 — 3] : :2 — 5]
0 2 1 O
[5] (b) A few steps in rowreducing the following matrix have been performed. Specify the values of
r,s,t and w on your answer sheet. 13—2g—1 13—35—1 —410§9—>013 r§s 610129 0—8t§w
l4] 3
(c) Evaluate: Z n!(n +1)
n=1 [5] W
Answer problem 11, on answer sheet # 11 11. (a) On your answer sheet, graph the region determined by the following constraints. Label the solution
region l6]
7 r ~4x+ y is 16
x+ 3y 5 15
x _>_ 0
y 2 0 (b) Determine the vertices of the region. [5] (c) A Holiday Concert is being presented by a community association, and the proceeds are to be donated
to a charity. Each adult ticket costs $10 and each student ticket costs $5. The number of adult tickets
sold must be at least twice the number of students tickets sold. And, the concert hall can accommodate
no more than 600 people. They need to sell at least 75 student tickets and at least 200 adult tickets. The association wants to maximize the revenue for the concert.
Assume that x represents the number of adult tickets sold, and y represents the number of student tickets sold. [4] (i) Write an equation that represents the objective function, specifying clearly what your variables
represent. [6] (ii) Set up the system of constraints. NOTE: You do NOT have to graph the system of constraints. The End
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 Fall '08
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 Math, Derivative

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