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Unformatted text preview: MATH 1 13
FINAL EXAM December 15, 2005 Total Points: 200 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.
0 Mark your answers clearly. You must show all appropriate work in order to receive credit for an answer. You
may use graphing calculators, but show work algebraically and give exact answers where indicated.
0 Clearly indicate your answer, and write answers in simpliﬁed form.
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 IF DIRECTIONS ARE NOT FOLLOWED. Answer problem 1 on answer sheet # 1 2 —
3 g(x)=x2—10 , h(x)=x 10 mob—rm ’ '5—
[12] (i) Find: [igjoo (ii) Find: (f og)(x) —1
[4] (b) h(x) = (—27) x + 32_x Find 11(3) , writing your answer as an exact answer. [4] (c) f(x) = log7 x + me“ Find: (f)(7) 2
[4] ((1) Write —3— In x3 1n 2x as a single log expression. 1. (a) Let: (iii) What is the domain of f .7 Answer problem 2 on answer sheet # 2
2. Find the exact values of the xintercepts for each function below, showing your work algebraically.
[5 ea.] (a) h(x) = x2 — 6x — 27 3 +9
(b) f(x)= x
x—9 (c) g(x) = 11fl(x 10)
(d) h(x) = e"+1 — 40 Answer problem 3 on answer sheet # 3 3. A line passes through the points A (10,  2) and B (8, 4). [5 ea.] (a) Find the distance from point A to point B.
(b) Find the equation of the line that passes through points A and B.
(c) Find the equation of the circle whose center is the midpoint of line segment AB, and whose radius is 9 units.
(d) Give the equation of the vertical line that passes through point B Answer problem 4 on answer sheet # 4 4. (a) Determine the zeros of the following function, stating the multiplicity of each zero: f (x) = (x — 3)2 (x — l)
[16] (b) Evaluate the following, correct to 3 decimal places, by ﬁrst writing the expression in terms of natural logarithms: 10g 7 36 (c) Simplify, writing the result with only pesitive exponents: — + 5 y 3
((1) Find the sum: 201 + 4)!
n=1 Answer problem 5 on answer sheet # 5 f(x) 5. For the function, f, whose graph is on the right:
[15] (a) State the domain of f.
(b) Determine the interval(s) over which f is increasing.
(c) Determine the zeros of the function f.
((1) Does the function f have an inverse ? Explain how you can tell. ‘5 (6) State the value of f (4) Answer problem 6 on answer sheet # 6 6. [4] (a) State the domain of the function f (x) = ln(x — 18) . [4] (b) Write the equation of the function that represents reﬂecting the graph of y = x about the x—axis and shiﬁing it 2 units to the left and 9 units down. [4] (c) The vertex of a parabola which opens downward is the point (300, 2000). The function represented by
this parabola describes the proﬁt, P, in dollars, that a company makes when x units are sold. What is the
maximum possible proﬁt the company could make ? [4] (d) If a system of equations is made up of two equations that represent parallel lines, how many solutions
will the system of equations have ? Answer problem 7 on answer sheet # 7 4 + x — 3x
7 (a) Solve for x: —— 3 —— =
x—2 x—2
[6 ea.]
3
5 (b) Solve for x: (x + 5) = 8 (c) Solve the system of equations represented by the following matrix by ﬁrst replacing row 3 with the matrix
row operation 2R2 + R3 . Write the new matrix, then use back substitution to solve the system. 1 3 0 g1
0 —2 5§—1 0 4 —3§9 Answer problem 8 on answer sheet # 8
8. The accompanying diagram represents a corral that is to be constructed. The amount of fencing needed for the corral is 500 Pt.
X m y n I n
[4] (a) Write a quadratic function that expresses the area of the corral in terms of its Width, x. [4] (b) Find the dimensions that will give the maximum possible area of the ﬁeld.
[5] (c) Solve the following system of equations, stating clearly what the solution is. Show your work algebraically. 2x _§
3 y 3
4x—6y=10 Answer problem 9 on answer sheet # 9
9. (a) A rectangular lot whose perimeter is 640 ft. is fenced along all 4 sides. An expensive fencing along
the lot’s 2 lengths costs $40 per foot. A less expensive fence along the 2 widths costs $15 per foot.
[6] The total cost of the fencing along the three sides comes to $21,600. Determine the dimensions of the lot. [4] (b) Explain what is meant by the statement: “The halflife of radium is 1620 years”.
(c) The population of a city is given by P = 200,000e kl , where t is the time in years with t = 0
corresponding to the year 1998. In 2000, the population was 230,000. [3] (i) What was the population in 1998 ?
[4] (ii) Find the value of k correct to 5 decimal places.
[4] (iii) Determine how many years after 1998 it will take for the population to double. [6] (d) Solve for x: (a) log3 (x — 8) + 10g3 x = 2 Answer problem 10 on answer sheet # 10
5x —1
x — 3 10. [4] (a) State the equation(s) of the horizontal asymptote(s) of the graph of the function g(x) = (b) The accompanying graph represents a function, f.
[4] Complete the following: i) Asx—)oo, f(x)—> . ii)Asx——)2+, f(x)—> :3 . f(x+h)—f(x) MO
h 9 [5] (c) f (x) = 5x Find and simplify the difference quotie t‘ Answer problem 11 on answer sheet # 11
1 1. (a) Draw a set of axes on your answer sheet and sketch the graph of the following system of inequalities. Label the
solution region “ S “ and label all boundary lines with their equations.
[6] 2x + 5y > 10
y S 3
x < 6
(b) The region of the graph labeled “S” represents the solution region of the system of constraints listed below.
The objective function z = 3x + 4y
is subject to the constraints:
x 2 2
2x + 2y S 12
x + 2y S 8 yZO [6] (i) Determine the coordinates of vertices A and B. Show your work algebraically. SJ
[4] (ii) Find the maximum value of the objective function , z , given above, subject to the given
constraints. [2] (iii) Suppose that in the above problem x represents the number of model X plasma TV’s sold, in 100’s, and y
represents the number of model Y plasma TV’s sold, in 100’s. Suppose the objective function z above
represents the proﬁt, in 1000’s of dollars, a company makes on selling model A plasma TV’s and model B
plasma TV’s. Explain the meaning of the maximum value of the objective function, z which you found in
(b) above, in relation to the number of TV’s sold. THE END
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This note was uploaded on 04/04/2010 for the course MATH 113 taught by Professor Staff during the Spring '08 term at Maryland.
 Spring '08
 staff
 Algebra

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