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NAME. ID .SECTION
Nov. 1, 1995 3
MATH 115 EXAM 2
INSTRUCTIONS: Write your name, section number, and ID number on this test sheet and each of the answer sheets. Number the answer sheets from 1 to 4. Read the exam carefully, answer the problems on the sheets as indicated in the instructions for each problem and mark your answers clearly. You must show all appropriate work in order to receive credit for an answer. Good luck! NO CALCULATORS OR AUDIO DEVICES PERMITTED.
Answer Problem 1 on this sheet. (2i) 1. Graph of each of the following equations. Name the graph (circle, ellipse, hyperbola, parabola, polynomial, rational, exponential, logarithm). Label points such as centers or vertices with their coordinates. Asymptotes should be clearly shown (and should have appropriate slope). a. 4x2 + y2 = 36________ __ b. 16y2 -9x2 =
c. 2x = 4 + y
PLEASE TURN OVER FOR THE REMAINING PROBLEMS. 3-
Answer Problem 2 on this sheet. (14) 2. Sketch the graph of each of the following functions. Your graph should show x and y intercepts labeled with their values and any asymptotes. a. y = Iog2 (x + 1) - 3 b. y = -^ - + 4
Answer Problem3 and 4 on answer sheet 1.
do) 3. Let f(x) = -s^ (x f!V-3)(x + 4) Find each of the following for f and its graph. You do not need to sketch the graph. X. •"•* 1 a. x-intercept(s): b. y-intercept: c. vertical asymptote(s): d. horizontal asymptote(s):
(«) 4. Sketch the graph of a parabola that has its vertex at (1, 2) and passes through the points (-2,0) and (4,0). Then write an equation for that parabola.
Answer Problem 5 on answer sheet 2. (18) 5. Simplify each of the following:
a. 3 c. ln(enVe)
Answer Problem 6 on answer sheet 3. (20) 6. Solve each of the following equations: a 2x = (4x+l)
b. Iog2x + Iog2 (x + 1) - Iog 2 3= 2
Answer Problem 7 on answer sheet 4. (9) 7. Find two numbers (not necessarily integers) whose difference is 10 so that the sum of the larger number and the square of the smaller number is a minimum. You must clearly identify any variables used, write equations which satisfy the conditions of the problem and use the techniques of chapters 3 and 4 in solving this problem.
3-
- Sketch the graph of each of the following functions. Your graph should show x and y intercepts labeled with their values and any asymptotes. a. y = Iog2 (x + 1) - 3
For each part—7 points (14 points total) -5 pts incorrect type of graph (e.g., exponential rather than logarithm, y =x type instead of y = x4 type) -2 pts incorrect or omitted reflection -2 pts incorrect or omitted translation -1 pt each incorrectly placed intercept -1 pt intercepts not labeled with their values
- Let f(x) = (x - 3)(x + 4) -f———. Find each of the following for f and its graph. You do not need to sketch the graph.
a. x- intercept(s): x = 3, - b. y-intercept: y = 12 c. vertical asymptotes: x = 1, x = - d. horizontal asymptotes: y = 1
10 points total -2 pts each incorrect or omitted or extraneous answer asymptotes must be expressed as equations (not numbers)
- Sketch the graph of a parabola that has its vertex at (1, 2) and passes through the points (-2,0) and (4,0). Then write an equation for that parabola.
8 points -3 pts incorrect graph -5 pts incorrect general form for equation -2 pts each incorrect value in equation (a, h, or k) (up to -4 pts)
y = a(x - 1) + 2 0 = a(-3)2 + 2 2
"5. Simplify each of the following: 6 points each = 18 points total
(3 pts) so 8 a
(3 pts)
= 16 (1 Pl) _ ~—4^ (2 pts)
3a = -4 or a = -T (3 pts)
c. In (enVe) = In e71 + In Ve (3 pts) = 7t + r (3 pts)
-3 pts each incorrect use of a log or exponential rule -2 pts any other algebraic error -1 pt each ASMD or copying error
- Solve each of the following equations:
2X _ , 2X = (22 ) x= 4x + 4 -3x = 4 v _^4 3
b. Iog2x + Iog2 (x + 1) - Iog 2 3= 2 ,Iog
x + x - 12 = 0 (x + 4)(x - 3) = 0 x = ^, 3 Check: Iog2 (-4) not defined Iog2 3 + Iog2 (4) - Iog2 3 = 2 okay solution: x = 3
10 points each = 20 points total either problem: -3 pts each incorrect use of a log or exponential rule -2 pts any other algebraic error -1 pt each ASMD or copying error -2 pts do not check for extraneous solution in log equation
- Find two numbers (not necessarily integers) whose difference is 10 so that the sum of the larger number and the square of the smaller number is a minimum. You must clearly identify any variables used, write equations which satisfy the conditions of the problem and use the techniques of chapters 3 and 4 in solving this problem. x = larger number 9 points y = smaller number (1 pt) x-y=10 (Ipt) x + y2 = S (2 pts) S(x) = x + (x-10) = x2 - 20x + 100 + x = x2 - 19x + 100 (1 pt) min S for x = "^ = ~ ~2~ = ~2~ (3 pts)
-1 pt each omitted identification of a variable -4 pts each omitted or incorrect equation -3 pts do not use techniques of chapter 3 -3 pts incorrect formula (-b/2a) -1 pt each incorrect substitution into correct formula -2 pts any other algebraic error
' = x-10 = -± (Ipt)
