MATH 1040 Test 2 Spring 2016 Version A QP 16, 17, 20, 25, Calc 1.5, 1.6, , App D. Student s Printed Name:

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1 Student s Printed Name: Instructor: CUID: Section # : You are not permitted to use a calculator on any portion of this test. You are not allowed to use any textbook, notes, cell phone, laptop, PDA, or any technology on either portion of this test. All devices must be turned off while you are in the testing room. During this test, any communication with any person (other than the instructor or his designated proctor) in any form, including written, signed, verbal, or digital, is understood to be a violation of academic integrity. No part of this test may be removed from the testing room. Read each question very carefully. In order to receive full credit for the free response portion of the test, you must:. Show legible and logical (relevant) justification which supports your final answer.. Use complete and correct mathematical notation. 3. Include proper units, if necessary. 4. Give exact numerical values whenever possible. You have 90 minutes to complete the entire test. On my honor, I have neither given nor received inappropriate or unauthorized information at any time before or during this test. Student s Signature: Do not write below this line. Free Response Problem Possible Points Points Earned Free Response Problem Possible Points Free Response Multiple Choice 33 Points Earned 8 6 Test Total 00 Page of 3

2 Multiple Choice. There are 3 multiple choice questions. Each question is worth 3 points and has one correct answer. The multiple choice problems will count as 33% of the total grade. Use a number pencil and bubble in the letter of your response on the scantron sheet for problems 3. For your own record, also circle your choice on your test since the scantron will not be returned to you. Only the responses recorded on your scantron sheet will be graded. You are NOT permitted to use a calculator on any portion of this test.. What is the equation for the function graphed below? ( pts.) a) f (x) = x+ + 4 b) f (x) = x+ 4 c) f (x) = x+ + 3 e) f (x) = x+ 3 d) f (x) = x 3. (3 pts.) Evaluate the following expression: sin sin 7π 6 + tan a) 3π c) π b) 5π d) 7π e) π Page of 3

3 3. Suppose f is such that for all values of x, 3x + 6x f (x) x 4x + 3. What is the value of lim f (x)? x ( pts.) a) b) 0 c) 3 d) - e) The limit cannot be determined from the given information. 4. Which of the following has the same solutions as log x ( pts.) a) x( x +) = 4 b) x ( x +) = 4 c) x ( x +) = e d) e) x x + = 4 x x + = e log ( x +) =? 5. Over which of the following intervals is the function f (x) = (x ) one-to-one? (3 pts.) a) (,) b) (, ) c) ( 0, ) d) (,) e) (, ) 6. Solve 4 5 x+ = 8 3x. (3 pts.) a) x = b) x = c) x = d) x = e) x = Page 3 of 3

4 7. ( pts.) Given that sinθ = 3 5 and tanθ = 3, determine cosθ. 4 a) 4 5 b) 3 5 c) 4 5 d) e) 0 8. According to the Intermediate Value Theorem, on which interval will (3 pts.) f (x) = x 5 + 4x 7x have a y-value of y = 3? a) (, ) b) (, 0) c) (, ) d) All of the above e) None of the above 9. What value for c should be chosen to make f(x) a continuous function? x +3x + 7 x 7 f (x) = x 4 c x = 7 (3 pts.) a) c = 5 b) c = 5 c) c = 8 d) c = 8 e) c = 4 Page 4 of 3

5 0. A function f x ( pts.) a) 5 b) 3 c). is defined by the table below. Evaluate f f ( 5) x f ( x) d) f ( f ( 5) ) is undefined e) There is insufficient information to answer the problem.. The displacement in centimeters of an oscillating particle is given by s t (3 pts.) = sinπt + 3cosπt, where t is measured in seconds. What is the average velocity of the particle over the time interval 0,? a) cm/sec b) cm/sec c) cm/sec d) cm/sec e) cm/sec Page 5 of 3

6 . Evaluate sin( arctan(3) ). (3 pts.) a) 0 b) c) 3 d) e) Of the four statements below, which must be true in order for lim f (x) to exist. x. lim f (x) exists x. lim f (x) exists x + ( pts.) 3. lim x 4. f () exists f (x) = lim f (x) x + a) only,, and 4 b) only and c) only,, and 3 d) only 4 e) all of,, 3 and 4 The Free Response section follows. PLEASE TURN OVER YOUR SCANTRON while you work on the Free Response questions. You are welcome to return to the Multiple Choice section at any time. Page 6 of 3

7 Free Response. The Free Response questions will count as 67% of the total grade. Read each question carefully. In order to receive full credit you must show legible and logical (relevant) justification which supports your final answer. Give answers as exact answers. You are NOT permitted to use a calculator on any portion of this test. Follow these special directions for all limits in the Free Response: Show all work. You will not be given any credit for using L Hopital s Rule to find the limits. Infinite limits should be answered with = or =, whichever is appropriate. If the limit does not exist (and cannot be answered as or ), state DNE.. (5 pts.) Given that sinθ = 5 3 and 3π θ π, determine the value of sin ( θ ). 3 5 sinx = sin xcos x = = 0 69 the "second side" of can be calculated with the Pythagorean Thm in the fourth quadrant, tangent and sine are negative sin x = 5 3 cos x = 3 cosine positive Triangle second side calculation cosine Double angle formula Correct answer (award this last point only if correct) Page 7 of 3

8 . (7 pts.) Solve cos x + cos x = cos x for [ 0,π ]. cos x + cos x = 0 ( cos x ) ( cos x +) = 0 cos x = x = π 3, 5π 3 cos x = x = π All terms to one side and combined Factor Set each factor equal to zero 0.5 for each factor Statement with solutions correct ( pt. for first quadrant solution) Statement with solution correct -0.5 incorrect notation -0.5 PER incident extra angles or angles outside of [0,pi] or extra solutions at least -4 attempting to dividing all terms by cosx -0.5 incorrect notation -0.5 even if all solutions are written down, but only one solution is boxed, or circled as the answer -5 never factored -6 guess and check (no supporting work) Page 8 of 3

9 3. (5 pts.) Determine the inverse function f ( x) of the function f ( x) = 3 x +. y = 3 x + xy + y = 3 xy = 3 y x = 3 y y f ( x) = 3 x = 3 x x 4. (6 pts.) Solve the following for x : ln(x 3) + ln(x ) = ln(x + 4) ln( x 3) ( x ) = ln( x + 4) ln x 5x + 6 = ln x + 4 x 5x + 6 = x + 4 x 7x 8 = 0 ( x 9) x + x = 9 = 0 x = not in domain of ln x Solve for x 3.5 Interchange variables.5 No penalty for not using f (x) notation instead of y OK to interchange first and then solve -/ final answer labeled f(x) instead of f (x) or y -/ no label on final answer -/ missing parentheses -/ excessive equals Points Awarded Combine logs Simplify arguments and set equal Solve 3 Eliminate solution not in domain -0.5 for any sign error -0.5 missing parentheses -0.5 excessive equals -5 combining logs with sum instead of a product (makes rest trivial) -4 combining logs on left w division instead of product 5. (5 pts.) Based on the graph of f (x) below, evaluate the following limits. lim x f (x) =.5 lim f (x) = lim f (x) dne + x x lim f (x) = x lim f (x) = 0 x 0 each part right or wrong -0.5 or incorrect/missing = - more than incorrect/missing = Page 9 of 3

10 6. (6 pts.) On the axes below, sketch the graph of an example of a function f( x ) that satisfies all of the given conditions. Indicate any asymptotes with dotted line(s). lim f (x) = f (0) does not exist lim f (x) = lim f (x) = x 0 x x pts. 0.5 pts. pt. pt. + f (x) = lim f (x) = f () = lim f (x) = lim x x pt. pt. 0.5 pts. 0.5 pts. and f( x) is defined everywhere except at x = and x = 0 - fails vertical line test (- for excessive places where not a function) - not defined everywhere - i.e. left off a chunk of graph -0.5 open circle at x=3-0.5 no dotted line at VA Answers can vary for example: left most side does not have to have horizontal asymptote, right side need not be a line, etc x + x + 7. (5 pts.) Evaluate the following limit: lim x x x + x + 4 lim = = x x small neg Shows evidence of understanding that denominator gets small Recognizes limit goes to negative infinity (correct work must be shown; no credit for correct answer with no proper work) -0.5 to - poor notation (-0.5 if 0 is ever written in denominator; - if no limit is written in problem; - if there is no limit of function as first step) - mixing x with small neg in denominator - never plugs in x=value to numerator (and has x even after limit is dropped) - specific x-values plugged in inline with the work -5 small/small -.5 incorrect sign on infinity (no other mistakes).5.5 Page 0 of 3

11 8. (6 pts.) Evaluate the limit: lim lim 3+ x 3 = lim x 3 = lim = lim x 3+ x 3( 3+ x) x 3 3+ x 3 x 3 = lim 3 x 3( 3+ x) ( x 3) = lim 3( 3+ x) = 8 9. (4 pts.) Evaluate the limit: lim x π cos 6 (3+ x) 3 3+ x x 3 3 x 3( 3+ x) x 3 x 4 sin x 6 ln e x Common denominator in numerator Flip denominator Simplification Cancel with - Correctly finds limit (award only if correct) - including 0/0 in work - missing equals (-0.5 for only one missing equals) - missing limits (-0.5 for only one missing) Excused missing = sign ONLY if it was at a line break -0.5 limit notation carried too far -0.5 lack of limit rewrite after cancellation at least -3 negative not distributed resulting in problem that cannot be finished properly cos lim x π x 4 sin x 6 ln e x cos π 4 sin π = 6 ln e π = π = π = π Simplify trig each Simplify denominator 0.5 Simplify whole thing 0.5 Correct answer (award only if correct) - missing equals (-0.5 for only one missing =) - missing limits (-0.5 only one missing) -0.5 limit notation carried too far Excused missing = sign ONLY if it was at a line break Page of 3

12 ( 0. (6 pts.) Use a limit to show that f (x) = x 5 ) 3 x x = 5. lim x 5 OR lim x 5 ( x 5 ) 3 x x 5 ( x 5 ) 3 x x 5 x 5 x + 5 ( x + 5 = lim x 5) ( 3 x) x 5 ( x 5) x + 5 ( x 5 )( 3 x) = lim x 5 ( x 5 ) x + 5 = lim x 5 = lim x 5 has a removable discontinuity at 3 x x + 5 = = x x + 5 = = Conjugate Simplify numerator.5 Cancel.5 Limit evaluation Correct answer (award only if correct) - including 0/0 in work - missing equals (-0.5 for only one missing equals) - missing limits (-0.5 for only one missing) Excused missing = sign ONLY if it was at a line break -0.5 limit notation carried too far up to - for notational errors (such as omitting the parentheses in the denominator or poor limit notation) - for stating the conjugate correctly but not multiplying the denominator by the conjugate (- total if remainder of problem is correct, additional - for incorrect solution) -0.5 lack of limit rewrite after cancellation Page of 3

13 . (6 pts.) Use the ε-δ definition to prove the limit: lim x 3 7 x = 3 i. Let ε > 0 be given. Find an appropriate δ > 0 Want δ > 0 so that if 0 < x 3 < δ then ( 7 x) 3 < ε x 6 < ε x + 3 < ε x + 3 < ε ii. Prove the limit. Determines a value for delta 3 Proof 3 -/ for each unique invalid statement like = or > instead of < switching epsilon and delta dropping constants algebra mistakes dropping absolute value assuming conclusion -/ for not stating 0<abs(x-a)<delta in proof Given ε > 0, let δ = ε then 0 < x 3 < δ implies ( 7 x) 3 = x 6 = x + 3 < δ = ε = ε. (5 pts.) Identify the discontinuities. List the x-values and state the type of discontinuity for each x = 4 x = x = x = removable infinite removable jump 0.5 each 3. ( pt.) Check to make sure your Scantron form meets the following criteria. If any of the items are NOT satisfied when your Scantron is handed in and/or when your Scantron is processed one point will be subtracted from your test total. My scantron: is bubbled with firm marks so that the form can be machine read; is not damaged and has no stray marks (the form can be machine read); has 3 bubbled in answers; has MATH 040 and my Section number written at the top; has my Instructor s name written at the top; has Test No. written at the top; has Test Version A both written at the top and bubbled in below my CUID; and shows my correct CUID both written and bubbled in (bubble in a 0 in place of the C). Page 3 of 3 Point Reason No extras OK to say hole and vertical asymptote 0.5 each

Student s Printed Name: KEY_&_Grading Guidelines_CUID:

Student s Printed Name: KEY_&_Grading Guidelines_CUID: Instructor: Section # : You are not permitted to use a calculator on any portion of this test. You are not allowed to use any textbook, notes, cell