Mathematics Midterm Exam 2. Midterm Exam 2 Practice Duration: 1 hour This test has 7 questions on 9 pages, for a total of 50 points.

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1 Mathematics Mierm Exam Mierm Exam Practice Duration: hour This test has 7 questions on 9 pages, for a total of 50 points. Q-Q5 are short-answer questions[3 pts each part]; put your answer in the boxes provided. Q6 and Q7 are long-answer and each worth 0 pts; you should give complete arguments and explanations for all your calculations; answers without justifications will not be marked. Use the two blank pages is you need extra space, but you must leave a clear note that this has been done. This is a closed-book examination. None of the following are allowed: documents, cheat sheets or electronic devices of any kind (including calculators, cell phones, etc.) Please circle your course and section: I don t know. PRINT your name and ID # very clearly. Failure to do so may result in a grade of 0: First Name: Last Name: Student-No: Signature: Student Conduct during Examinations. Each examination candidate must be prepared to produce, upon the request of the invigilator or examiner, his or her UBCcard for identification.. Examination candidates are not permitted to ask questions of the examiners or invigilators, except in cases of supposed errors or ambiguities in examination questions, illegible or missing material, or the like. 3. No examination candidate shall be permitted to enter the examination room after the expiration of one-half hour from the scheduled starting time, or to leave during the first half hour of the examination. Should the examination run forty-five (45) minutes or less, no examination candidate shall be permitted to enter the examination room once the examination has begun. 4. Examination candidates must conduct themselves honestly and in accordance with established rules for a given examination, which will be articulated by the examiner or invigilator prior to the examination commencing. Should dishonest behaviour be observed by the examiner(s) or invigilator(s), pleas of accident or forgetfulness shall not be received. 5. Examination candidates suspected of any of the following, or any other similar practices, may be immediately dismissed from the examination by the examiner/invigilator, and may be subject to disciplinary action: (i) speaking or communicating with other examination candidates, unless otherwise authorized; (ii) (iii) (iv) (v) purposely exposing written papers to the view of other examination candidates or imaging devices; purposely viewing the written papers of other examination candidates; using or having visible at the place of writing any books, papers or other memory aid devices other than those authorized by the examiner(s); and, using or operating electronic devices including but not limited to telephones, calculators, computers, or similar devices other than those authorized by the examiner(s)(electronic devices other than those authorized by the examiner(s) must be completely powered down if present at the place of writing). 6. Examination candidates must not destroy or damage any examination material, must hand in all examination papers, and must not take any examination material from the examination room without permission of the examiner or invigilator. 7. Notwithstanding the above, for any mode of examination that does not fall into the traditional, paper-based method, examination candidates shall adhere to any special rules for conduct as established and articulated by the examiner. 8. Examination candidates must follow any additional examination rules or directions communicated by the examiner(s) or invigilator(s).

2 Mathematics Mierm Exam Short-Answer Questions. Put your answer in the box provided. Full marks will be given for a correct answer placed in the box, while part marks may be given for work shown. Unless otherwise stated, calculator ready answers are acceptable. ( cos(x). (a) Let f(x) = x ). Find f (x). Solution: Answer: ( x ( ) cos(x) y = x ( ) cos(x) ln y = ln x ( ) ln y = cos x ln x ) cos(x) [sin ] x ln(x ) cos x x ln y = cos x ( ln(x )) y cos x = sin x ln(x ) y x [ y = y sin x ln(x ) cos x ] x (b) A particle moves along a straight line and its position at time t is given by s(t) = t t3 5 t +. Find the time interval on which the velocity of the particle is increasing. Answer: 3 < t < 5 Solution: The velocity v(t) = s (t) = 3 x3 +4x 5x and v (t) = x +8x 5 = (x 3)(x 5). It follows that v (t) > 0 iff 3 < t < 5. Hence, the time interval we want is (3, 5).

3 Mathematics Mierm Exam. (a) Sketch the graph of a function f(x) whose domain is all real numbers except 0, and such that lim f(x) =, lim f(x) = 3, lim f(x) =, lim f(x) =, and the slope of the x + x x 0 x tangent line to the curve at (3,) is. y x Solution: Anything with these properties. Note that the green dotted line should be tangent to curve at (3,). (b) Find the absolute extremal values of the function f(x) = 3( 3 x) x on the interval [ 8, 7]. Answer: min = 0, max = 4 Solution: Check endpoints: f( ) = 3 ( 8 ) + = 7, f(7) = = 0. Now check the critical numbers: f (x) = 3 3 x Set f (x) = 0 or DNE for the critical numbers: 3 = 3. x 3 x = 0 3 x = x = 8 We have f(8) = 3 8 = 4. Also, f (x) is not defined at 0, so we need to check f(0) = 0 0 = 0. Thus, the absolute maximum of f(x) is at x = 8, and the absolute minimum is at x = 0 and x = 7.

Mathematics 04-84 Mierm Exam 3. (a) The graph of f (x) (the second derivative of f(x)) is displayed below. For what x value(s) does f(x) have an inflection point?

4 Mathematics Mierm Exam 3. (a) The graph of f (x) (the second derivative of f(x)) is displayed below. For what x value(s) does f(x) have an inflection point? f 00 (x) x Answer: x = 0 and x = 8 Solution: There will be a POI when f (x) changes sign. (b) The number of people q willing to ride the ferry at price p is determine by the relationship ( ) q 3000 p =. 600 Find the price elasticity of demand E d if there are currently 800 people riding the ferry each daily at $4 per ticket.. Answer: E=-/3 Solution: Differentiating with respet to price give us: ( ) q 3000 = dq dp. Subbing in the values gives us: ( ) 00 = 600 dq dp = dq dp 600. = 50 = E = dq dp p q = = 3

5 Mathematics Mierm Exam 4. (a) Find the equation of the line tangent to y = arccos(x) at x =. Answer: y = π 4 ( x ) Solution: Let f(x) = arccos(x). The equation for the tangent line is y = f(a) + f (a)(x a), where a =. ( ) f (x) = x, so f = = ( ) As for the point, we have: f(a) = arccos = π. 4 So the tangent line is given by y = π 4 ( x ). (b) Find the asymptotes of the curve y = 3x + 7x 89 8x 4x + 6 Answer: y = 3, no VA. 8 Solution: We need the roots of the denominator: x = ( 4)± 6 4(8)(6) 8 which has no solutions. So, there aren t any vertical aymptotes. For the horizontal asympotes, we need to limits to ±. 3x + 7x 89 lim x 8x 4x + 6 = lim 3 + 7/x 89/x x 8 4/x + 6/x = 3 8 It s exactly the same for the limit to.

6 5. (a) Below is a graph of f(x). Mathematics Mierm Exam x i. What is the global maximum value? Answer: 3 ii. State the x value(s) where a local minimum occurs. Answer:, 7 (b) The price (in dollars) p and the quantity demanded q for a TV are related by the equation: p = f(q), where f is a differentiable function. When p = $00, the cost function C for producing q TVs is increasing at a rate of 4 dollars per month and the quantity demanded q is increasing at a rate of 8 TVs per month. Find the marginal cost for this kind of TV at p = $00. Answer: 3 dc Solution: Given: = 4 and dq = 8; we know it s since it is changing with time. Ask for: dc dq So we need to relate dc dc to and dq somehow! dq This is precisely what the Chain Rule does: dc = dc dq dq Hence, the marginal cost for this kind of TV at p = $00 is given by dc dc dq = dq = 4 8 = 3.

7 Mathematics Mierm Exam Full-solution problems: Justify your answers and show all your work. Place a box around your final answer. Unless otherwise indicated, simplification of answers is required in these questions. 6. A plane flying with a constant speed of 4 km/min passes over a ground radar station at an altitude of 9 km and climbs at an angle of 30 degrees. At what rate is the distance from the plane to the radar station increasing minutes later? Hint: the cosine law is given by c = a + b ab cos C. Solution: x 0 y 9 Using the notation of the picture, we have x = 8 two minutes after the plane passes over the radar station. So we want to know dy when x = 8, given that dx = 4. We use the law of cosines: Differentiating with respect to t, which gives y = x x cos(0 ) ( ) y = x x y = x + 9x + 8 y dy = xdx + 9dx, dy dx x = + 9 dx. y We can fill in x and dx immediately, but we need to know y when x = 8. Again using the law of cosines, when x = 8 = y = = 7, Thus, two minutes after the plane passes over the radar station, the distance between the plane and the radar station is changing at a rate of dy = 50 7 km/min.

8 Mathematics Mierm Exam 7. Letf(x) = 3 x 3 x. (a) Find the x-intercept(s). Solution: For x-intercepts, we set y = f(x) = 0. so ( 3 x 3 x = 0 3 x ) 3 x 3 = 0 3 x = 0 x = 0 or 3 x 3 = 0 x 3 = 3 x = 3 3 = 7 x = ± 7. (b) Find the intervals on which f is increasing or decreasing. Solution: We start by taking derivative: f (x) = 3 x 3 3 = 3x 3 f (x) = 0 x 3 3x 3 3 = x 3 3x 3 f (x) not defined 3x 3 = 0 x = 0. = 0 x 3 = x = 3 = x = ±. To classify the critical points, we determine the sign of f (x). Denominator is always positive, and numerator is positive for [, ] and negative otherwise. so x = is local minimum, x = + is local maximum. x = 0 is a critical point, but it is not a local extremum. The function is increasing on [, ] and decreasing on (, ]U[, ). (c) Find the intervals on which f is concave up. Solution: f (x) = 9 x 5 3 = f (x) = 0 9x 5 3 9x 5 3 = 0 no solution f (x) = not defined 9x 5 3 = 0 x = 0 For any x < 0, f > 0 and for x > 0, f < 0. Therefore x = 0 is an inflection point. The function is concave up on (, 0] and concave down on [0, ).

9 (d) Sketch the graph of f. Mathematics Mierm Exam Solution: It may be helpful to summize the data (but this isn t required). As for the graph, you should plot those 3 roots, and the extrema.

The University of British Columbia Final Examination - December 6, 2014 Mathematics 104/184 All Sections

The University of British Columbia Final Examination - December 6, 2014 Mathematics 104/184 All Sections Closed book examination Time: 2.5 hours Last Name First Signature MATH 104 or MATH 184 (Circle one)