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1 DEPARTMENT OF MATHEMATICS AND STATISTICS QUEEN S UNIVERSITY AT KINGSTON MATH DEC 2014 CDS/Section 700 Students ONLY INSTRUCTIONS: Answer all questions, writing clearly in the space provided. If you need more room, continue your answer on the back of the previous page, providing clear directions to the marker. For full marks, you must show all your work and explain how you arrived at your answers, unless explicitly told to do otherwise. Only CASIO FX-991, Gold Sticker or Blue Sticker calculators are permitted. Write your student number clearly at the top of each page. You have three hours to complete the examination. Wherever appropriate, include units in your answers. When drawing graphs, add labels and scales on all axes. PLEASE NOTE: Proctors are unable to respond to queries about the interpretation of exam questions. Do your best to answer exam questions as written. I II III IV V VI VII VIII IX Total This material is copyrighted and is for the sole use of students registered in MATH 121/124 and writing this examination. This material shall not be distributed or disseminated. Failure to abide by these conditions is a breach of copyright and may also constitute a breach of academic integrity under the University Senate s Academic Integrity Policy Statement.

2 Section I. Multiple Choice (10 questions, 2 marks each) Each question has four possible answers, labeled (A), (B), (C), and (D). Choose the most appropriate answer. Write your answer in the space provided, using UPPERCASE letters. Illegible answers will be marked incorrect. You DO NOT need to justify your answer. (1) Find the slope of a tangent line to the graph defined by x 2 y + xy 2 = 1 at the point (1,1). (A) Slope is -3. (B) Slope is -1. (C) Slope is 1. (D) Slope is 3. (2) Identify a formula that corresponds to the graph shown at the right. ( (A) y = sin 2x π ) 4 ( (B) y = sin 2x π ) 2 1 ( x (C) y = sin 2 π ) 4 ( x (D) y = sin 2 π ) 2 3π 2π 2 π π 2 1 π 2 π 3π 2 2π 5π 2 (3) Suppose that g(t) is odd, with h(t) is even, with g(t) dt = 3, and h(t) dt = 8. What is the value of I = 2 2 (4g(t) + 2h(t)) dt? (A) I < 20. (B) 20 I < 30. (C) 30 I < 40. (D) I 40.

3 (4) Suppose f(t) = 1+cos(t). Which of the following is the second degree Taylor polynomial approximation to f(t) at t = π/2? ( (A) 1 x π ) 2 (B) 1 1 ( x π ) 2 2 ( (C) 1 x π ) ( x π ) ( (D) 1 x π ) 1 ( x π ) (5) The function f(x) is defined as follows: x 2 x x < 4 f(x) = 12 x = 4 3x x > 4 Which of the following statements best describes the function around x = 4? (A) The function is continuous at x = 4; the limit lim x 4 f(x) exists. (B) The function is NOT continuous at x = 4; the limit lim x 4 f(x) exists. (C) The function is continuous at x = 4; the limit lim x 4 f(x) does NOT exist. (D) The function is NOT continuous at x = 4; the limit lim x 4 f(x) does NOT exist. (6) The anti-derivative of g(x) = 2 x + cos(x) is given by what function below? (A) 2 x ln(2) + sin(x) + C (B) 2 x ln(2) sin(x) + C 2 x (C) ln(2) + sin(x) + C (D) 2 x ln(2) sin(x) + C (7) What is the domain of the function y = ln(2 3x)? (A) x < 2/3 (B) x < 3/2 (C) x > 2/3 (D) x > 3/2

4 (8) Evaluate the limit lim x (A) 4 (B) 1 5 (C) 0 (D) The limit does not exist. 2 3x + 4x x g(2 + h) g(2) (9) What is defined by the formula lim? h 0 h (A) The first derivative of g(t) at t = 2. (B) The first derivative of g(t) at t = h. (C) The second derivative of g(t) at t = 2. (D) The second derivative of g(t) at t = h. (10) How many solutions does the equation below have? (Hint: Sketching graphs may help) sin(x) = x 2 (A) No solution. (B) One solution. (C) Two solutions. (D) Three solutions.

5 Section II. Related Rates [/8] A baker starts with a cylinder of dough 10 cm long and with radius 2 cm. She then rolls the cylinder back and forth, which makes the cylinder narrower but longer; the volume remains constant. 2 cm 10 cm 15 cm (a) When the length of the cylinder of dough reaches 15 cm, what is its radius? (b) If the length of the cylinder is growing at 0.5 cm/s, what is the rate of change of radius when the length reaches 15 cm?

6 STUDENT NUMBER: Section III. Linearization and Optimization [/20] Consider the function f(x) = x 2. (a) Find the formula for the tangent line to f(x) = x 2 at x = (b) Your formula defines the tangent line as shown on the diagram. If we build a triangle with the boundaries defined by the tangent line from part (a), the line y = 0 (or x axis), and the line x = 4, then we obtain the triangular region shown. Find the area of this triangle, based on your answer from part (a) [Section III continued on next page]

7 STUDENT NUMBER: Section III continued (c) We are now going to generalize the results from the first part of the question to an arbitrary point (not specifically x = 3). Find the formula for the tangent line to f(x) = x 2 at the point x = a. 16 (d) We define a triangle using the boundaries. the tangent line from part (c), the line y = 0 (or x axis), and the line x = 4, then we obtain the triangular region shown. (Only one value for x = a is shown, though any value between a = 0 and a = 4 would be possible.) x = a Find the area of this triangle. Your answer will be a function of the point a.

8 Section III continued (e) Find the value of a where the tangent line to f(x) = x 2 drawn at x = a will result in the triangle with the largest area, where the triangle is defined by the tangent line to f(x) = x 2 at x = a, the line y = 0 (or x axis), and the line x = 4. Note that your answer must be between a = 0 and a = 4. You do not need to show your answer is a global maximum for the triangle area. You should use your answers from previous parts of this question.

9 Section IV. Velocity, Distance and Gas Consumption [/8] A car s velocity, v(t) was tracked for one hour, with the following results. t (hrs) v(t) (km/hr) (a) Use a right sum with 5 intervals to estimate the distance travelled by the car between t = 0 and t = 1 hour. Give units in your answer. (b) A car s mileage or gas consumption depends on its speed. For the same car as in part (a), the table below shows the relationship between the speed the car is travelling (v) and the resulting gas consumption (reported in liters required to travel 100 km). v (km/hr) Efficiency (liters per 100 km) Use any appropriate method over 5 intervals to estimate the amount of gas consumed by the car between t = 0 and t = 1 hour. Give units in your answer.

10 Section V. Families of Functions [/8] Consider the parabolic family of functions defined by where a, b and c are constants. Find the member of this family that: passes through the point (0,1), and has a local maximum at (3, 7). g(x) = ax 2 + bx + c, Final values: a = b = c =

11 Section VI. Integration [/10] Evaluate the following definite and indefinite integrals. You must show how you arrived at your answer: calculator-computed values are not sufficient. (a) xe x2 dx (b) π 0 t sin(t) dt (c) y 2 y dy y 3

12 Section VII. Limits and Graphs [/10] For the function h(x) = xe 2x, (a) Find lim x h(x). (b) Find lim h(x). x 0 (c) Find the critical point(s) of h(x), and classify them as local maxima or minima. (c) Use two steps of Newton s method to find an approximate solution to xe 2x = 0.1, starting at an initial estimate of x 0 = 0.

13 Section VIII. Sales [/8] A company is selling toasters at a rate given by the function where f is in thousands of toasters per week, and t is measured in weeks. f(t) = t (a) Based on the rate given by f(t), find the exact number of toasters sold by the company over the 12 weeks between t = 0 and t = 12, rounded to the nearest unit. (b) Using your answer to part (a), find the average number of toasters sold per week over the same 12 weeks. Include units in your answer.

14 Section IX. Radioactive Dating [/8] The half-life of potassium-40, a radioactive isotope of potassium, is 1.25 billion years. Due to the volatility of the products of its radioactive decay, samples specifically of lava flow (a rock formation that was once liquid) can be measured, and the fraction of original vs. decayed potassium-40 can be used to date the lava flow. (a) Give a formula for the fraction of potassium remaining in a lava sample after t million years. (b) A fossil has been found in an undisturbed rock layer, between two layers of old lava. Old Lava 82.0% original potassium-40 present; 18.0% decayed Rock layer with fossil Old Lava 80.1% original potassium-40 present; 19.9% decayed Find the range of possible ages for the fossil. Give units in your answer.

15 Space for additional work. space. Indicate clearly which Section you are continuing if you use this

16 Space for additional work. space. Indicate clearly which Section you are continuing if you use this