Part A: Short Answer Questions

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1 Math 111 Practice Exam Your Grade: Fall 2015 Total Marks: 160 Instructor: Telyn Kusalik Time: 180 minutes Name: Part A: Short Answer Questions Answer each question in the blank provided. 1. If a city grows from an initial population of 1000 at a constant relative growth rate of 0.02, then the population function of the city will be given by: P (t) = 2. A critical number of a function f is an x-value in the of a function such that either f (x) =. or f (x) is 3. Newton s method uses the x-intercept of the to the graph of a function to approximate the x-intercept of the function itself. Newton s method will work as long as this x-intercept found is to the actual x-intercept than. 4. The differential dy represents the change in the y-coordinate on the while y represents the change in the y-coordinate on the. 5. The limit below can be interpreted as the derivative of a function f(x) at a point x = a. Find f(x) and a: x 2 4 lim x 2 x 2 f(x) = a =.

2 6. Suppose the position function of a particle is given by the formula y = f(t). The average velocity of that particle over the time interval [1, 2] will be given by the formula:. The instantaneous velocity of the particle at time 1 will be given by the limit:. 7. To test a function for symmetry we replace x with. If we get the negative of the function we started with, the function has symmetry. If we get the same as the function we started with, the function has symmetry. 8. Use the graph of a function f(x) given below to sketch the graph of the derivative f (x) on the same coordinate axes. Part B: Limit Evaluation Questions For each of the the following limits, find its value if it exists. If limit does not exist, explain why. If the limit is infinite, specify whether it s + or. Show all work and justify your answer using algebra and/or calculus. [4] 9. lim x 0 x x 2 [4] 10. lim x 1 2x 2 2 2x 2 + 4x lim x 3 x 2 9 x lim x 1 (ln x) 2 x 2 1

3 [6] 13. lim x x2 e x 14. lim x x2 + 1 x 2 x Part C: Differentiation Questions Find the derivative of each function below. Make any obvious simplifications. You do not need to show any work, but may be able to receive part marks for work shown if your final answer is incorrect. 15. f(x) = x + 3e x 2 sin 1 x [4] 16. y = x 5 2 x 17. g(θ) = sin2 θ cos 5 θ [6] 18. u = t log 2 t 19. h(x) = sec(x 2 e x ) Part D: Miscellaneous Computational Questions Solve each question in the space provided. Show all work and be sure to justify your answer using algebra and/or calculus. Correct answers alone will not receive full marks. Give exact answers; no decimal approximations please. 20. Find the most general antiderivative of the function below: [4] f(x) = 2x 3 4 sin x + 1 x 21. Find the equation of the tangent line to the graph of f(x) = tan x (0, 0). x+1 at the point [6] 22. Use Newton s method to approximate 205 to four decimal places. Use x 1 = 15 as your initial approximation. Feel free to use your graphing calculator to do the computations for you, but be sure to write both what you typed into your graphing calculator and what you got as output.

4 23. Find the numbers a and b which make the function below continuous everywhere: ax + 2 : x < 1 f(x) = x 2 : 1 x 1 2x b : x > 1 [6] 24. Use the Intermediate Value Theorem and Rolle s Theorem to show that the equation e x + x 3 = 2 has exactly one root. 25. Consider the function f(x) = x + x 2. (a) Find the intervals on which the function f(x) = x+x 2 is concave up and concave down. [1] (b) What are the coordinates of the inflection point of this function? [4] 26. Find the absolute maximum and minimum values of the function f(x) = e x2 on the interval [ 1, 1]. 27. Use the limit definition of the derivative to find f (x) if f(x) = x Make sure to show all steps and justify your answer algebraically. DO NOT use l Hospital s rule. 28. Find d2 y dx 2 if: y 3 + y = x 2 + x Part E: Graph Sketching Question Answer the questions below to determine a number of properties of the graph of a function. You will then be instructed to use these properties to sketch the graph. Make sure to justify all your answers using calculus when appropriate, and make sure to give exact answers. Note that you will not receive marks for a correct graph if that graph does not match the properties found, or if the locations of points found are not labelled on the graph. 29. Consider the function f(x) = 9 x 2. (a) What is the domain of f(x)? (b) Find the x and y intercepts of the graph of f(x). (c) Find the intervals on f(x) is increasing and decreasing.

5 [1] (d) f(x) has either a local maximum or a local minimum. Specify which it is and give its coordinates. (e) Find the concavity of this function (HINT: the concavity is always the same - it is either always concave up or always concave down). (f) Sketch the graph of this function, being sure to locate all the features found above. Part F: Word Problems Solve each problem, being sure to show all work. Please answer each question in a sentence. Decimal approximations are acceptable as answers to these questions. 30. A company expects a revenue of R(x) = 300x x 2 when x items are sold and a cost of C(x) = x when x items are produced. Find the profit function P (x) (HINT: profit is revenue minus cost), and then find the instantaneous rate of change of profit with respect to the number of items produced when 150 items are produced (this is called the marginal profit.). 31. One ship is located 10km North of an island and is sailing East at 20km/h while another is 25 km East of the island and is sailing North at 15km/h. Find the instantaneous rate of change of the distance between the two ships at this time. [8] 32. A box with a square base and open top is to be made from 300cm 2 of cardboard. Find the dimensions of such a box that maximize the volume. Be sure to justify your answer using calculus, and be sure to show that the critical number you ve found does in fact maximize the volume rather than minimizing the volume.