MAT 145: Test #2 (Part II: 31 points)

Transcription

1 MAT 145: Test #2 (Part II: 31 points) Part 2: Calculator OK! Name _ Calculator Used Score Use f (x) = 4x for questions 16 through 18. Express all solutions using exact values. Show appropriate x + 3 evidence to support your responses. 16. Determine an equation for the tangent line to f when x = 3. Show all appropriate steps to justify your result and express your equation in the form y = mx + b. (2 pts) 17. Lynette created the following limit statements about the function f (x) = 4x x + 3. (I) lim f (x) = 0 (II) lim f (x) = (III) lim x 0 x x 3 (a) Circle each limit statement that is TRUE. (1 pt each) f (x) = (b) Select ONE false limit statement from above and explain how you know it is a false statement. It is not sufficient to simply state, The limit is not correct. Be sure to indicate which limit statement you are discussing. (1 pt) 18. For f (x) = 4x, Tyler stated that there are no values of x for which f is undefined, but Alfonso disagreed. x + 3 Examine the function f and then respond to (a) through (d) below. Be clear, precise, and specific. (a) Who is correct? Tyler Alfonso (Circle one.) (1 pt) (b) Describe the basis for your response to (a). Provide factual mathematics evidence as part of a clear and concise response. (1 pt) (c) Write a new function, h(x), for which you know the function is defined for all real numbers. (1 pt) (d) Finally, create another new function, k(x), that has these two properties: (i) the function k(x) is defined for all real numbers; (ii) there is exactly one x value at which the derivative of k is undefined. Explain how you know k' is undefined at that x value. (2 pts)

2 A particle moves along the x axis of a coordinate plane so that its position in relation to the origin, in feet, at time t is s(t) = 1 3 t t 2 + 2t +1, t in minutes, t any real number. Use this information for questions 19 through 23. Please carefully check the units of measure you use with your responses! (1 pt each) 19. Calculate the velocity function, v(t), at time t minutes. Include units. 20. Calculate the acceleration function, a(t), at time t minutes. Include units. 21. Determine the instantaneous rate of change of the particle s position at time t = 2 minutes. Include units. 22. Heather stated that the particle was moving to the left at precisely the time t = 1. Is Heather correct? Refer to calculus-based evidence to support or refute Heather s claim and explain how you know based on that evidence. 23. For all values t 0, state all time intervals for which the particle is slowing down. If no such time intervals exist, explain why not. Use the graph of the function y = f (x) shown here as you respond to questions 24 through 29. (1 pt each) 24. lim f (x) x lim x 5 f (x) 26. lim f (x) 27. lim f (x) + x 3 x f (0) 29. f!( 2)

3 30. Here are the number N of homeless students identifiable through US public schools, 2008 through School Year Ending: N 795, , ,948 1,065,794 1,166,339 1,258,182 1,360,747 Source (9/15/15): tables- report.cfm When a function is generated to model this data set, H(x) = 472.4x x x represents the number of homeless students, H, where x represents the number of years after 2008 (i.e., x = 0 represents 2008). (a) Determine H!(x) and use it to calculate the instantaneous rate of change in the number of homeless publicschool students in the year Be sure to label appropriately your value for H!. (2 pts) (b) Examine the value of H! for each year represented in the data table. Write one or two sentences to describe how H! is changing during the time span 2008 to Be specific and precise. (1 pt) (c) Based on your analysis of the values of H! for the time span 2008 to 2014, does the pattern you described in response (b) seem encouraging or discouraging? Explain. (1 pt) When its drain is opened, the water remaining in a residential swimming pool can be modeled by the function R(t) = t, with R measures in gallons and t in minutes. We know that at time t = 0 min, the pool is full, containing 20,000 gal of water: R(0) = In precisely two hours and five minutes (t = 125 min), the pool is empty: R(125) = 0. Use this information for questions 31 through In this context, what is the meaning of R(20) = 12000? (1 pt) 32. Explain the meaning of R!(80) = 100 for this situation. (1 pt) 33. Graph R(t) on your graphing calculator (adjust your window!) and imagine its tangent lines and their slopes for 0 < t < 125. Use this graph and your visualized tangent lines to complete the following statement. Circle one best response from among A through C. (1 pt) Water is draining from the pool _?_ as time passes. (A) more slowly (B) more quickly (C) at a constant rate 34. Show here a pencil-and-paper calculation of R!(t) and use it to determine the exact value of R!(20). Include appropriate units. (2 pts)

4 BONUS! BONUS! BONUS! (B) Isaac Newton is credited as one of the two inventors/discoverers of Calculus. For extra-credit points, either (i) state the name of one scientific field Newton studied or (ii) write one sentence to describe a discovery Newton is credited with making. Be very specific! (2 pts) (C) In the figure here, line l is tangent to the graph of y = x x2 500 at the point Q. (a) Determine the x-coordinate of point Q. (3 pts) (b) Write an equation for line l, in the form y = mx + b. (1 pt) (c) Suppose the graph of y, with x and y measured in feet, represents a hill. There is a 50-foot-tall tree growing vertically at the top of the hill. Does a spotlight at point P, directed along line l, shine on any part of the tree? Provide complete and appropriate evidence to support your conclusion. (2 pts) (D) There exists at least one line that contains the point (2, 1) and is tangent to the curve y = x 3 x 2. Determine the ordered pair, on the graph of y, that is on such a tangent line. If more than one such point exists, state the ordered pair whose x coordinate is closest to 0. State exact values for x and y, or if that isn t possible, round your ordered-pair coordinates to the nearest ten-thousandth of a unit. Provide complete, clear, and appropriate calculus-based evidence. (4 pts)

5 Part I: No Calculators (19 points) (1)-(10): 1 pt each; no partial credit (11) (12) 1 pt each Calculus I MAT 145 Test #2: 50 points Evaluation Criteria (13) (14) 2 pts each; note the instructions provided with some derivative requests (15) 3 pts: correct and complete algebraic evidence showing determination of derivative; correct format Bonus (A) 3 pts: Show correct response and include clear and accurate explanation Part II: Calculators May Be Used (31 points) (16) 2 pts: correct slope, correct equation, evidence included (17) 5 pts: (a) 3 pts (correct T/F assessment of each limit statement; (b) 2 pts (correct, clear, and complete explanation for a false limit statement) (18) 5 pts: (a) 1 pt (correct choice); (b) 1 pt (clear and accurate description); (c) 1 pt (accurate function); (d) 2 pts (1 pt correct function, 1 pt clear and accurate explanation) (19) 1 pt (correct response with appropriate units) (20) 1 pt (correct response with appropriate units) (21) 1 pt (correct response with appropriate units) (22) 1 pt (correctly agree or disagree with student; correctly use calculus-based evidence to support decision) (23) 1 pt (correct response with appropriate notation) (24)-(29): 1 pt each; no partial credit (30) 4 pts: (a) 2 pts (1 pt each response); (b) 1 pt (accurate one-to-two-sentence explanation); (c) 1 pt (statement regarding encouraging/discouraging with explanation) (31) 1 pt: accurate, clear, and precise explanation (32) 1 pt: accurate, clear, and precise explanation (33) 1 pt: correct choice (34) 2 pts: 1 pt: correct R!(t) with evidence; 1 pt: correct value for R!(20) ; check units Bonus (B) 2 pts: correct, clear, and specific response Bonus (C) 6 pts: (a) 3 pts (evidence required); (b) 1 pt (correct linear equation); (c) 2 pts (correct determination with evidence/explanation) Bonus (D) 4 pts: correct ordered pair; complete, clear, appropriate evidence