MAT 145: Test #3 (50 points)

Transcription

1 MAT 145: Test #3 (50 points) Part 2: Calculator OK! Name Calculator Used Score 21. For f (x) = 8x 3 +81x 2 42x 8, defined for all real numbers, use calculus techniques to determine all intervals on which f is increasing and on which f is decreasing. Show all calculations. Explain your Increasing on: Decreasing on: 22. Use calculus techniques to determine the location of any local maxima and any local minima of f (x) = 8x 3 +81x 2 42x 8, where f is defined for all real numbers. Show all calculations. Explain your Local Maxima at: (exact x values) Local Minima at: (exact x values) 23. For g(x) = x 3 3x 2 + x 2, defined for all real numbers, use calculus techniques to determine all intervals on which g is concave up and on which g is concave down. Show all calculations. Explain your Concave Up on: Concave Down on: 24. Use calculus techniques to determine ordered pairs for any points of inflection of g(x) = x 3 3x 2 + x 2, where g is defined for all real numbers. Show all calculations. Explain your answer, using calculus, in a sentence. (2 pts) Points of Inflection: (ordered pairs: exact values)

2 25. Use the information here to sketch the graph of a function f that meets ALL the following requirements. Label your graph to help me identify these requirements. (12 pts) lim f (x) = 2 lim f (x) = 0 x x lim f (x) = lim f (x) = + x 2 x 2 There is a local minimum at x = 2. There is an absolute minimum at x = 5. f!(x) > 0 on ( 2,0) ( 0,2) ( 5, ) f!(x) < 0on (, 2) ( 2,5) f!! (x) > 0 on ( 4, 1) ( 0,2) (2,6) f!! (x) < 0 on (, 4) ( 1,0) 6, ( ) The y-intercept is (0, 2). Exactly two x-intercepts: ( 2, 0) and (3, 0) A point on the graph is ( 4, 1). A point on the graph is (5, 2)

3 26. Choose ONE of the three optimization problems described below. Solve the problem, showing complete evidence and calculus justification. Include a drawing or a graph to represent the situation. (5 pts) (A) Determine all ordered pairs on the curve y = ½x 2 that are closest to the point (6,0). (B) A segment through the point (2,7) has an endpoint A = (0,y) on the positive y axis and another endpoint B = (x,0) on the positive x axis. If the origin is labeled point P, determine the ordered pairs A and B so that triangle ABP has the smallest area possible. (C) A rectangular field is to be fenced. If we represent the field as rectangle ABCD, the cost of the fence for side AB is $10 per foot, the cost of the fence for side CD is $10 per foot, the cost of the fence for side BC is $3 per foot, and the cost of the fence for side DA is $7 per foot. We have $700 available to pay for fence materials. Determine the dimensions of the rectangle ABCD that will fence in the largest area for $700.

4 BONUS! BONUS! BONUS! (A) For the function f (x) = αx 3 + βx 2 +1, determine exact values of α and β so that the ordered pair ( 1,2) is a point of inflection. Include complete and appropriate justification. (3 pts) (B) There exists at least one line that contains the point (4,8) and is tangent to the curve y = x 3. 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 with the smallest x coordinate. Round your ordered-pair coordinates to the nearest thousandth of a unit. Provide complete, clear, and appropriate calculus-based evidence. (4 pts) (C) LuEllen still falls asleep thinking about Calculus! She was dreaming again about inverse operations, and she remembered: We can reverse the process of calculating a derivative! Another example, LuEllen told her classmates, if we have the derivative f!(x) = 4 3 cos(2x)esin(2x), then the function it came from must have been f (x) =!. State a function f LeEllen must have been referring to and explain how you determined that. (3 pts)

5 Part I: No Calculators (25 points) (1)-(10): 1 pt each; no partial credit (11) (12): 2 pts each; partial credit possible (13) (16): 1 pt each; no partial credit (17): 2 pts: 1 pt for counterexample; 1 pt explanation (18) (19): 4 pts total ( 1 / 2 pt each); no partial credit (20): 1 pt: clear and accurate explanation Calculus I MAT 145 Test #3: 50 points Evaluation Criteria Part II: Calculators May Be Used (25 points) (21) (24): 2 pts each: must show correct exact-value responses with appropriate and correct calculus evidence (25) 12 pts: 10 pts for correct components; 2 pts overall structure and flow of your graph (26) 5 pts: 1 pt for required sketch, drawing, or graph; 1 pt for statement of function to be optimized; 1 pt for use of and reference to a constraint; 2 pts for correct solution with calculus evidence BONUS! Provide complete and accurate solutions with justifications.