MAT 146 Test #1B Part I I Name _________________ Section: 02 07 (circle one) 50 points (Part I I : 20 points) Calculator Used ______________ Impact on Course Grade: approximately 10% Score ________________

15. Region B in Quadrant I is bordered by the y-axis and the functions f x( ) = 3x + 2 and g x( ) = x2 + 2 . Determine the exact area of region B. Include all necessary calculus-based evidence to justify your response. (4 pts) _______________ 16. A solid is to be built on region R in the xy-plane. Region R is bordered by the x-axis and the function h x( ) = 4! x2 . Any cross section of the solid, perpendicular to the y-axis, is a square. Set up, but do not calculate, an integral expression to determine the volume of this solid. (4 pts) ________________________ 17. Set up and calculate a definite integral to determine the average value of the function k(x) = 6

x + 4

for –3 ≤ x ≤ 0. (2 pts) _______________

18. Banyon decided to use integration by parts (IBP) to evaluate the integral 3exx3!4

3e

! dx . He couldn’t

decide, however, the substitution to use. (a) State the initial IBP substitution Banyon should use. (b) Explain why that is the most appropriate IBP substitution to use for this integral. (4 pts) 19. Allister claims that the following statement is always true: Suppose that a continuous function y = f (x) has domain [a,b], and that at a point x = c, where a < c < b, f ʺ″(c) = 0. Claim: A point of inflection of f occurs at x = c.

(a) If you agree with Allister’s claim, explain how you know he is correct. If you disagree with Allister’s claim, provide one counterexample to show the statement is not always true. (3 pts)

Allister’s study buddy, Clovenius, read the above statement incorrectly. He thought the statement was the converse of the original: Suppose that a continuous function y = f (x) has domain [a,b], and that at a point x = c, where a < c < b, we know there is a point of inflection of f. Claim: f ʺ″(c) = 0.

(b) If you agree with Clovenius’s claim, explain how you know he is correct. If you disagree with Clovenius’s claim, provide one counterexample to show the statement is not always true. (3 pts)

BONUS!

Consider the graphs of y =23x and y = cos3 x( ) shown in the picture

here. The two curves intersect at point P. Region R is bounded by the two curves and the y-axis. Region S is bounded by the two curves and the x-axis. Create and calculate the ratio of the area of region S to the area of region R. Express your ratio as a decimal value, rounded to the nearest ten-thousandth of a unit. Show all appropriate calculus-based evidence and explanation to justify your result.

Calculus II MAT 146

Test #1 Total Points: 50 Impact of Exam on Semester Grade: Approximately 10%

Evaluation Criteria Part I: No Calculators 30 points Questions 1 through 10 1 pt each with no partial credit. No need to show any work on these. Questions 11 through 14 5 pts each. Partial credit is possible. Show all steps leading to each solution. Be clear, complete, and accurate. Use appropriate symbolism. For the definite integral, carry out appropriate simplification. BONUS! 5 pts: Correct, complete, and clear response to Dexterinia’s result, including description, explanation, and example. No credit earned if there is any reference to or exhibit of specific numerical average-value calculations. Part II: Calculators Allowed 20 points Question 15: 4 pts: Present a clear, complete, and accurate solution, including exact area. Include complete and appropriate calculus justification. Question 16: 4 pts: Show a complete, accurate, and correctly symbolized integral expression. Question 17: 2 pts: Show correct set up (1 pt) and correct exact solution (1 pt). Question 18: 4 pts: (a) Show appropriate IBP substitution (2 pts) and (b) provide clear and accurate explanation for your substitution choice (2 pts). Question 19: 6 pts

(a) 3 pts: Accurately and clearly justify or refute the claim. A counterexample must accompany any refutation.

(b) 3 pts: Accurately and clearly justify or refute the claim. A counterexample must accompany any refutation.

BONUS! 5 pts: Correct, complete, and clear presentation of your solution. Calculus-based evidence required. Ratio rounded as indicated.