e. [3 points] For which values oftisV(t) concave up? For which values is it concave down?f. [4 points]Find an estimate forintegraldisplay2I(t)dtusing Mid(5). Write all the terms in the sum.0g. [3 points]Suppose instead of the functionO(t) shown in the graph above, the water ispumped out of the tank at a constant rate ofrm3per hour. What must the value ofrbe so thatV(2) = 20? Your answer may involve a definite integral ofI(t).University of Michigan Department of MathematicsFall, 2012 Math 116 Exam 1 Problem 3 (tank)

Math 116 / Exam 1 (October 10 , 2012)page 95. [12 points]Oil leaks from a tank on the shore of a lake to form a semicircular slick on thesurface of the water (as shown in the figure below). A team of environmentalists is trying toestimate the amount of oil spilled. They took measurements of the densityPof oil (in kg perm2) in the slick and found that it was a function of the distancer(in m) from the source ofthe oil.The values ofP(r) measured by the environmentalists are shown in the table below.r050100150200P(r)1004012108a. [6 points]Write an expression involving integrals for the exact value of the mass of theoil in the lake inside a semicircle centered at the oil leak with a radius of 200 meters (seethe figure above). Include units.b. [4 points]Find approximations to your answer in part (a) using Left(4) and Right(4).Show your work by writing all the terms of the sums.c. [2 points] The environmentalists notice that the densityP(r) of oil is a decreasing function.Does this observation guarantee that one of the approximations in part (b) yields anoverestimate? If so, which one? Justify.University of Michigan Department of MathematicsFall, 2012 Math 116 Exam 1 Problem 5

Math 116 / Exam 1 (February 8, 2010)page 65. [10 points] For each statement below, circle TRUE if the statement isalwaystrue; otherwise,circle FALSE. There is no partial credit on this page.a. [2 points] The functionsinxxhas an anti-derivative.

b. [2 points]ddxintegraltextx2xet2dt= 4x3ex4-2xex2.

c. [2 points] The average of the functionf(x) =1xfromx= 1 tox= 3 is ln(√3).

d. [2 points]integraltextbaf(x)dxis greater than or equal to at least one of LEFT(n), RIGHT(n),TRAP(n), or MID(n) regardless of whatf(x) ornis.TrueFalse

e. [2 points] Ifintegraltextbaf(x)dx >0 thenf(b)> f(a).

University of Michigan Department of MathematicsWinter, 2010 Math 116 Exam 1 Problem 5

Math 116 / Exam 1 (October 13, 2010)page 2