Math 116 First Midterm October 14, 2009

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1 Math 116 First Midterm October 14, 9 Name: EXAM SOLUTIONS Instrctor: Section: 1. Do not open this exam ntil yo are told to do so.. This exam has 1 pages inclding this cover. There are 9 problems. Note that the problems are not of eqal difficlty, so yo may want to skip over and retrn to a problem on which yo are stck. 3. Do not separate the pages of this exam. If they do become separated, write yor name on every page and point this ot to yor instrctor when yo hand in the exam. 4. Please read the instrctions for each individal problem careflly. One of the skills being tested on this exam is yor ability to interpret mathematical qestions, so instrctors will not answer qestions abot exam problems dring the exam. 5. Show an appropriate amont of work (inclding appropriate explanation) for each problem, so that graders can see not only yor answer bt how yo obtained it. Inclde nits in yor answer where that is appropriate. 6. Yo may se any calclator except a TI-9 (or other calclator with a fll alphanmeric keypad). However, yo mst show work for any calclation which we have learned how to do in this corse. Yo are also allowed two sides of a 3 5 note card. 7. If yo se graphs or tables to find an answer, be sre to inclde an explanation and sketch of the graph, and to write ot the entries of the table that yo se. 8. Trn off all cell phones and pagers, and remove all headphones. Problem Points Score Total 1

2 Math 116 / Exam 1 (October 14, 9) page 1. [1 points] Sppose that f(x) is an odd fnction, g(x) is an even fnction, and 8 3 f(x)dx = 4 8 f(x)dx = g(x)dx = 3 3 g(x)dx = 5 Determine each of the following qantities, if possible. If there is not enogh information to determine the answer, then write NI in the space provided. Yo do not need to show yor work for this page. a. [ points] Evalate 8 3 (f(x) 3g(x))dx. 1 b. [ points] Evalate 8 g(x)dx..5 c. [ points] Evalate 8 3 f(x)g(x)dx. NI d. [ points] Evalate f(4x)dx. -.5 e. [ points] Evalate 3 (f(x) + 4)dx. 18

3 Math 116 / Exam 1 (October 14, 9) page 3. [1 points] A poplation of creatres is placed on a small preservation space. Ten creatres are initially placed on the preservation. The time it takes for a poplation to reach C creatres is given by C dx T (C) = x(4 x), where T is measred in years after the creatres were first placed on the preservation. 1 a. [6 points] Find a fnction for T (C) by analytically solving the integral given above. Be sre to show all appropriate work. Soltion: First we se partial fractions to rewrite the integrand. x(4 x) = A x + B 4A Ax + Bx = 4 x x(4 x) This gives s the conditions A = B = 1 =.5. We then have T (C) = 1 C 1 dx x + 1 C 1 dx 4 x = 1 ln x C 1 1 ln 4 x C 1 = 1 1 ln C ln 1 1 ln 4 C + 1 ln 39 = 1 1 ln 39 + ln C 4 C b. [ points] How long does it take for the creatres to reach a poplation of 5? State yor answer in a complete sentence and inclde nits in yor answer. Soltion: T (C) = 1 ln ln It takes approximately.8588 years (or approximately 1.36 months) for the poplation of creatres to reach 5. c. [4 points] Determine if the integral T (4) = 4 dx 1 x(4 x) converges or diverges. What does yor conclsion mean in terms of the creatres on the preservation? Soltion: 4 T (4) = 1 dx 1 x + lim 1 dx b x = 1 ln 4 + lim ( 1 ln 4 b + 1 ) ln 39 b 4 We know that lim b 4 ( 1 ln 4 b ) diverges, so the integral diverges. This means that the time to reach 4 creatres is infinite, so the poplation will never reach 4 creatres. b

4 Math 116 / Exam 1 (October 14, 9) page 4 3. [1 points] Let g(x) and h(x) be nonnegative fnctions defined for x, let h be differentiable for x and sppose that g(x) 4 when x 1 lim x h(x) = h(x) lim x x = h(x)dx and g(x)dx diverges h(x) dx both converge x Indicate whether yo think the following integrals converge, diverge, or whether there is not enogh information to determine convergence. Yo do not need to show yor work for this page. a. [3 points] g(x)h(x)dx CONVERGES DIVERGES CANNOT TELL b. [3 points] 1 g(x)h(x)dx CONVERGES DIVERGES CANNOT TELL c. [3 points] g(x)+h(x) dx CONVERGES DIVERGES CANNOT TELL d. [3 points] 1 h(x) x dx CONVERGES DIVERGES CANNOT TELL

5 Math 116 / Exam 1 (October 14, 9) page 5 4. [8 points] The graphs of f(x) and g(x) are shown below. Sppose that f(x) is a linear fnction. Estimate f(x)g (x)dx. Be sre to show appropriate work to spport how yo derived yor answer. y 6 4 y g(x) (,3) x f(x) 4 6 (5,) x Soltion: We can se integration by parts, letting = f(x), d = f (x)dx, dv = g (x)dx, and v = g(x). We then have f(x)g (x)dx = f(x)g(x) 5 f (x)g(x)dx. We know that f(x) is a linear fnction, so from the graph we determine f(x) = 3 5 x + 3 and f (x) = 3 5. We also know f(5) =, f() = 3, g(5) =, and g() =. We can se this to solve in or expression above. f(x)g (x)dx = f(5)g(5) f()g() g(x)dx = g(x)dx. By conting boxes, we can approximate g(x)dx, noting that each box has an area of. We approximate g(x)dx 13, and so we are left with f(x)g (x)dx (13) =

6 Math 116 / Exam 1 (October 14, 9) page 6 5. [1 points] A right isosceles triangle is a right triangle whose sides containing the right angle are of eqal length. The length from the triangle s hypotense to its right-angle vertex (opposite of the hypotense) is half the length of the hypotense. Consider the solid whose cross sections perpendiclar to the x-axis are right isosceles triangles, where the hypotense of each trianglar cross-section is contained in the region of the xy-plane bonded by the crves y = sin(x) and y = sin(x) between x = and x = π. a. [3 points] Find the volme of the cross-sectional slice located at x = x i with thickness x. Soltion: The triangle has base length sin(x i ) and the height length is sin(x i ). The area of the triangle is then sin (x i ), so the volme of the slice is sin (x i ) x. b. [3 points] Write a Riemann sm that approximates the volme of the entire solid sing n cross-sectional slices. Soltion: We can approximate the area of the solid by finding the volme of n crosssectional slices of depth x, and then adding these slices. volme n sin (x i ) x i=1 c. [6 points] Find the exact volme of the solid by sing a definite integral. Soltion: As we let n in or Riemann sm, we approach the exact volme with a definite integral. We have volme = sin (x)dx. We se integration by parts and some algebraic maniplation to solve the integral. Let = sin(x), d = cos(x)dx, dv = sin(x)dx, v = cos(x). Then we have sin (x)dx = sin(x) cos(x) + sin (x)dx = = sin(x) cos(x) π + = + sin (x)dx = x π dx dx cos (x)dx sin (x)dx (1 sin (x))dx sin (x)dx = π

7 Math 116 / Exam 1 (October 14, 9) page 7 6. [1 points] The distance between two points (x 1, y 1 ) and (x, y ) is given by D = (x x 1 ) + (y y 1 ). Consider the crve described by y = 3x 3, over the domain x 4. What is the average distance of the points on this crve to the point (, )? Soltion: A point on the crve has coordinates (x, 3x 3), so the distance from an arbitrary point on the crve to the point (, ) is given by D = (x ) + ( 3x 3 ) = x 4x x 3 = 4x 4x + 1 = (x 1) = x 1 We can se a definite integral to find the average distance over the domain x 4. avg. distance = (x 1)dx = 1 (x x) 4 = 1 (1 ) = 5 The average distance between a point on the crve y = 3x 3 over the domain x 4 is 5.

8 Math 116 / Exam 1 (October 14, 9) page 8 7. [11 points] A land srveyor is hired to measre the area of a plot of land to be sold. The srveyor ses two main highways as points of reference while measring the property. Highway 116 is soth of the property and rns perfectly in the east-west direction. Highway 1 is west of the property and rns perfectly in the north-soth direction. The srveyor starts at Highway 1 and moves eastward for the entire for-mile width of the property as he measres the distances of the northern and sothern borders of the property from Highway 116. Let n(x) and s(x) be the distances, in miles, of the northern border and sothern borders, respectively, from Highway 116 when he is x miles east of Highway 1. The srveyor s measrements are recorded in the table below. x n(x) s(x) a. [4 points] Estimate the area of the property sing the midpoint rle with for sbintervals. Be sre to show all appropriate work and don t forget to inclde appropriate nits. Soltion: Using for sbintervals, we have x = 1. To determine the length of each sbinterval, we consider the distance n(x) s(x). If we se the midpoint rle to approximate the area, we then have MID(4) = 1[(n(.5) s(.5)) + (n(1.5) s(1.5)) + (n(.5) s(.5)) + (n(3.5) s(3.5))] = 1( ) = 14 Using the midpoint rle with for sbintervals, we find that the area of the land is approximately 14 sqare miles. b. [4 points] Estimate the area of the property sing the trapezoid rle with for sbintervals. Be sre to show all appropriate work and don t forget to inclde appropriate nits. Soltion: Again, we have x = 1 and the lengths are determined by the distance n(x) s(x). The trapezoid rle ses the average of the left-hand and right-hand sms. RIGHT(4) = 1[ ] = 1.7 LEFT(4)1[ ] = TRAP(4) = = 13.6 Using the trapezoid rle with for sbintervals, we find that the area of the land is approximately 13.6 sqare miles. c. [3 points] Becase he took calcls, the srveyor knows that he can determine x, the niform distance at which he shold make measrements in order to ensre the measred area is within a desired level of accracy. Given that n(x) is a decreasing fnction and s(x) is an increasing fnction, determine the vale of x the srveyor shold se in order to measre the area within.5 sqare miles if he is sing left- and right-hand Riemann sm approximations for the area. Soltion: We solve for x nder the condition x [n(4) s(4)] [n() s()].5, which gives x.86 miles.

9 Math 116 / Exam 1 (October 14, 9) page 9 8. [13 points] Let C() be a fnction that satisfies C () = cos( ), C() = 3, and let S() be a fnction that satisfies S () = sin( ), S() = 1. a. [4 points] Write expressions for C(t) and S(t) that satisfy the above conditions. Soltion: C(t) = t cos( ) t d + 3 S(t) = sin( ) d 1 b. [5 points] A particle traces ot the crve given by the parametric eqations x(t) = C(ln(t)), y(t) = S(ln(t)) for t 1. What is the speed of the particle at time t? Yo may assme that t 1. Soltion: speed = = = ( d dt (1 1 t ln(t) ln(t) t cos(ln(t)) ln(t) cos( ) ( d d) + dt ln(t) ) ( ) 1 + t sin(ln(t)) ln(t) sin( ) ) d c. [4 points] For t 1, is the crve given by the parametric eqations in part (b) of finite or infinite length? Jstify yor answer. Soltion: The crve has infinite length. The arc length for t 1 is given by integrating the speed over the interval [1, ). With a -sbstittion = ln(t) we have which diverges by the p-test. arc length = 1 dt t ln(t) = ln(1) d

10 Math 116 / Exam 1 (October 14, 9) page 1 9. [1 points] The primary objective of most manfactring companies is to prodce and sell the nmber of nits that will generate the maximm profit for the company. Let R() define the revene income the company earns when selling nits, and let C() define the cost of prodcing nits. Then the profit, P, of selling and prodcing nits is determined by P () = R() C(), where profit, revene, and cost are all measred in dollars. a. [4 points] When trying to determine if it is beneficial to prodce and sell additional goods, companies will often consider the marginal revene, defined by R (), and the marginal cost, defined by C (). Below is a sketch of one company s marginal revene and marginal cost, as a fnction nits. On the same axes, sketch a graph of the company s marginal profit, P (). R () C () a b b. [4 points] Using yor answer to part (a), sketch a graph of P () on the axes provided below, given the conditions that P () = P and P (b) >. a b P c. [4 points] Given that b a R ()d = $135,, b a C ()d = $64,, and the company s profit when selling b nits is $5,, determine the company s profit when selling a nits. Does the company make or lose money when selling a nits? Soltion: P (b) P (a) = b a P ()d = b 5, P (a) = 71, P (a) = 19, The company loses $19, when selling a nits. a R ()d b a C ()d

Math 116 Second Midterm March 20, 2013

Math 116 Second Midterm March 20, 2013 Math 6 Second Mierm March, 3 Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until you are told to do so.. This exam has 3 pages including this cover. There are 8 problems. Note that the