Math 116 First Midterm February 6, 2012

Transcription

1 Math 6 First Midterm Februar 6, Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until ou are told to do so.. This exam has pages including this cover. There are 9 problems. Note that the problems are not of equal difficult, so ou ma want to skip over and return to a problem on which ou are stuck. 3. Do not separate the pages of this exam. If the do become separated, write our name on ever page and point this out to our instructor when ou hand in the exam. 4. Please read the instructions for each individual problem carefull. One of the skills being tested on this exam is our abilit to interpret mathematical questions, so instructors will not answer questions about exam problems during the exam. 5. Show an appropriate amount of work (including appropriate explanation) for each problem, so that graders can see not onl our answer but how ou obtained it. Include units in our answer where that is appropriate. 6. You ma use an calculator except a TI-9 (or other calculator with a full alphanumeric kepad). However, ou must show work for an calculation which we have learned how to do in this course. You are also allowed two sides of a 3 5 note card. 7. If ou use graphs or tables to find an answer, be sure to include an explanation and sketch of the graph, and to write out the entries of the table that ou use. 8. Turn off all cell phones and pagers, and remove all headphones. Problem Points Score Total

2 Math 6 / Exam (Februar 6, ) page. [ points] Indicate if each of the following is true or false b circling the correct answer. No justification is required. a. [ points] If 3f(x) + dx = 8, then f(x) dx =. 3f(x)+ dx = 3f(x)dx+ dx = 3 f(x)dx+ = 8 then f(x) dx = b. [ points] If b a f(x)dx = and b a g(x)dx = 3 then b a f(x)g(x)dx = 6. For example: If f(x) = and g(x) = 3 x with a = and b =, then b a f(x)dx = dx = and b a g(x)dx = 3 xdx = 3. But b a f(x)g(x)dx = 3 xdx = 3 6. c. [ points] If f(x) = x + t 4 dt then f(x) is increasing. Since f (x) = + ( x) 4 ( ) = + 6x 4 >, then f(x) is increasing. d. [ points] If f(x)dx g(x)dx then f(x) g(x) for x. For example: f(x) = x and g(x) = x, then f(x)dx = xdx = and g(x)dx = xdx =. But f() = g() =. e. [ points] If g(x) is odd and 3 g(x)dx =, then 3 g(x)dx =. Since g(x) is odd, then g(x)dx = and a b g(x)dx = b a g(x)dx. Hence 3 g(x)dx = 3 g(x)dx + g(x)dx = 3 g(x)dx = 3 g(x)dx =. f. [ points] If f(t) is measured in dollars per ear, and t is measured in ears, then b a f(t)dt is measured in dollars per ears squared. The units for b a f(t)dt are dollars (dollars per ear (units for f(t)) times ear (units for t)).

3 Math 6 / Exam (Februar 6, ) page 3. [4 points] Let F (x) = x f(t)dt, where the graph of f(t) is given below. In each blank space below, determine whether the number on the left is greater than, less than, or equal to the number to the right, and fill the blank with the smbol >, <, or = accordingl. If there is not enough information to compare the given pair of numbers, write none in the blank space F ( ) F () F ( ) F () F () F (3) F () 8 F ( ) F () F ( ) F () 3 5 f(t)dt Average of f(x) on [, ]. F ( ) < F () F ( ) < F () F () > F (3) F () < 8 F ( ) = F () F ( ) > F () 3 5 f(t)dt < Average of f(x) on [, ].

4 Math 6 / Exam (Februar 6, ) page 4 3. [ points] A boat travels in a straight line toward an island d km awa. The velocit v(t) (toward the island is positive velocit) in km/hr, t hours after departing from its starting position. The velocit v(t) during the first three hours is recorded at half hour intervals, and is given in the table below: t v(t) a. [8 points] Find an estimate for how far the boat is from the starting point after 3 hours using the four approximations LEFT, RIGHT, MID and TRAP. Use the maximum number of subintervals possible. Write each sum, and justif whether the sum is an underestimate or an overestimate. Assume the velocit is alwas decreasing and has no inflection points. Circle our answers. LEFT(6) = ( ) = 5 overestimate. RIGHT(6) = ( ) = 94 underestimate. TRAP(6) = (5 + 94) = 4.5 underestimate. MID(3) = ( ) = 6 overestimate. since v(t) is decreasing and concave down. b. [4 points] If it takes the boat 5 hours to reach the island, write an expression involving integrals for the distance S(t) between the island and the boat for t 5. S(t) = d t v(x)dx.

5 Math 6 / Exam (Februar 6, ) page 5 4. [ points] Given the graph of f(x), sketch the graph of g(x). Make sure our graph accuratel shows the intervals where g(x) is increasing or decreasing and its concavit. a. [5 points] Let g (x) = f(x) with g() =. b. [5 points] Let g(x) = x f(t)dt.

6 Math 6 / Exam (Februar 6, ) page 6 5. [ points] The graph of f(x) and a table of values for the continuous functions g(x) and h(x) are given below. The function h(x) is an antiderivative of g(x) x 3 4 g(x) h(x) Compute the exact value of each of the following expressions: 7 a. [ point] f(x) dx 7 f(x) dx = = 9. b. [4 points] e f(ln x) x dx If u = ln x, then e f(ln x) dx = x f(u)du = c. [7 points] Find xg (x)dx If w = x, then xg (x)dx = 4 w g (w)dw = 4 Integration b parts with u = w and v = g (w) ields 4 wg (w)dw = wg(w) 4 xg (x)dx = g(w)dw wg (w)dw. = 4g(4) g() (h(4) h()) = 4(9) (5) (7 3) =. 4 wg (w)dw = () = 3. 4

7 Math 6 / Exam (Februar 6, ) page 7 6. [ points] The lateral faces of a tank are determined b the curve = x 4 and the x-axis (where x and are measured in meters). The length of the tank is meters. Be sure to include units in our answers. x 4 a. [5 points] The tank is filled with water to a height of one half a meter. If the densit of water is, kg/m 3, write an expression that approximates the mass of one slice of water meters above the ground and meters thick. Mass is densit times volume. The densit is constant, and the volume of one slice at height with thickness is Therefore, the mass of the slice is ( ) /4 m 3., ( ) /4 kg. b. [ points] Write a definite integral that represents the total mass of water in the tank. We add up all of the slices from to.5:.5, ( ) /4 d kg. c. [4 points] Write a definite integral that represents the amount of work required to pump the water to the top of the tank. Work is force times distance. The force on a slice at height with thickness (or d) is 9.8 times the mass, which we computed above. This slice travels a distance ( ) under this force. Therefore the work done on one slice is So we add up all of those slices to get.5 9.8, ( ) /4 ( ) N. 9.8, ( ) /4 ( ) d J or N m.

8 Math 6 / Exam (Februar 6, ) page 8 7. [8 points] Let S be the solid whose base is the region bounded b the graph of the curve = x(+a ln(x)) (for some positive constant a > ), the x-axis, the lines x = and x = e. The cross-sections of S perpendicular to the x-axis are squares. Find the exact volume of S. Show all our work to receive full credit. x a ln x As the cross-section of each slice is a square with sidelength, the volume of one slice is x, Therefore the total volume is e ( ) dx = x( + a ln x) = a e x( + a ln x) dx +a u du = ln + a. a The second line comes from the u-substitution u = + a ln x.

9 Math 6 / Exam (Februar 6, ) page 9 8. [ points] A bullet, with constant densit δ = 3 g/cm 3, has the shape of a solid generated b rotating the region enclosed b piece of the ellipse x + 4 = 4 ling in the first quadrant, the x-axis and the -axis, around the x-axis. The variables x and are measured in cm. x 4 = 4 a. [5 points] Write an expression that approximates the mass of one slice of the bullet perpendicular to the x-axis at x cm from the -axis and x cm thick. Mass is densit times volume. The slice at x is a right circular clinder, so its mass is ( ) 3 π x 4 x grams. Let ( x, ȳ, z) be the coordinates of the bullet s center of mass. b. [3 points] Find a formula for x. The center of mass has x-coordinate x = x dm M ( ) 3πx x 4 dx = ( ) 3π x 4 dx c. [ points] What can ou sa about ȳ. Justif. Because of the rotational smmetr and constant densit, we have ȳ =.

10 Math 6 / Exam (Februar 6, ) page 9. [ points] In the following problems show all our work to receive full credit. a. [7 points] The population of an invasive aquatic plant in a circular lagoon has densit given b δ(r) = ( e r ) kg/m, where r is the distance in meters from its center. The lagoon has radius R meters. Find the exact amount of plants living at the lake. Since densit is a function of radius, we will slice the biomass into rings with thickness r. The mass of one ring is approximatel δ(r) πr r. So now we add up all the slices, and we get total mass R ( e r) πr dr = π = π R R ( r e r) dr ( e u ) du = π ( u + e u) R = π(r + e R ). b. [4 points] Let F (x) = x e t dt. Find the exact value of the length of the curve on x. The formula for length of an arc is L = + (F (x)) dx. B the second FTC, F (x) = e x. Therefore the length of the arc is L = + ( e x ) dx = = = e. + (e x ) dx e x dx