Math First Midterm Februar 9, 06 EXAM SOLUTIONS. Do not open this eam until ou are told to do so.. This eam has pages including this cover. There are 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.. Do not separate the pages of this eam. If the do become separated, write our initials (not name) on ever page and point this out to our instructor when ou hand in the eam. 4. Note that the back of ever page of the eam is blank, and, if needed, ou ma use this space for scratchwork. Clearl identif an of this work that ou would like to have graded.. Please read the instructions for each individual problem carefull. One of the skills being tested on this eam is our abilit to interpret mathematical questions, so instructors will not answer questions about eam problems during the eam. 6. Show an appropriate amount of work (including appropriate eplanation) for each problem, so that graders can see not onl our answer but how ou obtained it. 7. The use of an networked device while working on this eam is not permitted. 8. You ma use an calculator that does not have an internet or data connection ecept a TI- 9 (or other calculator with a qwert 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 note card. 9. For an graph or table that ou use to find an answer, be sure to sketch the graph or write out the entries of the table. In either case, include an eplanation of how ou used the graph or table to find the answer. 0. Include units in our answer where that is appropriate.. Turn off all cell phones, smartphones, and other electronic devices, and remove all headphones, earbuds, and smartwatches.. You must use the methods learned in this course to solve all problems. Problem Points Score 0 4 4 0 0 6 Problem Points Score 7 0 8 4 9 7 0 9 Pre-Test Total 00
Math / Eam (Februar 9, 06) page. [0 points] A portion of the graph of a function f is shown below. 6 4 = f() 4 6 Note: You ma assume that pieces of the function that appear linear are indeed linear. Use the graph above to evaluate each of the epressions below, and write our answer on the answer blank provided. If an of the quantities do not eist (including the case of limits that diverge to or ), write dne. a. [ point] f() f. [ point] lim h 0 f(4. + h) f(4.) h Answer: b. [ point] lim f() Answer: 4 f(p) g. [ point] lim p 0. p Answer: 0 c. [ point] lim q f(q) Answer: 4 h. [ point] lim t f(t)f(t + ) Answer: d. [ point] lim z f() DNE Answer: 0 i. [ point] lim + f(f()) Answer: e. [ point] lim r 6 f(r) Answer: j. [ point] lim s f(f(s)) Answer: Answer:
Math / Eam (Februar 9, 06) page. [ points] In Townsville, USA, a vat of Chemical Z is spilled into Lake Townsville. Let c(d) be the concentration of Chemical Z (in mg/l) at a depth of d meters below the surface in Lake Townsville. Assume that c(d) is differentiable for 0 < d <. A portion of the graph of Z = c(d) is shown below. Z 4 Z = c(d) 4 a. [ point] What is the concentration (in mg/l) of Chemical Z at the surface of Lake Townsville? Answer: 4 b. [ points] Circle all of the intervals below for which c (d) is positive over the entire interval. Circle none if there are no such intervals. 0. < d < 0.8. < d <.8. < d <.8. < d <.8 4. < d < 4.8 none d c. [ points] What is the average rate of change of the concentration of Chemical Z over the interval from d = to d =? Remember to include units. ( ) mg/l ( ) m = (mg/l)/m Answer: (mg/l)/m d. [ points] Suppose that c(d) is linear for. < d <. Find c (.). Answer: e. [ points] Using our answer to part (d), circle the appropriate choice and fill in the blank in the sentence below. Remember to include units. Answer: If we go from a depth of.00 meters to a depth of.498 meters below the surface of Lake Townsville, the concentration of Chemical Z will (circle one) increase decrease b approimatel 0.004 mg/l.
Math / Eam (Februar 9, 06) page 4. [4 points] Let h() = ( + )e. Then the derivative of h is given b the formula h () = ( + 7)e. Find an equation for the tangent line to the graph of = h() at =. Because h() = ( + )e () = 4 h () = (() + 7)e () = 9 the tangent line has slope 9 and goes through the point (, 4), so to get the formula for the tangent line: 4 = 9( ) = 9 Answer: = 9 e if < 4. [0 points] Consider the function g defined b g() = cos( ) if < if. ( )(6 ) a. [ points] Use the limit definition of the derivative to write an eplicit epression for g (). Your answer should not involve the letter g. Do not attempt to evaluate or simplif the limit. Please write our final answer in the answer bo provided below. ( g cos ( + h) +h ) cos ( ) () = lim h 0 h Answer: g () = ( cos ( + h) +h ) cos ( ) lim h 0 h b. [ points] Find all vertical asmptotes of the graph of g(). If there are none, write none. Note that = is *not* a vertical asmptote because the third piece of the formula for g() is onl valid for. The vertical asmptotes are = 0 and = 6. Answer: = 0, = 6 c. [ points] Determine lim g(). If the limit does not eist, write dne. lim g() = lim ( )(6 ) = lim + 7 6 =. Answer:
Math / Eam (Februar 9, 06) page. [0 points] Vikram takes a non-stop train ride from Chennai straight to New Delhi. Let g(t) be the distance (in km) of Vikram s train from Chennai t hours after his ride begins. Assume that the function g is increasing and invertible, and that g and g are differentiable. Several values for g(t) are shown in the table below. t 0 6. 0 6 8 g(t) 0 46 448 69 74 80 a. [ points] Estimate the instantaneous velocit of Vikram s train 6 hours after his ride begins. Show our work and include units. We estimate using average velocit based on nearb measurements: 448 46 6. = 68 So we estimate the instantaneous velocit of Vikram s train 6 hours after his ride begins to be about 68 km/h. Answer: 68 km/h b. [ points] Suppose (g ) (700) = C, where C is some constant. (i) Using the data in the table above, find the best possible estimate of C. Show our work. We estimate the derivative based on nearb measurements: 0 = 0.0 h/km 74 69 So we estimate C 0.0. 700 has units km, and C has units h/km. Answer: 0.0 (ii) In interpreting the equation (g ) (700) = C, what are the units on 700 and C? Answer: Units on 700 are km Answer: Units on C are h/km c. [ points] Let R(t) be the total rainfall (in cm) in New Delhi during the first t hours of Vikram s train ride. Epress the following statement with a single mathematical equation: Over the first 900 km of Vikram s train ride, it rained.6 cm in New Delhi. Answer: R(g (900)) =.6
Math / Eam (Februar 9, 06) page 6 6. [ points] On the aes provided below, sketch the graph of a single function = h() satisfing all of the following: h() is defined for all in the interval < <. h () > 0 for all <. lim h() = 0. h( ) =. The average rate of change of h() between = and = is. h() =. h() is linear between = and =. h () =. lim h() =. 4 lim h() does not eist. 4 h () < 0 for all > 4. Make sure that our sketch is large and unambiguous. 4 4 4 4
Math / Eam (Februar 9, 06) page 7 7. [0 points] Note that the situations described in parts a. and b. on this page are not related to each other. a. [6 points] A dose of a total of. milliliters of a drug is injected into a patient steadil for 0. seconds. At the end of this time, the quantit of the drug in the bod starts to deca eponentiall, decreasing b 0.8 percent per second. Let Q(t) be the quantit of the drug in the bod, in milliliters, t seconds after the injection begins. The function Q(t) can be described using a piecewise-defined formula, as shown below. Use the description above to fill in the four answer blanks provided below with appropriate formulas and bounds so that the function Q(t) is continuous for all t > 0. Answer: Q(t) = 4t if 0 < t 0..(0.998) t 0. if 0. < t. b. [4 points] Suppose that someone studing parking habits at U-M during the 0-6 school ear makes the following statement: During this school ear, the number of cars that arrive on campus before 8 am has increased b % ever thirt das. Let C(d) be the number of cars that arrive on campus before 8 am on the dth da of the school ear. Which of the formulas below model the situation described in the quote above, where K is some positive constant? (Circle all correct answers. Or circle none of these.) C(d) = K(0.) d/0 C(d) = K(./0) d C(d) = K + (0./0) d C(d) = K(.) d/0 C(d) = K(0.8) d/0 C(d) = K + (./0) d C(d) = Ke.d C(d) = K(4) d/0 C(d) = K + 0.d C(d) = Ke 0.d C(d) = Kd. C(d) = K + 0.d/0 C(d) = Ke ln(.)d/0 C(d) = Kd 0. C(d) =. sin( πd ) + K C(d) = Ke ln(0.)d/0 C(d) = K + (0.) d/0 C(d) =. cos( πd ) + K C(d) = K(0./0) d C(d) = K + (.) d/0 none of these
Math / Eam (Februar 9, 06) page 8 8. [4 points] Let A() be a sinusoidal function, a portion of which is shown in the graph below. 7 6 4 Write a formula for A(). = A() 4 6 7 8 9 There are man possible formulas. Among the possbilities are the following: A() = 4 cos ( π ( )) + A() = 4 cos ( ( π )) + A() = 4 sin ( π ( 4)) + A() = 4 sin ( π ( )) 4 + Answer: A() = 4 cos ( π ( )) +
Math / Eam (Februar 9, 06) page 9 9. [7 points] Consider the function f() defined b e A + B if < f() = C( ) if 0 if >. Suppose f() satisfies all of the following: f() is continuous at =. lim f() = + lim f(). + lim f() = 4. Find the values of A, B, and C. Show our work. You must give eact answers. Do not use decimal approimations. For eample, 0. would not be an acceptable answer if the answer were. Because f() is continuous at = (the first propert), lim f() = lim f(). + So we have e A + B = C( ) = 0 and thus e A = B (*). Now, b the second propert, we have lim f() = + lim f(), so + Thus C = 6. 0 = + C( ) 6 = + 4C 4 = 4C 6 = C Note that if lim ea eists, then it is equal to 0 (and A < 0). B the third propert, we therefore see that 4 = lim f() = lim (ea + B) = 0 + B = B. So, B = 4, and using equation (*) above, we see that e A = ( 4) so e A = 4 and A = ln(4/). Answer: A = ln ( ) 4, B = 4, and C = 6
Math / Eam (Februar 9, 06) page 0 0. [9 points] Suppose data is collected at a U-M basketball game held at Crisler Center. Let E(t) be the total amount of electricit, in megawatt-hours (MWh), that has been used b Crisler Center during the first t minutes of the basketball game, which starts at eactl 7:00 pm. Assume that E is invertible and that both E and E are differentiable. a. [ points] Suppose b and c are positive constants. Use a complete sentence to give a practical interpretation of the equation E(0 + b) = E(0) + c in the contet of this problem. Your sentence should involve the constants b and c but not E. Be sure to include units. In the b minutes after 7:0 pm, Crisler Center uses c MWh of electricit. b. [ points] Fill in the two answer blanks below to write a single mathematical equalit involving the derivative of either E or E which supports the following claim: During the basketball game, Crisler Center uses about.8 MWh of electricit during the first seconds after 7:4 pm. Answer: E (4) = 6 c. [ points] Which of the sentences below best epresses the meaning of the equation E (0) =.E () in the contet of this problem? (Circle the one best choice.) A. Crisler Center uses 0% more electricit during the first 0 minutes after the game starts than during the first minutes after the game starts. B. It takes half as long for Crisler Center to use the first MWh of electricit during the game than for it to use the net 8 MWh. C. Crisler Center uses 0% as much electricit during the first 0 minutes after the game starts than during the first minutes after the game starts. D. It takes 0% longer for Crisler Center to have used a total of 0 MWh of electricit during the game than for it to use the first MWh. E. Crisler Center uses twice as much electricit during the first 0 minutes after the game starts than during the net minutes. F. It takes 0% less time for Crisler Center to have used a total of MWh of electricit during the game than for it to use the first 0 MWh.
Math / Eam (Februar 9, 06) page. [ points] A portion of the graph of a function g is shown below. In each of parts a. d. on this page, the corresponding portion of the graph of a = g() function obtained from g b one or more transformations is shown, together with a list of possible formulas for that function. In each case, circle the one correct formula for the function shown. a. [ points] = U() b. [ points] = M() U() =? Circle the one correct choice below. g() g() + g(). g(0.) g() g( + ) 0.g() g() g( ) M() =? Circle the one correct choice below. g() g() + g(). g(0.) g() g( + ) 0.g() g() g( ) c. [ points] = A() d. [ points] = R() A() =? Circle the one correct choice below. g() + g() g( + ) g(0.) + g(0.) 0.g( + ) g( ) g( ) 0.g( ) R() =? Circle the one correct choice below. g( ) + g( + ) g( ) + g( ) g( ) g( + ) g( ) + g( + ) + g( +)+ e. [ points] A portion of the graph of the derivative of one of the five functions above is shown on the right. Which derivative is shown? Circle the one correct choice below. g () U () M () A () R ()