Math 115 First Midterm February 8, 2017

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1 EXAM SOLUTIONS Math First Midterm Februar 8, 07. Do not open this eam until ou are told to do so.. Do not write our name anwhere on this eam.. 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 UMID (not name) on ever page and point this out to our instructor when ou hand in the eam.. 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. 6. 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. 7. 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. 8. The use of an networked device while working on this eam is not permitted. 9. You ma use an one 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 single notecard. 0. 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.. Include units in our answer where that is appropriate.. Problems ma ask for answers in eact form. Recall that = is a solution in eact form to the equation =, but =.67 is not.. Turn off all cell phones, smartphones, and other electronic devices, and remove all headphones, earbuds, and smartwatches. Put all of these items awa.. You must use the methods learned in this course to solve all problems. Problem Points Score Problem Points Score Total 00

2 Math / Eam (Februar 8, 07) page. [9 points] The graphs of the functions f() and g() are shown below. 6 = f() = g() Note that the graph of f() is linear for < and >, and g() is linear on < < and < <. For each of the following parts, find the given limit. If an of the quantities do not eist (including the case of limits that diverge to or ), write dne. If the limit cannot be found based on the information given, write not enough info. You do not need to show an work. a. [ points] Find lim f(). b. [ points] Find lim (f(t) ). t c. [ points] Find lim f()g(). DNE d. [ points] Find lim f(e ). e. [ points] Find lim + g (). f( + h) f() f. [ points] Find lim. h 0 h 0.

3 Math / Eam (Februar 8, 07) page The graphs of the functions f() and g() are included here for our convenience. 6 = f() = g() g. [ points] Find all the values of with < < at which the function f() is not continuous., 0, h. [ points] What is the range of = g()? [0,] i. [ points] For which of the following values of is f () > 0? Circle all that appl. = = =. = e none of these

4 Math / Eam (Februar 8, 07) page. [ points] Let K(p) = ( + cos(p)) +p. Use the limit definition of the derivative to write an eplicit epression for K (). Your answer should not involve the letter K. Do not attempt to evaluate or simplif the limit. Please write our final answer in the answer bo provided below. K () = lim h 0 ( + cos( + h)) +(+h) ( + cos()) +() h

5 Math / Eam (Februar 8, 07) page. [0 points] A group of students planted a pine tree and an oak tree alongside the Diag. Let P (t) and O(t) be the height (in feet) of the pine and the oak t ears after the were planted, where P (t) = 70 6A 0.0t 0 and O(t) = + 00e 0.t where A > is a constant. For this problem, our answers should be in eact form or accurate up to the first two decimal places. a. [ points] How tall (in feet) were each of the trees when the were planted? Pine: Oak: b. [ points] Ten ears after the trees were planted, the height of the pine was 8 ft. Find the value of A. Find our answer algebraicall and show all our work. 70 6A 0. = 8 A 0. = 6 = 0.8 A = (0.8) A = (0.8).0 c. [ points] How man ears after being planted does it take the oak to be 8 ft? Find our answer algebraicall and show all our work. 0 = e 0.t 0 = e 0.t e 0.t = ( ) 6 0.t = ln 800 t = 0. ln ( ). ( ) 6 0. ln ears

6 Math / Eam (Februar 8, 07) page 6. [0 points] After the students planted the pine and the oak, the universit has been monitoring the growth and health of the trees. Fifteen ears after being planted, an invasion of cankerworms (a tpe of caterpillar) is found on the oak. It is predicted that the number of cankerworms (in hundreds) in the oak s weeks after the pest was detected is given b C(s) = e 0.s. a. [ points] B what percent is the population of cankerworms epected to grow ever week? Since b = e 0., then r = e 0.. The population grows b 00r% = 00(e 0. )% ever week. The population grows b 00(e 0. )%.9% ever week b. [ points] Let F (m) be the number of cankerworms in the oak (in thousands) m das after the pest was detected. Find a formula for F (m) in terms of m onl. F (m) = 0.e 0.0m c. [ points] A population of weevils (another insect) invades the pine. It is estimated that the population of weevils increases b percent ever weeks. How man weeks does it take for the population of weevils to triple? Show all our work and round our answer to the nearest week. If the population of weevils is given b W (t) = ab t, then the fact that it increases b percent ever weeks ields W () =.W (0). In other words ab =.a b =. =.. The time T it takes the population to triple satisfies W (T ) = W (0). Hence a(.) T = a (.) T =. T ln(.) = ln() T = ln() ln(.) ln() ln(.) 6 weeks

7 Math / Eam (Februar 8, 07) page 7. [7 points] A portion of the graph of the function f() is shown below. Note that f() is linear for >. 0 a. [ points] Let the quantities I V be defined as follows: = f() 8 I. The number 0. 6 II. f( ) f(). III. f ( ). IV. The slope of the secant line between the points on the graph at = and =. V. The slope of the tangent line at =. Rank the quantities in order from least to greatest b filling in the blanks below with the options I V. You do not need to show our work. V < IV < 0 < II < III b. [ points] There are four graphs below. Circle the one graph that could be the graph of the derivative of f(). Note that the graphs are not all drawn at the same scale. I. III. II. 0 0 IV. 0 0

8 Math / Eam (Februar 8, 07) page 8 6. [ points] A compan designs chambers whose interior temperature can be controlled. Their chambers come in two models: Model A and Model B. a. [ points] The temperature in Model A goes from its minimum temperature of C to its maimum temperature of C and returning to its minimum temperature three times each da. The temperature of this chamber at 0 am is C. Let A(t) be the temperature (in C) inside this chamber t hours after midnight. Find a formula for A(t) assuming it is a sinusoidal function. The amplitude of A(t) is ( ) = 9. The period is 8 hours and the midline is = +( ) = 6. ( We know that ) A(0) = a maimum value of A(t). Then π A(t) = 9 cos (t 0) A(t) = ( ) π 9 cos (t 0) b. [6 points] Let B(t) be the temperature (in C) inside Model B t hours after midnight, where ( ) B(t) = cos 7 t +. Find the two smallest positive values of t at which the temperature in the chamber is 6 C. Your answer must be found algebraicall. Show all our work and give our answers in eact form. ( ) cos 7 t + ( = 6 cos ( ) 7 t + = cos t = 7 ( (cos ) ) ) 7 t + =. ( t + = π cos ) 7 t = 7 ( (π cos ) )

9 Math / Eam (Februar 8, 07) page 9 7. [0 points] Two housecats, Jasper and Zander, escape from their house at the same time and travel along a straight line between their house and a tree. Let J(t) (respectivel Z(t)) be Jasper s (respectivel Zander s) distance, in feet, from the tree t seconds after escaping. The table below shows some of the values of J(t) and Z(t). Assume that J(t) is invertible. t J(t) Z(t) 9 6 a. [ points] What is Jasper s average velocit for 6 t? Be sure to include units. 6 = 0 6 = =. ft/sec. b. [ points] Estimate Z (). Remember to show our work. You can approimate it using either: 6 Average rate of change in [, 7]: 7 = 7/6. 6 Average rate of change in [, ]: = /9. Averaging these two: (7/6 + /9) = 0.86, or Average rate of change in [, 7]: 7 = 0.8. (picking one) 7/6 c. [ points] Circle the one statement below that is best supported b the equation Z(J (8) ) =. i. seconds after escaping, Zander is feet closer to the tree than Jasper was 8 seconds after escaping. ii. Four seconds before Jasper is 8 feet from the tree, Zander is feet from the tree. iii. When Jasper is feet further from the tree than he was 8 seconds after escaping, Zander is feet from the tree. iv. When Jasper is feet closer to the tree than he was 8 seconds after escaping, Zander is feet from the tree. v. Four seconds after Jasper is 8 feet from the tree, Zander is feet from the tree. d. [ points] Circle the one statement below that is best supported b the equation (J ) () = 0.. i. In the third second after leaving the house, Jasper travels about 0. feet. ii. When Jasper is feet from the tree, he is traveling about 0. feet/second slower than he was one foot earlier. iii. Jasper gets about. feet closer to the tree during the third second after leaving the house. iv. It takes Jasper about one-tenth of a second to go from feet to. feet from the tree. v. One-half of a second before Jasper was feet from the tree, he was about.9 feet from the tree.

10 Math / Eam (Februar 8, 07) page 0 8. [ points] On the aes provided below, sketch the graph of a single function = f() satisfing all of the following conditions: The domain of f() is the interval 8 < 6. f() is continuous for all in the interval 8 < <. f ( 7) = 0. f() is decreasing and concave up for all in the interval 6 < <. The average rate of change of f() is equal to 0. between = and =. f(0) = and f (0) =. lim f() = f() and lim f() < lim f(). + f() has constant rate of change on the interval 6. Make sure that our graph is large and unambiguous

11 Math / Eam (Februar 8, 07) page 9. [9 points] A pharmaceutical compan just released a new medication to reduce the cold smptoms in children between 8 months old and ears of age. Let D(z) be the dose (in ounces) recommended for a child that is z ears old. A table with some values of D(z) is shown below. z. 8 0 D(z) a. [ points] Find a formula for D(z) on z assuming it is a linear function in this interval Using the points (,.) and (, 8.6), we can find the slope m = =.7. Appling the point slope formula for the linear function we get D(z) =. +.7(z ). D(z) =. +.7(z ) or.7z + 0. b. [ points] Suppose that D(z) is invertible. Give a practical interpretation of the equation D (9) = 6.. A child whose recommended dose is 9 ounces is si and a half ears old. c. [ points] Below is the first part of a sentence that will give a practical interpretation of the equation D () =. in the contet of this problem. Complete the sentence so that the practical interpretation can be understood b someone who knows no calculus. Be sure to include the appropriate units in our answer. As the age of an child increases from ears to months, the recommended dose increases b about 0. ounces.

12 Math / Eam (Februar 8, 07) page 0. [ points] Find all real numbers B and positive integers k such that the rational function satisfies the following two conditions: H() = H() has a vertical asmptote at = lim H() eists. If no such values eist, write none. Justification: In order for lim 9 + k 6 B H() to eist, the degree of the polnomial in the numerator has to be smaller or equal to (the degree of the polnomial in the denominator). In order for = to be a vertical asmptote, ou need the denominator to be zero. Hence 6 B( ) = 6 8B = 0 which requires B =. In this case H() = 9 + k 6 with k =, or. Since 9 + k 0, then H() has a vertical asmptote at = when B =. B = k =,, or. [ points] A part of the graph of a function h() is given below. In each of the following parts, the corresponding portion of the graph of a function obtained from h b one or more transformations is shown, together with a list of possible formulas for that function. In each case choose the one correct formula for the function shown. Note that the graphs are not all drawn at the same scale. a. [ points] = w() = h() b. [ points] = u() 6 8 A. h( ) F. h( ) A. h(0. + ) F. h( ) B. h() G. h( + ) B. h(0. ) G. h( + ) C. h() + D. h( ) + H. h( ) I. h( + ) J. h( ) C. h(0.( )) D. h(0.( + )) H. h(( )) I. h(( + )) J. h(0.( + 0.)) E. h( + ) K. none of these E. h(0.( 0.)) K. none of these

Math 115 First Midterm February 9, 2016

Math 115 First Midterm February 9, 2016 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,