MA 137 - Calculus I for the Life Sciences THIRD MIDTERM Fall 2013 11/19/2013 Name: Sect. #: Answer all of the following questions. Use the backs of the question papers for scratch paper. No books or notes may be used. You may use a calculator. You may not use a calculator that has symbolic manipulation capabilities. When answering these questions, please be sure to: check answers when possible, clearly indicate your answer and the reasoning used to arrive at that answer (unsupported answers may receive NO credit). QUESTION SCORE TOTAL 1. 10 2. 15 3. 10 4. 15 5. 10 6. 10 7. 10 8. 10 9. 10 Bonus. 10 TOTAL 100
Please make sure to list the correct section number on the front page of your exam. In case you forgot your section number, consult the following table: Sections # Lecturer Time/Location 001-004 Kate Ponto MWF 10:00 am - 10:50 am, CP 320 005-008 Alberto Corso MWF 01:00 pm - 01:50 pm, CB 102 Section # Recitation Instructor Time/Location 001 Dustin Hedmark TR 09:30 am - 10:20 am, CB 347 002 Michael Gustin TR 09:30 am - 10:20 am, RRH 0130 003 Dustin Hedmark TR 11:00 am - 11:50 am, CB 347 004 Michael Gustin TR 12:30 pm - 01:20 pm, CB 347 005 Liam Solus TR 12:30 pm - 01:20 pm, FB 213 006 Liam Solus TR 02:00 pm - 02:50 pm, FB 213 007 Joseph Lindgren TR 02:00 pm - 02:50 pm, CB 245 008 Joseph Lindgren TR 03:30 pm - 04:20 pm, FB 213
1. (a) Your cousin is coming to visit you for Thanksgiving and just called to give you an update on her arrival time. She told you that she left her house at 10AM and her speed has been between 50 mph and 70 mph. If her house is 400 miles from yours will she be on time for dinner at 6PM? Use the Mean Value Theorem to justify your answer. (b) Let f(x) = x 3 x on [0, 2]. By the Mean Value Theorem, there exists a c in (0, 2) such that f f(2) f(0) (c) =. 2 0 Find such a c.
2. Let f(x) = xe x2. (a) Find the first derivative of f. (b) Where are the critical points of f? (c) Where is f increasing? decreasing? (d) Classify the critical points of f.
(e) Find the second derivative of f. (f) Where is f concave up? concave down? (g) Where are the inflection points of f? pts: /15
6 5 4 3 2 1 0-1 -2-3 -4-5 -6-7 -8-9 -10-6 -5-4 -3-2 -1 0 1 2 3 4 5 6 3. Suppose g is a twice differentiable function defined on [ 2, 4] and g is decreasing on the interval [ 2, 1] and increasing on [ 1, 4]; g is concave up on [ 2, 0] and [1, 4] and concave down on [0, 1]. If g( 1) = 3, g(0) = 0 and g(1) = 2 sketch a possible graph for such a function g. y 3 2 1 2 1 1 2 3 4 x 1 2 3
10 9 8 7 6 5 4 3 2 1 0-1 -2-3 -4-5 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 4. The graph below is the graph of a function h. y 8 6 4 2 6 4 2 2 4 6 8 x 2 4 Find (please be as accurate as possible in identifying the values and intervals below): the critical points of h the local maxima of h the local minima of h
the global maxima of h the global minima of h where h is increasing where h is decreasing where h is concave up where h is concave down the inflection points of h pts: /15
Coughing: When a foreign object lodged in the trachea (windpipe) forces a person to cough, the diaphragm thrusts upward causing an increase in pressure in the lungs. This is accompanied by a contraction of the trachea, making a narrower channel for the expelled air to flow through. For a given amount of air to escape in a fixed time, it must move faster through the narrower channel than the wider one. The greater the velocity of the airstream, the greater the force on the foreign object. X rays show that the radius of the circular tracheal tube contracts to about two-thirds of its normal radius r 0 during a cough. 5. According to a mathematical model of coughing, the velocity v of the airstream is related to the radius x of the trachea by the equation v(x) = k(r 0 x)x 2 r 0 2 x r 0, where k and r 0 are constants. The restriction on x is due to the fact that the tracheal wall stiffens under pressure and a contraction greater than r 0 is prevented (otherwise 2 the person would suffocate). [ ] r0 Determine the value of x in the interval 2, r 0 at which v has an absolute maximum. What is the absolute maximum of v on the interval?
6. (a) Find the general expression for the sequence a n based on the values a 0 = 0 a 1 = 1 2 a 2 = 2 5 a 3 = 3 10 a 4 = 4 17 a 5 = 5 26. (b) Consider the sequence a n = Using your calculator, compute the terms ( 1 + 1 n) n defined for n 1. a 1 = a 10 = a 100 = a 1,000 = a 10,000 = a 100,000 = ( Do the above values suggest that lim 1 + 1 n exist? n n) If so, what does the limit seem to be?
7. Find the limits as n (if they exist) of the following sequences: (a) a n = ( 1) n (b) b n = 2 + ( 1)n 4n 1 + n 2 (Hint: Use the Sandwich/Squeeze Theorem) (c) c n = n(1 3n2 ) n 3 + 1
8. (a) Write the recursion for a population N t that quadruples in size every unit of time and that has 10 individuals at time t = 0. What is an explicit formula for the population N t discussed above? (b) Consider the sequence given by the recursion rule a n+1 = 3 a n 2. Find all fixed points (equilibria) of the recursion.
9. Consider the recursion a n+1 = 1 4 a n(10 a n ). It has fixed points (equilibria) a = 0 and a = 6. (1) Use the stability criterion to determine the stability of these two fixed points. a = 0 is stable/unstable a = 6 is stable/unstable (Circle your answer) (2) Use the cobwebbing method to illustrate your answer from (1) by choosing an initial value a 0 = 0.88 (plot this one on the left) and an initial value of a 0 = 7 (plot this one on the right)..a n+1.a n+1..a n a 0 = 0.88 a 0 = 7..a n
Bonus. The officers of the UK Biology Club are planning an event and they want to advertise it widely on campus by means of several posters. The top and bottom margins of each poster are 8 cm and the side margins are each 4 cm. If the area of printed material on the poster is fixed at 288 square centimeters, find the dimensions of the poster with the smallest area. (The officers want to print the posters on high quality paper but the budget of the UK Biology Club is, unfortunately, tight!) 8 4 4 8