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1 MA 37 Calculus I with Life Science Applications FINAL EXAM Fall 206 2/5 /206 Name: Sect. #: Do not remove this answer page you will return the whole exam. No books or notes may be used. Use the backs of the question papers for scratch paper. You may use a graphing calculator during the exam, but NO calculator with a Computer Algebra System (CAS) or a QWERTY keyboard is permitted. Absolutely no cell phone use during the exam is allowed. The first part of the exam consists of 2 multiple choice questions, each worth 5 points. Record your answers on this page by filling in the box corresponding to the correct answer. For example, if (a) is correct, you must write a b c d e Do not circle answers on this page, but please do circle the letter of each correct response in the body of the exam. It is your responsibility to make it CLEAR which response has been chosen. You will not get credit unless the correct answer has been marked on both this page and in the body of the exam. The second part of the exam consists of four open-response questions and one bonus question. When answering these questions, check your answers when possible. Clearly indicate your answer and the reasoning used to arrive at that answer. Unsupported answers may receive NO credit.. a b c d e 2. a b c d e GOOD LUCK! 3. a b c d e 4. a b c d e 5. a b c d e 6. a b c d e 7. a b c d e 8. a b c d e 9. a b c d e QUESTION SCORE OUT OF Multiple Choice 60 pts 3. 0 pts 4. 0 pts 5. 0 pts 6. 0 pts Bonus. 0 pts TOTAL 00 pts 0. a b c d e. a b c d e 2. a b c d e

2 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 Alberto Corso MWF 0:00 am - 0:50 am, CB 8 Section # Recitation Instructor Time/Location 00 Luis Sordo Vieira TR 08:00 am - 08:50 am, CB Luis Sordo Vieira TR 09:00 am - 9:50 am, CB Kalila Sawyer TR 0:00 am - 0:50 am, CB Kalila Sawyer TR :00 am - :50 am, CB Stephen Deterding TR 2:00 pm - 2:50 pm, CB Stephen Deterding TR 0:00 pm - 0:50 pm, CB Matias von Bell TR 02:00 pm - 02:50 pm, CB Matias von Bell TR 03:00 pm - 03:50 pm, CB 339

3 . Our thyroid absorbs most of the iodine in our body. As a result radioactive iodine is used to treat thyroid disorders since it will attack the thyroid and no other parts of the body. If a patient has to avoid contact with other people until less than 0 percent of the original dose of iodine remains in his/her body, how long will the patient have to wait after the iodine is administered? It is known that this form of radioactive iodine has a half-life of 3.22 hours. [Hint: recall the expression W (t) = W 0 (/2) t/t hl, where T hl is the half-life of the element.] (a).32 hours (b) hours (c) 32.2 hours (d) hours (e) None of the above. 2. Let { 2x + c x < f(x) = 3x 2 + 2x + 5 x For what value(s) of c is this function continuous? (a) c = 4 (b) c = 8 (c) c = 2 (d) c = 6 (e) c = 20

4 3. The following log-log plot gives resting heart rates for various animals. 0 3 pulse rate (beats/minute) mass (grams) The functional relationship between the mass M of an animal and its pulse rate P best describing the above data is: (a) P =, M (b) P =, 700 M 0.29 (c) P = 0.29M +, 700 (d) P =, 700 M 0.29 (e) None of the above. 4. Suppose f(x) = x 3 2x + 5x. Find an equation of the tangent line to the graph of y = f(x) at the point (, 4). (a) y = 4x + 8 (b) y = 4x (c) y = x + 3 (d) y = 4 (e) y = 3x +

5 5. Let g(x) be a differentiable function such that lim x 0 g(x) = 2 and lim x 0 g (x) = 4. Use l Hôpital s rule to compute the limit g(x) 2 lim x 0 x cos(x). (a) 2 (b) 3 (c) 4 (d) 5 (e) 6. Suppose f(x) = x 2 [g(x)] 3, g(4) = 2, and g (4) = 3. Find f (4). (a) 256 (b) 288 (c) 425 (d) 640 (e) 696

6 7. Let f(x) = x 3 2x 2. Find the largest open interval on which f is decreasing. (a) ( 2, 0) (b) (0, 2) (c) (, 2/3) (d) (2/3, ) (e) (0, 4/3) 8. The volume of a cone of radius r and height h is given by V = 3 πr2 h. If the radius and the height both increase at a constant rate of 2 centimeters per second, at what rate, in cubic centimeters per second, is the volume increasing when the height is 9 centimeters and the radius is 6 centimeters? (a) (b) (c) (d) (e) 2π 0π 24π 54π 96π

7 9. Consider the function f(x) = 2x x 2 defined on the interval [, 3]. Estimate 3 (2x x 2 ) dx using right endpoints for n = 4 approximating rectangles all having bases of the same length, as shown in the picture. y x 2 3 (a).333 (b) 2.5 (c) 0 (d).75 (e) Suppose y(x) satisfies the differential equation y (x) = 7y(x), with initial condition y(0) = 5. The solution y(x) is (a) (b) (c) (d) (e) y(x) = 7e 5x y(x) = 05e 7x y(x) = 5e 7x y(x) = 5e 7x y(x) = 5e 5x

8 . The function f that verifies the following equation is 2 + x a f(t) t 7 dt = 4x 3 (a) f(x) = 4x 3 (b) f(x) = 2x 3 (c) f(x) = 4x 4 (d) f(x) = 2x 3 (e) f(x) = 4x 3 2. Suppose h is a continuous function such that h() = 2 h () = 2 h () = 2 h(2) = 6 h (2) = 5 h (2) = 3 and h and h are continuous everywhere. Evaluate: 2 h (t) dt. (a) 3 (b) 4 (c) 7 (d) 8 (e)

9 3. When an new species is introduced into an environment there may be no natural predators. In this case, the population may grow very rapidly. Suppose such an invasive species is introduced into a region and the population is measured at several times. Time in months Population 504 4,032 32, ,048. Plot this data (as accurately as possible) in the semi log plot below Find a functional relationship between population and time. pts: /0

10 4. (a) Nina has one more gift to wrap for her family. Her gift is rather small, but it has a very recognizable shape so she wants to keep her family guessing by using the largest possible box. She doesn t have any boxes left. It is late on Christmas Eve and all the stores are closed. She decides to make a box (with no lid) out of a 00 cm by 00 cm square sheet of cardboard. She cuts a square from each corner of the sheet so that she can fold up the sides to make her box. Write the expression of the volume V (x) of the box as a function of the size x of the square she cuts from each corner. (b) What is the volume of the largest possible box that Nina can make? (c) What s the length of the side of the square she cuts from each corner? pts: /0

11 5. (a) Evaluate 2 (2x 2 + 2x 2 ) dx. (b) Graph y = x. Use the graph and a geometric argument to evaluate 3 5 ( x) dx. (c) Graph y = + x 2. Use the graph and a geometric argument to evaluate ( + x 2 ) dx. pts: /0

12 6. Use the Fundamental Theorem of Calculus (part ) to evaluate the following derivatives: (a) If F (x) = x u 3 + du then F (x) equals: (b) If G(x) = x s 2 s 2 + ds then dg dx equals: pts: /0

13 Bonus. (a) Find all the fixed (equilibrium) points for the recursive sequence x t+ = x t e 2 ( x3 t ). (b) What [ does the Stability Criterion say about the fixed (equilibrium) points found in part ] (a)? It is given that f (x) = ( 3 2 x3 ) e 2 ( x3) is the derivative of f(x) = x e 2 ( x3). (c) Sketch a cobweb graph starting at x 0 = 0.5 and x 0 =.5, respectively, on each of the figures below. Use it to determine lim t x t in each case. 2 x t+ 2 x t+ x t+ = x t x t+ = x t x 0 = 0.5 x t x 0 =.5 pts: /0 x t

Fall /16/2013

Fall /16/2013 MA 37 - Calculus I for the Life Sciences FINAL EXAM Fall 3 /6/3 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.