MA 37 - Calculus I for the Life Sciences FIRST MIDTERM Fall 3 9/4/3 Name: 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.. 3. 4. 5. 6. 7. 8. 9.. Bonus. TOTAL
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 -4 Kate Ponto MWF : am - :5 am, CP 3 5-8 Alberto Corso MWF : pm - :5 pm, CB Section # Recitation Instructor Time/Location Dustin Hedmark TR 9:3 am - : am, CB 347 Michael Gustin TR 9:3 am - : am, RRH 3 3 Dustin Hedmark TR : am - :5 am, CB 347 4 Michael Gustin TR :3 pm - : pm, CB 347 5 Liam Solus TR :3 pm - : pm, FB 3 6 Liam Solus TR : pm - :5 pm, FB 3 7 Joseph Lindgren TR : pm - :5 pm, CB 45 8 Joseph Lindgren TR 3:3 pm - 4: pm, FB 3
. (a) The graph of a function f is below. What is the domain of f? What is the range of f? 8 7 6 5 4 3 3 4 5 6 7 8 9 (b) Isf invertible? Why? If so, draw the graph of the inverse off. 8 7 6 5 4 3 3 4 5 6 7 8 9 pts: /
- - -3 - - - - -3 - - - - -3 - - - - -3 - -. If the graph off(x) is y x match the functions f(x), f(x + ) and f(x) with the graphs below y y y x x x pts: /
3. (a) If log 3 (log x) = what isx? (b) 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 percent of the original dose of iodine remains in her body, how long will she have to wait after the iodine is administered? It is known that this form of radioactive iodine has a half-life of 3. hours. pts: /
4. There are many possible functional relationships between height and diameter of a tree. One particularly simple model is given by H = AD 3/4 where A is a constant that depends on the species of tree, H is the height, and D is the diameter. IfA = 5 plot this relationship in the double log plot below. 4 3 3 Is your graph a straight line? If so, what is its slope? pts: /
5. 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 5 5 Population 54 4,3 3,56 58,48. Plot this data (as accurately as possible) in the semi log plot below. 6 5 4 3 5 5 5. Find a functional relationship between population and time. pts: /
- - -3 - - - - -3 - - 6. The graphs of f and g are given below. y y x x graph of f graph of g Use the properties of limits and the given graphs to evaluate each limit, if it exists. If the limit does not exist, explain why. (a) lim [f(x)+g(x)] x (b) lim [f(x)+g(x)] x (c) lim f(x)g(x) x (d) f(x) lim x g(x) (e) lim x 3 f(x) x pts: /
7. Evaluate the following limits (a) lim x [ln(3x ) ln(x +)] (b) x 4 lim x (x ) (c) lim x 5x sin(x) pts: /
8. A very large tank contains 5, L of pure water. Brine that contains 3 g of salt per liter of water is pumped into the tank at a rate of 5 L/min. It can be shown that the concentration of salt aftertminutes (in grams per liter) is C(t) = 3t +t. (a) How long does it take for the concentration to reach the level of5 grams per liter? (b) What happens to the concentration ast? Explain why this makes sense. pts: /
9. (a) For what value of the constant c is the function f defined below { cx f(x) = +x if x < x 3 cx if x continuous on (, )? (b) Suppose f is continuous on the interval [,5] and the only solutions of the equation f(x) = 6 are x = andx = 4. Iff() = 8, explain why we must have that f(3) > 6. (Hint: use the Intermediate Value Theorem. It also helps to draw a picture.) pts: /
. Genes produce molecules called mrna that go on to produce proteins. High concentrations of protein inhibit the production of mrna, leading to stable gene regulation. This process has been modeled to show that the concentration of mrna over time is given by the equation m(t) = e t (sint cost)+ (a) Evaluate lim m(t). t (b) Use the Squeeze (Sandwich) Theorem to evaluate lim m(t). t (Hint: notice that e t m(t) +e t for every real numbert.) pts: /
Bonus. (a) The number of new infections produced by an individual infected with a pathogenlike influenza depends on the mortality rate that the pathogen causes. This pathogeninduced mortality rate is referred to as pathogen s virulence. Extremely high levels of virulence result in very little transmission because the infected individual dies before infecting other individuals. Under certain assumptions, the number of new infections N is related to virulence v by the function N(v) = 8v +v +v, where v is the mortality rate (that is, virulence) andv. Evaluate and interpret your result. lim v 8v +v +v (b) In Einstein s theory of relativity, the massmof a particle with velocity v is m = m v /c, wherem is the mass of the particle at rest andcis the speed of light. What happens to the mass m of the particle as the speed v tends to the speed of light c? That is, evaluate: m lim v c v /c. pts: /