May 9, 2018 MATH 255A Spring Final Exam Study Guide. Types of questions

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1 May 9, 18 MATH 55A Spring 18 Final Exam Study Guide Rules for the final exam: The test is closed books/notes. A formula sheet will be provided that includes the key formulas that were introduced in the course. No electronic devices except an approved model of graphing or scientific calculator. All cellphones must be off and put away completely for the duration of the exam. Show all your work. Types of questions 1. Exponential growth/decay. Doubling time and half-life. (Sections 1.4, 1., online homework). Continuity; piecewise functions. 3. Limits of functions and sequences. Difference equations. Cobwebbing. 4. Calculate a derivative; use derivative rules. Use derivatives of basic functions: power, exponential, logarithmic, trigonometric. Derivatives of implicit functions. Derivatives in the Fundamental Theorem of Calculus. Interpret the derivative (state units for dimensional quantities, discuss interpretation as rate of growth/decay). 5. Linear and quadratic approximations of functions. Tangent lines. Sensitivity and elasticity. When is the linear approximation an underestimate or an overestimate?. Graphing: find asymptotes, critical points, intervals of increase/decrease, inflection points and intervals of concavity up/down. Graph a function given a list of properties. 7. Optimization: local and global extrema; First and Second Derivative Tests. Closed and Open Interval Methods. Applications. 8. Compute an integral using one of the techniques: use properties of integral and interpretation as area; use Riemann sum to approximate; use the Fundamental Theorem of Calculus to evaluate. 9. Find indefinite integral of a function. Find antiderivative that satisfies certain condition. 1. Applications of integration: accumulated change, population growth, growing grapes (degreedays) examples. The following list provides examples of questions for each type. Some of these questions appeared on past final exams, others were taken from the textbook, homework problems, or midterm reviews. You can find more examples of questions in worked textbook examples, online homework, past midterms and practice problems from the course schedule.

2 Review Problems 1. If a bacterial population initially has twenty individuals and doubles every 9.3 hours, then how many individuals will it have after three days?. The size of the population of humpback whales is modeled by the function N(t) = 4(1.5) t, where t is measured in years since the beginning of 198. (a) How long does it take for the population to double in size? (b) Find N (1), state the units, and write a sentence to interpret its value. 3. Carbon-14 has a half-life of 5,73 years. How much is left of 15 g Carbon-14 after t years? 4. Find dy dx for each of the following: (a) y = x tan(3x x) (b) y =.1 x x + x + 1 (c) xy 3 = y 3x (d) y = x e t3 t + t + 1 dt. 5. (a) Find the limit of the sequence a n as n : a n = 4n n + 1 (b) Determine how large n needs to be to ensure that a n L <.1 where L is the value of the limit in part (a).. Consider the sequence defined by the following difference equation: a n+1 = 1 +.5a n, a 1 = 5. (a) Find the first five terms of the sequence. (b) Find all equilibria of the difference equation. Sketch a cobwebbing diagram. (c) Determine the limit of a n as n. Justify your answer! 7. Find the value that needs to be assigned to d, if any, to guarantee that f will be continuous for all x > : 3x + 1 f(x) = x, x 3 5x + d, x > 3.

3 8. (a) Find the linear approximation of the function y = 4 + x around x =. (b) Sketch a graph of the function and the linear approximation near x =. (c) If x =.3 use the linear approximation to estimate y. (b) Using second-order derivative, determine whether the linear approximation overestimates or underestimates the values of the function near x =. 9. A certain cell is modeled as a sphere. If the formulas S = 4πr and V = 4 3 πr3 are used to compute the surface area and volume of the sphere, respectively, estimate the effect on S and V produced by a 1% increase in the radius r. 1. A drug is given to a patient in order to lower the blood pressure level. The blood pressure in mm Hg is a function of elapsed time (in minutes) since the drug was administered, that is, B = f(t). Interpret the meaning of the statement f (3) = Given the function y = x 4 98x. Determine the domain, critical points, intervals where the function is increasing or decreasing, inflection points, intervals of concave up or down, intercepts where possible and asymptotes where applicable. Sketch a graph of the function. 1. (a) Sketch a graph of a function y = f(x) with all of the following properties: f(x) is continuous on (, ) y = 1, y = are horizontal asymptotes. f (x) < for x < 1 and x > 1; f (x) < for x < 1; f (x) > for x > 1; f(1) =.5. (b) Which of the following are correct (circle all that apply): A. The value x = 1 has to be a critical point for f(x). B. The value x = 1 could be a critical point for f(x). C. The point (1, f(1)) is an inflection point for f(x). D. The value f(1) is a local extremum. E. The value f (1) must be <. 13. Find the global maximum and the global minimum of f(x) = x ln x on [., ]. 3

4 14. The function C(t) = 1(e.5t e.5t ) [ mg/l ] is used to describe the concentration of a drug in the bloodstream t hours after the drug is administered. (a) Find C (t). Compute C () and C (4) and interpret. (b) Use C (t) to determine the intervals of increase and decrease of C(t). (c) Find the maximal concentration in [mg/l] and the time it occurs. 15. Compute definite integrals: (a) (b) x (x 3 + 1) dx. 5, e.t 1 + e.t dt. (c) (d) π/4 (e x e x ) dx. tan x dx. 1. If F (x) = 4 x find F so that F () =. (b) Find a constant C so that the largest value of F (x) + C is. 17. Using the Riemann sum for a uniform subdivision with n = 8 and left endpoints, approximate the area under the curve y = 3 + x over the interval [1, 5]. Geometrically, explain whether this approximation underestimates or overestimates the true area. 18. Use properties of definite integrals and appropriate formulas from geometry to calculate the integral exactly: 5 (7 x 5 5 x ) dx. 19. The population of Mariposa, CA is changing at the rate P (t) = 1 + 3t 1/ people per month. The current population is, people. Obtain a prediction for the size of the population sixteen months from now.. Sweet corn in Western Oregon has a lower developmental threshold of 5 F and requires approximately 1,597 degree-days to reach maturity. Suppose the temperature in the fields is given by where t is time in days. T (t) = sin(πt), 4

5 Answers: Given the formula for the number of growing degree-days (GDD) that accumulate from t = to t = x, GDD = x (T (t) T ) dt. (a) Calculate the number of degree-days that accumulate from t = to t = 1. (b) Estimate (rounding to the nearest integer) the time x required for the corn to reach maturity. 1. 4,81.. (a) About 14. years. (b) N (1) = [whales/year]. At the beginning of 199 years the whale population was growing at a rate of whales per year t/573 t ln()/573, or 15e 4. (a) tan(3x x) + (x x) sec (3x x); (b).1(1 x ) (1+x+x ) ; (c) y = 1 3 y3 +x 1 xy ; (d) 5. (a) 4; (b) n 4. e x 3. 1+x+x. (a) a 1 = 5, a = 3.5, a 3 =.75, a 4 =.375, a 5 =.1875; (b) a = is the only equilibrium. The cobwebbing diagram shows the lines y = x and y =.5x + 1; the sequence a n is monotonically decreasing towards the equilibrium a =. (c) lim n a n =. The Monotone Convergence Theorem applies on the interval I = [, 5] since f(x) =.5x + 1 is continuous, increasing, and transforms the interval I into itself. 7. d = (a) y = 1x + ; (c) y.75 (d) f (x) = 1 <, the function is concave down, so the graph lies below the tangent line. The linear approximation is 4 4(4+x) 3/ an overestimate. 9. S increases by %; V increases by 3%. [The elasticity of S to r is, and the elasticity of V to r is 3, no matter what the value r is.] 1. After 3 minutes the blood pressure decreases at a rate.1 mm Hg per minute. 11. Domain: (, ); asymptotes: none; intercepts: (, ), (, ±7); critical points: x =, x = ± 7, inflection points: x = ± 7 ; increases on ( 7, ) and ( 7, ) decreases on (, 7 ) and (, 7 ). Concave up on (, 7 ) and ( 7, ); concave down on ( 7, 7 ). The graph is a W shaped curve, similar to the one in Example, Section 4.1 5

6 1. (a) The graph could be a sigmoidal decay curve as discussed in Example 3 Section 3.. (b) Only B and C are correct. x = 1 could be a critical point if f(x) would fail to be differentiable at x = 1. For smooth sigmoidal decay curves x = 1 is not a critical point. 13. Global minimum 1 (1 + ln ).8457 at x = 1. Global maximum 4 ln 3.39 at x =. 14. (a) C (t) = 5e.5t + 5e.5t. C () = 3.3, C (4) =.43. After hours the concentration is increasing at a rate of 3.3 [mg/l/hr]; after 4 hours the concentration is decreasing at a rate of.43 [mg/l/hr]. (b) C(t) is increasing on (, 4 ln()), decreasing on (4 ln(), ). (c) The maximum concentration of 5 [mg/l] is achieved when t = 4 ln().77 hours. 15. (a) 7/9; (b) 5, ln 1+e; (c) 1 (e4 e 4 ) 4; (d) 1 ln. 1. (a) F (x) = 4x x. (b) C = This is an underestimate, since y = 3 + x is increasing on [1, 5] and the left end point of each interval I i produces the least value of the function on this interval (13 5π) (a) 18 degree-days; (b) 89 days.