Calculus for the Life Sciences. Midterm 1

MATH 255A Midterm 1: Study Guide Textbook coverage: 1.1: Real Numbers and Functions. Interval notation, domains, ranges, intervals of increase and decrease. Examples 2, 3, 7. 1.2: Data Fitting with Linear and Periodic Functions; Examples 2-6. 1.3: Power functions and scaling laws. Equations of lines, linear regression, trigonometric functions, transformations. Proportionality notation. Geometric similarity. Examples 1-5. 1.4: Exponential functions. Exponential growth and decay. Doubling time and half-life. Compound interest. Solving exponential equations with logarithms. Examples 1, 2, 5, 6. 1.5: Function building. Transformations of graphs, compositions. Examples 1, 2, 4, 5, 7, 8. 1.6: Inverse functions and logarithms; Examples 3, 4, 6-12. Review Questions 1. (see also 1.1: 13-18) Find the domains of the functions and compute the indicated values (or state that the corresponding values are not in the domain) (a) f (x) = (1 − 3x)−1/3 , f (0), f ( 13 ), f (3). ( 3x + 5, x < 2 (b) f (x) = , f (1), f (2), f (3). 3x2 − 1, x > 2 Sketch the graphs and determine the ranges of the functions. 2. (see also 1.1: 44-47) Friend’s rule is a method for calculating pediatric drug dosages in terms of child’s age (up to 12 21 years). If A is the adult dose (in mg) and n is the age of the child (in years) then the child’s dose is given by D(n) = 2 nA. 25 (a) What is the domain of the function defined by (n, D)? (b) Graph this function for A = 600 mg. (c) If a 3-year-old child receives 150 mg of a certain drug, what is the corresponding dose for a 7 year old? What is the adult dose? 3. (see also 1.2: 1-6, 11-18) 1 (a) Find a linear function (y = mx + b) that passes through the points (−1, 4) and (1, 3). (b) Find an exponential function (y = abx ) that passes through the same points. (c) Find an equation for a straight line through the points (−1, 4) and (−1, 3). Does this equation represent a function y = f (x)? Does it represent a function x = g(y)? 4. (see also 1.2: 47) The following Table presents the data on the average carbon dioxide (CO2 ) level in the atmosphere, measured in parts per million (ppm), at Mauna Loa Observatory from 2004 to 2012. Year CO2 level (in ppm) 2004 377.5 2006 381.9 2008 385.6 2010 389.9 2012 393.8 (a) Use the linear regression method to find the best-fitting data line. Plot the line and the points. (b) Use the linear equation obtained in part (a) to obtain a prediction for the CO2 level at Mauna Loa Observatory in 2016. 5. (see also 1.3: 11-16, 34, 35) If x ∝ y 6 and y ∝ z 1/3 then how does z change as x decreases from 16 to 4? 6. (see also 1.3: 25-27) Consider a cylinder of radius r and height 5r. Express the volume V and the surface area S of this cylinder as a function of r. If r is doubled, what happens to V ? If V is decreased by a factor of 64, what happens to S? 7. (see also 1.3: 36-38) A sample based on nineteen mountain ash trees of different sizes yielded a relationship between the leaf area, A [m2 ], of the tree, and the stem diameter at breast height (DBH), d, [cm]. The relationship obtained was A ∝ d2.89 . If one of the points that this relationship passed through was (d, A) = (30, 78), find the equation and sketch the graph. 8. (see also 1.4: 31, 32) Assume $1, 000 is invested for t = 10 years at a rate r = 0.08 compounded (a) annually (b) daily (c) continuously. Calculate the future value. (Round your answers to two significant digits.) 2 9. (see also 1.4: 35, 36) The height of beer froth is modeled by the function H(t) = 17(0.94)t/15 where t is measured in seconds. Estimate (accurate to 1 second) at what time the froth is (a) one half of its original height (b) one tenth of its original height. 10. (see also 1.4: 33, 39-42, 1.5: 37, 38) If a bacterial population initially has 20 individuals and doubles every 9.3 hours then how many individuals will it have after three days? 11. (see also 1.5: 34, 35; 1.2: 31-36, 47, 48) The tides at La Selva Beach, California, on Saturday September 17, 2016 are given by the following table (actual data have been approximated for simplicity of calculation): Time Height (ft) Tide 11:30am 5.41 High 5:30pm 0.69 Low 11:30pm 5.41 High Let T (t) = A cos(B(t − C)) + D denote the height of the tide t hours after 12:00 noon. (a) Find values of A, B, C, and D such that the function fits the La Selva Beach data. (b) Sketch the function over the interval [−12, 12]. 12. (see also 1.6: 15-30; 1.3: 1-10) Write the expressions in terms of base e and simplify where possible: (a) 52x (b) 3x (c) log2 (x2 − 2) 2 (d) log10 (ex + e) 2 2 Note: 3x is usually interpreted as 3(x ) , while 3x
2 is the same as 32x . 13. Which one is greater: log5 127 or log10 999? Answer without using calculator! 14. Solve the equations: (a) log5 x + log5 (x − 2) = 1. (b) ex 2 −4x+5 = e. (c) 3e−3x = e−x . (d) e−x = x 2 + 1. [Solve graphically, do not use calculator!] 3 (e) e−x = x2 . [Use equation solving function on a graphing calculator, show work!] 15. (see also 1.6: 9-14) Find the inverses of the functions. State the domain and the range of the inverse. (c) y = exp(−x2 ) on [0, ∞). (a) y = x/(x + 1); (b) y = 0.1 e3x−5 ; Note: exp(x) is another notation for ex . 16. (see also 1.6: 31-34) (a) Sketch the indicated points on the logarithmic scale (use base 10): 0.01, 0.2, 10, 25,000. (b) Sketch a log-log graph of the function y = x10 1/2 over the interval [0.01, 100]. On the vertical axis, mark the values that correspond to x = 10−2 , 100 , 102 . 17. (1.6: 44, 46) From one individual to another within the same species of mammal it is found that the brain volume V varies with the body weight W according to the power law V = aW b . The following data are obtained from measurements of a number of adult chimpanzees. Show, using a log-log plot that the power law fits these data quite well, and obtain the values of a and b: W , kg 31 36 38 42 47 48 53 V , cm3 365 380 382 397 410 415 427 Answers: 1. (a) Domain: (−∞, 13 ) ∪ ( 31 , ∞). Values: 1, undefined, − 21 ; (b) Domain (−∞, 2) ∪ (2, ∞). Values: 8, undefined, 26. 2. (a) Domain: [0, 12.5]. Note: algebraically, the domain is R (i.e. (−∞, ∞)). However, it’s meaningless to apply the formula for n outside the interval [0, 12.5], therefore the answer above is correct. (b) The graph is a straight line that raises from 0 to 600 [mg] in the interval from 0 to 12.5 years. (c) 350 mg; 625 mg. 3. (a) y = −0.5x + 3.5. 4 (b) y = √ 12
3 t/2 4 a= √ 12, b = q
3 4 . (c) x = −1. Not a function y = f (x) (the vertical line test fails). Works OK as a function x = g(y) (g(y) is a constant −1). 4. (a) y = 2.03x − 3690.5. (b) 401.98. 5. x ∝ z 2 , x is decreased by a factor of 4, therefore z is decreased by a factor of 2. 6. If r is doubled, V increases by a factor of 8. If V is decreased by a factor of 64 then S is decreased by a factor of 16. 7. A = 0.0042 d2.89 ; the graph has appearance similar to the cubic parabola y = x3 , passes through (30, 78). 8. $2,158.92, $2,225.35, $2,225.54. 9. 168 seconds; 558 seconds. 10. 4,281. 11. A = 2.36; B = π6 , C = −0.5, D = 3.05. (Hint for the graph: plot the points of the data and match with a graph of cosine with a horizontal shift, vertical stretching and a vertical shift.) 12. (a) e2x ln 5 ; (b) ex 2 ln 3 ; (c) ln(x2 − 2)/ ln(2); (d) (1 + ln(x + 1))/ ln(10). 13. log5 127 > log10 999. √ 14. (a) 1 + 6; (b) 2; (c) log(3)/2; (d) 0; (e) x ≈ 0.852606. x 15. (a) y = 1−x (note that the inverse is expressed in the form y = f (x)); (b) y = √ 1 (5 + ln(10x)); (c) y = − ln x on (0,1]. 3 16. (b) The graph is a straight line Y = − 12 X + 1 that passes through the points (−2, 2), (0, 1), (2, 0). The values marked on the y axis are 100, 10, 1. 17. a = 130.62, b = 0.298. 5