1 Math 131 Week-in-Review #7 (Exam 2 Review: Sections 2.6-2.8, 3.1-3.4,...
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1 Math 131 WIR, copyright Angie Allen Math 131 Week-in-Review #7 (Exam 2 Review: Sections 2.6-2.8, 3.1-3.4, and 3.7-3.9) Note: This collection of questions is intended to be a brief overview of the exam material (with emphasis on sections 3.7-3.9). When studying, you should also rework your notes, the previous week-in-reviews for this material, as well as your suggested and online homework. 1. The number of reptile species found in a wetland area can be modeled by π R(m) = 17 + 8 cos m species 6 where m is the end of the mth month of the year. Find and interpret R(10) and R ′ (10). 2. Sketch the graph of the derivative of f (x) shown below. f(x) x 2 Math 131 WIR, copyright Angie Allen 3. The table below gives the average yearly health care costs (per consumer unit) for various years since 1990. Year Per capita expenditure ($) 1990 1480 1995 1732 1998 1910 2000 2066 2002 2350 a) Find the average rate of change in health care costs (per consumer unit) between 1998 and 2002 and interpret your answer. b) Estimate the rate of change in health care costs (per consumer unit) in 2002 and interpret your answer. c) Use a linear approximation to estimate health care costs (per consumer unit) in 2005. Is your answer an overestimate or an underestimate? Why? 3 Math 131 WIR, copyright Angie Allen 4. Use the graph below to answer the following questions. a) If the graph is f (x), where is f ′ (x) negative? b) If the graph is f ′ (x), where is f (x) increasing? −4 −3 −2 −1 1 2 c) If the graph is f ′ (x), where is f (x) concave up? d) If the graph is f ′′ (x), where is f (x) concave up? e) If the graph is f ′ (x), where does f (x) have local maxima/minima (if any)? f) If the graph is f ′ (x), where does f (x) have inflection points (if any)? g) If F(x) is an antidervative of f (x), and the graph of f (x) is shown above, where is F(x) concave up? h) If F(x) is an antidervative of f (x), and the graph of f (x) is shown above, where is f ′ (x) < 0? i) If F(x) is an antidervative of f (x), and the graph of F(x) is shown above, where is f ′ (x) < 0? 3 4 x 4 Math 131 WIR, copyright Angie Allen 5. Find the derivative of each of the following functions. Do not simplify your answers. 2 −π x a) f (x) = ln(4x b) g(θ ) = s 6 ) + 3e √ 5 3x4 (tan3 (π x + 2)) θ 2 − sec(2θ + π ) log23 (θ csc θ ) 6. Given k(p) = 5 dk and p(h) = 1 + 4e0.6h , find when h = 1. Use form 2 of the chain rule. p dh 5 Math 131 WIR, copyright Angie Allen 7. Sketch the graph of a function that satisfies all of the given conditions. f (−5) = 8 and f (2) = −2 f ′ (2) = f ′ (4) = 0 f ′ (x) > 0 on (−∞, −5), (2, 4) f ′ (x) < 0 on (−5, 0), (0, 2), (4, ∞) f ′′ (x) > 0 on (−∞, −2), (0, 3) f ′′ (x) < 0 on (−2, 0), (3, ∞) vertical asymptote x = 0 lim f (x) = 1 x→−∞ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 8. Find the equation of the line tangent to the curve y = sec x − 2 cos x at x = π /3. (Hint: sin(π /3) = x cot x = cos sin x .) √ 3/2, cos(π /3) = 1/2, tan(π /3) = √ 3, sec x = 1 cos x , csc x = 1 sin x , tan x = sin x cos x , and 6 Math 131 WIR, copyright Angie Allen 9. Find the derivative of the function f (x) = x using the definition of the derivative. x+2 √ 10. Find the linearization (i.e. linear approximation) of the function g(x) = 5 3 − x at a = −29 and use it to √ approximate 5 30.15. Is your approximation an overestimate or an underestimate? 11. Use calculus to determine where f (x) = ln x is decreasing. x Math 131 WIR, copyright Angie Allen 7 12. The position of a particle is given by s = t 3 − 10.5t 2 + 30t, t ≥ 0, where t is time in seconds and s is measured in meters. a) Find the velocity and acceleration after 4 seconds. b) When is the particle at rest? c) When is the particle moving forward (that is, in the positive direction)? d) Find the total distance traveled by the particle during the first 4 seconds. 8 Math 131 WIR, copyright Angie Allen e) Determine when the particle is speeding up and/or slowing down. Hint: Use the graphs of the position, velocity, and acceleration functions below. Be sure to explain your reasoning. 13. Fill in the blanks below. • Given a charge function Q(t) that gives the electric charge at time t, the is the rate of change of the with respect to and is found by calculating . • Given a mass function f (x) that gives the mass of a rod at length x, the rod’s is with respect to and is found by calculating the rate of change of the . • Given a volume function V (P) that gives the volume of a substance with pressure P, the substance’s is the rate of change of the with respect to , per unit volume, and is found by calculating . • Given a population function n(t) that gives the population at time t, the population’s is the rate of change of the and is found by calculating . P (R2 − r2 ), which gives the 4η l through a vessel or artery based on its distance from the is the rate of change of the with respect to . • Given the law of laminar flow v = with respect to of the blood , the blood’s and is found by calculating 9 Math 131 WIR, copyright Angie Allen 14. Use the information in the table below regarding the functions f (x) and g(x) to answer questions a) - c). x −6 0 5 8 64 f (x) 30 −6 19 58 4090 f ′ (x) −12 0 10 16 128 g(x) 24 0 35 80 4224 ′ g (x) −10 2 12 18 130 a) If h(x) = x2 − 3(g(x))4 , find h ′ (5). b) If j(x) = f (x)g(x2 ), find j ′ (8). c) If k(x) = f ( f (g(x))), find k ′ (0). Math 131 WIR, copyright Angie Allen 10 15. Write the formulas for the definition of the derivative at a number (both form 1 and form 2). 16. The mass of the part of a metal rod that lies between its left end and a point x meters to the right is 3x2 kg. a) Find the linear density when x is (a) 1 m, (b) 2 m, and (c) 3 m, and interpret each. b) Where is the density the highest? The lowest? Math 131 WIR, copyright Angie Allen 11 17. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference 3000 dynes/cm2 , and viscosity η = 0.027. a) Find the velocity along the centerline r = 0, at radius r = 0.005 cm, and at the wall r = R = 0.01 cm. b) Find the velocity gradient at r = 0, r = 0.005, and r = 0.01, and interpret each. c) Where is the velocity the greatest? Where is the velocity changing most? Math 131 WIR, copyright Angie Allen 18. Let y = 3x4 sin2 (π x). a) Find the differential dy. b) Evaluate dy and ∆y if x = 1/2 and dx = ∆x = 0.2. c) Sketch a diagram to illustrate the line segments with lengths dx, dy, and ∆y. 12
