Math 131 Week-in-Review #11 (Final Exam Review: All previous sections as well as sections 5.5, 6.1, 6.5, and 6.7)

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1 Math 131 Week-in-Review #11 (Final Exam Review: All previous sections as well as sections 5.5, 6.1, 6.5, and 6.7) Note: This collection of questions is intended to be a brief overview of the exam material with emphasis on sections 5.5, 6.1, 6.5, and 6.7. When studying, you should also look at your notes, the suggested homework problems from the textbook, as well as the other week-in-reviews for this material. 1. Compute the following indefinite integrals. 16xln(4x 2 + 3) a) 4x 2 dx + 3 b) sin(πx)(cos(πx) + e) 23 dx c) 3sin(x) 5cos(x) dx 1

2 2. Find the average value of the function f (x) = (x 4) 2 on the interval [0,3], and then find c such that f ave = f (c). 3. Find the number(s) b such that the average value of f (x) = 2 + 6x 3x 2 on the interval [0,b] is equal to 3. (Source: #12, pg. 463, Stewart) 2

3 4. Find the area between f (x) = x and the x-axis on the interval [ 4,4]. 5. Evaluate A 2 x 4 x 5 dx, where A is a constant and A >

4 6. A honeybee population starts with 200 honeybees and increases at a rate of n(t) = 50e t bees per week, where t is in weeks and t 0. a) Find the average rate of change of the number of honeybees during the first five weeks. Round to the nearest integer if necessary. b) Find the average number of honeybees during the first five weeks. Round to the nearest integer if necessary. 4

5 7. Find each of the following indefinite integrals: a) e 3x (e 3x 5) 7 dx b) 7x(x + 4) 5 dx c) 2x 6 x dx 5

6 8. The following table shows the number of heartbeats after t minutes of a patient after a surgery. t (min) Heartbeats a) Find the average heart rate between 38 and 42 minutes. Interpret your answer. b) Estimate the patient s heart rate (i.e. instantaneous heart rate) after 42 minutes. Interpret your answer. c) Find an appropriate model, h(t), for the data. (Round to four decimal places, if necessary). d) Use your rounded model to find the patient s heart rate after 42 minutes. How does this compare with your estimate above? e) Use your rounded model to find the average heart rate between 38 and 42 minutes. How does this compare with your answer above? 6

7 x 2 + 4x Find lim. x 7 + x Consider the graph of a function shown below x a) If the graph above is f (x), determine where any local and/or absolute extrema of f (x) occur. b) If the graph above is f (x), determine where f (x) is negative. c) If the graph above is f (x), determine where f (x) is decreasing. d) If the graph above is f (x), determine where f (x) is decreasing. 7

8 8 x 11. Find the domain of the function f (x) = 2 5x + 6 ( log 3 a 2 x 9). 12. Consider the function g(x) = x2 + 1 x 3. a) Find an expression for the slope of the secant line through (x,g(x)) and (x + h,g(x + h)). b) Find an expression for the slope of the tangent line at the point (x,g(x)). 8

9 13. Find the inverse of the function f (x) = 2x 3, if it exists. x Where is the function f (x) continuous? f (x) = sin(x + π) 3 x < 0 13 x 5 x 3 3e x 0 x 9 ln(x 7) x 2 7x 30 x > 9 ( ) Find the derivative of f (x) = log 3 (x 3 πx)csc(5x 2 ). 9

10 16. Compute the following integrals: a) A 2 8 dx, where A > 2. xlnx b) (2 +t) t 3 dt c) 4 3x 6 + lnx dx 10

11 17. Sketch the graph of a function that satisfies all of the following conditions: f ( 3) = f ( 5) = 0 f (x) < 0 if 5 < x < 3 f (x) > 0 if x < 5, 3 < x < 1, or x > 1 f (x) < 0 if x < 4 or x > 1 f (x) > 0 if 4 < x < 1 lim f (x) = x 1 + lim f (x) = x 1 lim f (x) = 2 x 18. Starting with the function g(x) = x, write the function f (x) which results from shifting g(x) to the left 2 x 3 units, then compressing horizontally by a factor of 2, then stretching vertically by a factor of 5, then reflecting about the y axis, and then shifting down by 18 units. 19. Find lim x 1 sinx x 2, if it exists. If it does not exist, use limits to describe the way in which it does not exist. + x 11

12 20. On Io, one of Jupiter s many moons, a volcanic eruption pushes a rock downward off a 175-meter cliff with an initial speed of 10 meters per second. If the acceleration due to gravity is m/s 2, find an equation s(t) that gives the rock s position above the ground t seconds after it was pushed off the cliff. 21. Mark each of the following curves as f (x), f (x), or f (x). 22. If g(3) = 2, g (3) = 8, g ( 2) = 1, f (3) = 2, f (3) = 4, f ( 2) = 7, and h(x) = 12 f (g(x))+ f (x) g(x), find h (3). 12

13 23. Find lim x 7x + π 16x 2 3x + e. 24. When finding (x + 4)e 3x2 +24x dx, write the integral that would be obtained after the appropriate u-substitution is made. 13

14 25. A fence is built to enclose a rectangular area of 800 square feet. The fence along 3 sides costs $6 per foot, and the fourth side costs $18 per foot. Find the dimensions of such a rectangle that will minimize the cost. 26. Find b 1 ( 5 x 4 (ln19)x 2 x 8 ) + 3 x πe x dx. 14

15 27. Consider the function f (x) = ex x. a) Use calculus to find any critical numbers of f (x), as well as the intervals where the function is increasing/decreasing, the values of any local extrema, the intervals where the function is concave upward/downward, and any inflection points. b) Find the absolute minimum of f (x) on the interval [0.5,3], if it exists. c) Find the absolute minimum of f (x) on the interval [ 1,3], if it exists. 15

16 28. The growth rate of a population of bacterial cells in a petri dish is given by f (t) = 0.2e 0.5t +10 cells per day, where t is the number of days since the start of the experiment, 0 t 20. After 3 days, there were 300 bacteria cells. a) Find the increase in the number of cells between days 8 and 15. b) How many cells were in the dish after 7 days? c) What is the average number of bacteria cells in the population from day 10 to day 15? 2e θ 29. Find g (θ) if g(θ) = cscθ u 2 u du. 16

17 30. Find the area bounded by the curves y = 0.6 x + 3 and y = x 2 3x + 1 on the interval [ 3,2]. 31. Find the derivative of g(a) = e seca a 2 +5a ln (2 sin(3a)). 32. Polonium-210 has a half-life of 140 days. If a sample has a mass of 180 mg, how long will it take for the mass to decay to 8 mg? 17

18 33. Find each of the following integrals: a) sin4x dx (1 + cos4x) 2 3 b) π (3 e t )e t dt c) 1 8x(lnx) 7 dx d) 3q 2 4 (3q 3 12q + 18) 2 dq 18

19 34. Solve the following equation for x: log b 2x = 3 2 log b 9 + log b 6 log b Find the equation of the tangent line to the curve y = secx + cos2x at x = π/3. 3 2, cos(π/3) = 1 2, and tan(π/3) = 3, and secx = 1 cosx, cscx = 1 sinx Hint: sin(π/3) = Also, sin(2π/3) = 3/2 and cos(2π/3) = 1/2., and cotx = 1 tanx. 19

20 t 36. Consider the function g(t) = f is [ 4,9]. 4 f (x) dx, where the graph of f (x) is shown below. Note that the domain of a) Find g(7). b) Where does g have an absolute max on [ 4,7]? c) Where is g increasing? 20

21 37. When using a Riemann sum with left endpoints to approximate the area under the curve f (x) = x 2 5x 3 on the interval from [6,10] using 16 rectangles, what is the height of the 4th rectangle? x 2 2x Use calculus to find the horizontal and vertical asymptotes of the function f (x) = x 4 x 3 x 2 5x 30, if any exist. If there are vertical asymptotes, describe the behavior near each asymptote. Hint: x 4 x 3 x 2 5x 30 = (x 3)(x 2 + 5)(x + 2). 21

22 39. Calculate the rate of flow in a small human artery where we take η = 0.03, R = cm, l = 3 cm, and P = 4000 dynes/cm 2. (Give your answer to three significant figures.) 40. Find the area bounded by the curves y 2 = x 2 and y2 = x The dye dilution method is used to measure cardiac output with 9 mg of dye. The dye concentrations, in mg/l, are modeled by f (t) = 1 5t(16 t), 0 t 16, where t is measured in seconds. Find the cardiac output. 22

23 42. The position of a particle is given by s = t t 2 +30t, t 0, where t is time in seconds and s is measured in meters. Determine when the particle is moving backward and speeding up. 23