Math 2413 General Review for Calculus Last Updated 02/23/2016

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1 Math 243 General Review for Calculus Last Updated 02/23/206 Find the average velocity of the function over the given interval.. y = 6x 3-5x 2-8, [-8, ] Find the slope of the curve for the given value of x. 2. y = x2 + x - 5, x = Use the graph to evaluate the limit. 3. lim x 0 f(x) 4. lim x 0 f(x)

2 5. Find lim x (-) - f(x) and lim f(x) x (-) + Use the table of values of f to estimate the limit. 6. Let f(x) = x - 4, find lim f(x). x - 2 x 4 x f(x) x - 7. Let f(x) =, find lim f(x). x2 + 4x - 5 x x f(x) Give an appropriate answer. 8. Let lim f(x) = 2 and lim g(x) =. Find lim x 8 x 8 x 8 9f(x) - 4g(x) 7 + g(x). Find the limit. 9. lim (x 3 + 5x 2-7x + ) x 2 0. lim x 0 + x - x Find the limit, if it exists.. lim x x4 - x - 2

3 2. lim x x 9 - x Provide an appropriate response. 3. The inequality - x 2 2 < sin x < holds when x is measured in radians and x <. x Find lim x 0 sin x x if it exists. For the function f whose graph is given, determine the limit. 4. Find lim f(x). x Find the limit. 5. lim x -2 x lim x 2 - (x - 2) 2 7. lim x -5-4 x lim x x2-5x + x3 + 2x x2 + 9x lim x - -3x2 + 6x + 4 4x 20. lim 3 + 4x 2 x - x - 5x 2 3

4 Divide numerator and denominator by the highest power of x in the denominator to find the limit. 3x 2. lim - - 5x -3 x 2x -2 + x lim t 6t 2-64 t - 4 Find all points where the function is discontinuous. 23. Provide an appropriate response. 24. Is f continuous at x = 0? f(x) = x 3, -3x, 3, 0, -2 < x 0 0 x < 2 2 < x 4 x = 2 Find the intervals on which the function is continuous. 25. y = x 2 + 0x + 35 x y = x 2-6x + 8 4

5 Provide an appropriate response. 27. Is f continuous on (-2, 4]? f(x) = x 3, -4x, 6, 0, -2 < x 0 0 x < 2 2 < x 4 x = Use the Intermediate Value Theorem to prove that -2x 3-7x 2 + 2x + = 0 has a solution between - and 0. Find numbers a and b, or k, so that f is continuous at every point. 29. f(x) = x2, x < - ax + b, - x 4 x + 2, x > 4 Solve the problem. 30. Select the correct statement for the definition of the limit: lim f(x) = L x x0 means that Use the graph to find a > 0 such that for all x, 0 < x - x0 < f(x) - L < y = 4x - f(x) = 4x - x0 = 2 L = 7 = NOT TO SCALE 5

6 32. f(x) = x0 = 3 L = 3 x y = x = NOT TO SCALE 33. f(x) = x - 3 x0 = 4 L = y = x - 3 = NOT TO SCALE 6

7 y = x2 f(x) = x2 x0 = 2 L = 4 = NOT TO SCALE A function f(x), a point x0, the limit of f(x) as x approaches x0, and a positive number is given. Find a number > 0 such that for all x, 0 < x - x0 < f(x) - L <. 35. f(x) = 4x 2, L =36, x0 = 3, and = 0. Prove the limit statement 36. lim x 5 x 2-25 x - 5 = lim x 7 3x 2-9x- 4 x - 7 = lim x 5 x = 5 Find an equation for the tangent to the curve at the given point. 39. y = x 2 + 2, (2, 6) 40. f(x) = 2 x - x + 3, (4, 3) Calculate the derivative of the function. Then find the value of the derivative as specified. 4. f(x) = 8 x + 2 ; f (0) 7

8 Solve the problem. 42. The graph of y = f(x) in the accompanying figure is made of line segments joined end to end. Graph the derivative of f. The figure shows the graph of a function. At the given value of x, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable? 43. x = - 8

9 44. x = Compare the right-hand and left-hand derivatives to determine whether or not the function is differentiable at the point whose coordinates are given. 45. y = x y = 2x Find the derivative. 46. y = 6x 2 + 7x + 2e x a. 2x + 2e x b. 2x e x c. 2x e x d. 6x + 2e x Find y. 47. y = (5x - 4)(2x3 - x2 + ) 48. y = (x2-5x + 2)(5x3 - x2 + 5) Find an equation of the tangent line at x = a. 49. y = x 2 - x; a = 2 9

10 50. y = x 3-25x + 5; a = 5 Find the second derivative. 5. y = 7x3-2x y = 8x3-6x2 + 6e x Find y. 53. y = (2x - 5)(2x3 - x2 + ) 54. y = x + x x - x Find the derivative of the function. 55. y = x 2 + 2x - 2 x2-2x y = (x + 4)(x + ) (x - 4)(x - ) Find the derivative. 57. s = 2e t 2et y = 5x 2 e -x Find the indicated derivative. 59. Find y if y = 3x sin x. The function s = f(t) gives the position of a body moving on a coordinate line, with s in meters and t in seconds. 60. s = 4t 2 + 2t + 9, 0 t 2 Find the body's displacement and average velocity for the given time interval. Solve the problem. 6. At time t, the position of a body moving along the s-axis is s = t 3-5t t m. Find the body's acceleration each time the velocity is zero. 62. Suppose that the dollar cost of producing x radios is c(x) = x - 0.2x 2. Find the marginal cost when 40 radios are produced. 0

11 Suppose that the functions f and g and their derivatives with respect to x have the following values at the given values of x. Find the derivative with respect to x of the given combination at the given value of x. x f(x) g(x) f (x) g (x) f(g(x)), x = 4 x f(x) g(x) f (x) g (x) f(x) + g(x), x = 3 Find the derivative of the function. 65. y = (x + )2(x2 + ) h(x) = cos x 6 + sin x Find y. 67. y = x y = sin(4x 2 e x ) Use implicit differentiation to find dy/dx. 69. x + y x - y = x 2 + y2 70. xy + x + y = x2y2 At the given point, find the slope of the curve or the line that is tangent to the curve, as requested. 7. x 5 y 5 = 32, tangent at (2, ) Use implicit differentiation to find dy/dx and d 2 y/dx y 2 - x 2 = xy + 3 = y, at the point (4, -) At the given point, find the line that is normal to the curve at the given point. 74. x 5 y 5 = 32, normal at (2, ) Find the derivative of y with respect to x, t, or, as appropriate. 75. y = ln(ln 2x)

12 76. y = ln - x (x + 3) 4 Find the derivative of y with respect to the independent variable. 77. y = 4 x Find the derivative of the function. 78. y = log 2 - x 79. f(x) = log5 (x 6 + ) Use logarithmic differentiation to find the derivative of y. x 80. y = x y = x(x + 3)(x + 4) Use logarithmic differentiation to find the derivative of y with respect to the independent variable. 82. y = (x + 2) x 83. y = (sin x) cos x Find the derivative of y with respect to x. 84. y = tan- 6x y = 2 sin - (4x 4 ) Find the value of df - /dx at x = f(a). 86. f(x) = x 3-3x 2-6, x 2, a = f(x) = x 2-4x + 7, x 2, a = 5 Solve the problem. 88. A wheel with radius 3 m rolls at rad/s. How fast is a point on the rim of the wheel rising when the point is /3 radians above the horizontal (and rising)? (Round your answer to one decimal place.) Solve the problem. Round your answer, if appropriate. 89. One airplane is approaching an airport from the north at 89 km/hr. A second airplane approaches from the east at 228 km/hr. Find the rate at which the distance between the planes changes when the southbound plane is 36 km away from the airport and the westbound plane is 5 km from the airport. 90. A man 6 ft tall walks at a rate of 7 ft/sec away from a lamppost that is 4 ft high. At what rate is the length of his shadow changing when he is 45 ft away from the lamppost? (Do not round your answer) 2

13 9. The volume of a rectangular box with a square base remains constant at 00 cm3 as the area of the base increases at a rate of 9 cm2/sec. Find the rate at which the height of the box is decreasing when each side of the base is 9 cm long. (Do not round your answer.) 92. The radius of a right circular cylinder is increasing at the rate of 6 in./sec, while the height is decreasing at the rate of 9 in./sec. At what rate is the volume of the cylinder changing when the radius is in. and the height is 7 in.? Determine all critical points for the function. 93. f(x) = x 2 + 2x f(x) = x 3-2x f(x) = -3x x - 4 Find the absolute extreme values of the function on the interval. 96. g(x) = -x 2 + x - 28, 4 x g(x) = 6-8x 2, -4 x 5 Find the absolute extreme values of the function on the interval. 98. f(x) = ln(x + 2) +, x 9 x 99. f(x) = e x - x, -2 x 2 Find the extreme values of the function and where they occur. 00. y = x3-2x y = x y = 8x x y = x 3-3x 2 + 5x - 6 x y = x 2 + 3x + 3 Solve the problem. 05. Sketch a continuous curve y = f(x) with the following properties: f(2) = 3; f (x) > 0 for x > 4; and f (x) < 0 for x < 4. 3

14 Find the largest open interval where the function is changing as requested. 06. Increasing f(x) = x Decreasing f(x) = x3-4x Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down

15 Graph the equation. Include the coordinates of any local extreme points and inflection points. 0. y = 0x2 + 20x. y = 8x x y = 2x3 + 3x2-2x 5

16 3. y = x 2 x2 + 2 Sketch the graph and show all local extrema and inflection points. 4. y = x 2-2x Solve the problem. 5. From a thin piece of cardboard 40 in. by 40 in., square corners are cut out so that the sides can be folded up to make a box. What dimensions will yield a box of maximum volume? What is the maximum volume? Round to the nearest tenth, if necessary. 6. A company is constructing an open-top, square-based, rectangular metal tank that will have a volume of 65 ft 3. What dimensions yield the minimum surface area? Round to the nearest tenth, if necessary. 7. A private shipping company will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 90 in. Suppose you want to mail a box with square sides so that its dimensions are h by h by w and it's girth is 2h + 2w. What dimensions will give the box its largest volume? 6

17 8. A window is in the form of a rectangle surmounted by a semicircle. The rectangle is of clear glass, whereas the semicircle is of tinted glass that transmits only one-fourth as much light per unit area as clear glass does. The total perimeter is fixed. Find the proportions of the window that will admit the most light. Neglect the thickness of the frame. 9. A long strip of sheet metal 2 inches wide is to be made into a small trough by turning up two sides at right angles to the base. If the trough is to have maximum capacity, how many inches should be turned up on each side? 20. The 9 ft wall shown here stands 30 feet from the building. Find the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall. 9' wall 30' 2. A rectangular field is to be enclosed on four sides with a fence. Fencing costs $5 per foot for two opposite sides, and $6 per foot for the other two sides. Find the dimensions of the field of area 850 ft2 that would be the cheapest to enclose. 7

18 22. Find the number of units that must be produced and sold in order to yield the maximum profit, given the following equations for revenue and cost: R(x) = 60x - 0.5x 2 C(x) = 9x Suppose a business can sell x gadgets for p = x dollars apiece, and it costs the business c(x) = x dollars to produce the x gadgets. Determine the production level and cost per gadget required to maximize profit. Find the linearization L(x) of f(x) at x = a. 24. f(x) = 4x 2-5x - 4, a = f(x) = 6x + 5, a = f(x) = x + x, a = 3 Solve the problem. 27. V = 4 3 r3, where r is the radius, in centimeters. By approximately how much does the volume of a sphere increase when the radius is increased from 2.0 cm to 2. cm? (Use 3.4 for.) 28. The diameter of a tree was 0 in. During the following year, the circumference increased 2 in. About how much did the tree's diameter increase? (Leave your answer in terms of.) Determine whether the function satisfies the hypotheses of the Mean Value Theorem for the given interval. 29. f(x) = x /3, -,3 30. g(x) = x 3/4, 0,3 Find the value or values of c that satisfy the equation the function and interval. 3. f(x) = x 2 + 3x + 2, [, 2] f(b) - f(a) b - a = f (c) in the conclusion of the Mean Value Theorem for 32. f(x) = x + 96, [6, 6] x a. 6, 6 b. 4 6 c. 0, 4 6 d. -4 6, f(x) = ln (x - 2), [3, 6] Round to the nearest thousandth. 8

19 Use l'hopital's Rule to evaluate the limit. 34. lim x /3 cos x - 2 x lim x 0 sin 2x tan 3x 36. lim x sin x x 37. lim x x 2 + 3x - x Find the limit. 38. lim x x 5 x 39. lim (ln x) 5/x x Use a calculator to compute the first 0 iterations of Newton's method when applied to the function with the given initial approximation. Make a table for the values. Round to six decimal places. 40. f(x) = x 3 + x - 9; x0 = Use Newton's method to approximate all the intersection points of the pair of curves. Some preliminary graphing or analysis may help in choosing good initial approximations. Round to six decimal places. 4. y = x 3-2x 2 and y = x 2 - Use Newton's method to find an approximate answer to the question. Round to six decimal places. 42. Where are the inflection points of f(x) = 5 4 x x3-0x located? A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the function. Use preliminary analysis and graphing to determine good initial approximations. Round approximations to six decimal places. 43. f(x) = x 2 - Find all the roots of the function. Use preliminary analysis and graphing to determine good initial approximations. Round to six decimal places. 44. f(x) = ln(2x) - 2x 2 + 3x + 9

20 Find the most general antiderivative. 45. (6x3-4x + 5) dx 46. x x + x x 2 dx x 2-5 x dx 7 x x dx Solve the initial value problem. d y = 5-5x, y (0) = 8, y(0) = 2 dx2 d y = 9; y (0) = -4, y (0) = 4, y(0) = 5 dx3 Provide an appropriate response. 5. The position of an object in free fall near the surface of the plane where the acceleration due to gravity has a constant magnitude of g (length-units)/sec 2 is given by the equation: s = - 2 gt2 + v0t + s0, where s is the height above the earth, v0 is the initial velocity, and s0 is the initial height. Give the initial value problem for this situation. Solve it to check its validity. Remember the positive direction is the upward direction. Estimate the value of the quantity. 52. The velocity of a projectile fired straight into the air is given every half second. Use right endpoints to estimate the distance the projectile travelled in four seconds. Time Velocity (sec) (m/sec) Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. 53. f(x) = x 2 between x = 0 and x = 2 using a right sum with two rectangles of equal width. 20

21 54. f(x) = between x = 2 and x = 7 using a right sum with two rectangles of equal width. x Graph the function f(x) over the given interval. Partition the interval into 4 subintervals of equal length. Then add to 4 your sketch the rectangles associated with the Riemann sum f(ck) xk, using the indicated point in the kth subinterval for ck. 55. f(x) = x 2-3, [0, 8], midpoint k= 56. f(x) = x 2-3, [0, 8], right-hand endpoint Solve the problem Suppose that f(x) dx = 3. Find f(x) dx and f(x) dx. Compute the definite integral as the limit of Riemann sums (3x - ) dx 2

22 Find the area of the shaded region Evaluate the integral. 6. /2 25 sin x dx /4 5 sec tan d - /4 Find the total area of the region between the curve and the x-axis. 63. y = 3 ; x 3 x3 Find the derivative. d x 64. dx 4t 9 dt x 65. y = 0 sin t d 66. dt 0 dt 2t u 2 du Find the average value of the function over the given interval. 67. y = x 2-6x + 6; [0, 6] 22

23 Evaluate the integral using the given substitution. dx 68. 4x + 8, u = 4x + 8 Evaluate the integral. x dx 69. (7x 2 + 3) x x 3 dx Use the substitution formula to evaluate the integral r dr r /2 0 cos x (5 + 4 sin x) 3 dx dx 9 - x 2 Find the area enclosed by the given curves. 74. Find the area of the region between the curve y = 2x/( + x 2 ) and the interval -4 x 4 of the x-axis. Evaluate the integral by using multiple substitutions sin 2 (x - 8) sin (x - 8) cos (x - 8) dx 23

24 Answer Key Testname: GENERAL CAL REVIEW slope is 3 3. does not exist ; x ; limit = 4.0 f(x) x ; limit = f(x) / Does not exist Does not exist x = 24. Yes 25. continuous everywhere 26. discontinuous only when x = 2 or x = No 28. Let f(x) = -2x 3-7x 2 + 2x + and let y0 = 0. f(-) = -6 and f(0) =. Since f is continuous on [-, 0] and since y0 = 0 is between f(-) and f(0), by the Intermediate Value Theorem, there exists a c in the interval (-, 0) with the property that f(c) = 0. Such a c is a solution to the equation -2x 3-7x 2 + 2x + = a = 3, b = if given any number > 0, there exists a number > 0, such that for all x, 0 < x - x0 < implies f(x) - L <

25 Answer Key Testname: GENERAL CAL REVIEW 36. Let > 0 be given. Choose =. Then 0 < x - 5 < implies that x 2-25 x = (x - 5)(x + 5) x = (x + 5) - 0 for x 5 = x - 5 < = Thus, 0 < x - 5 < implies that x2-25 x < 37. Let > 0 be given. Choose = /3. Then 0 < x - 7 < implies that 3x 2-9x- 4 x = (x - 7)(3x + 2) x = (3x + 2) - 23 for x 7 = 3x - 2 = 3(x - 7) = 3 x - 7 < 3 = Thus, 0 < x - 7 < implies that 3x2-9x- 4 x < 38. Let > 0 be given. Choose = min{5/2, 25 /2}. Then 0 < x - 5 < implies that x - 5 = 5 - x 5x = x 5 x - 5 < 5/ = Thus, 0 < x - 5 < implies that x - 5 < 39. y = 4x y = - 2 x f (x) = - ; f (0) = -2 (x + 2) Continuous but not differentiable 25

26 Answer Key Testname: GENERAL CAL REVIEW 44. Neither continuous nor differentiable 45. Since lim x 0 + f (x) = 2 while lim x 0 - f (x) =, f(x) is not differentiable at x = b x3-39x2 + 8x x4-04x3 + 45x2 + 6x y = 3x y = 50x x x e x 53. 6x3-36x2 + 0x x + 2 x 3-4x2 + 8x 55. y = (x2-2x + 2)2-0x 56. y = (x - 4) 2 (x - ) 2 2et 57. (2et + ) xe -x (2 - x) 59. y = 6 cos x - 3x sin x m, 0 m/sec 6. a(2) = -8 m/sec 2, a(8) = 8 m/sec $ (x + )(2x 2 + 3x - ) (x2 + ) cos 5 x ( + sin x) x x x x (x 2 + 4x + 2)e x cos(4x 2 e x ) - 6xe 2x (x 3 + 4x 2 + 4x) sin(4x 2 e x ) 69. x(x - y) 2 + y x - y(x - y)2 2xy2 - y x2y + x + 7. y = - 2 x dy dx = x y ; d2 y dx 2 = y2 - x 2 y 3 26

27 Answer Key Testname: GENERAL CAL REVIEW 73. dy dx = 3 ; d2 y dx 2 = y = 2x x ln 2x 76. 3x - 7 (x + 3)( - x) x ln ln 0 (2 - x) 6x (ln 5) (x 6 + ) x x - 4 x - x x(x + 3)(x + 4) x + x x (x + 2) x ln(x + 2) + x x (sin x) cos x (cos x cot x - sin x ln (sin x)) x x 3-6x m/s km/hr ft/sec cm/sec in.3/sec 93. x = x = -2 and x = x = absolute maximum is 9 4 at x = ; absolute minimum is 0 at 7 and 0 at x = absolute maximum is 6 at x = 0; absolute minimum is -94 at x = absolute minimum value is ln at x = 2; absolute maximum value is ln + 9 at x = absolute minimum value is at x = 0; absolute maximum value is e 2-2 at x = 2 27

28 Answer Key Testname: GENERAL CAL REVIEW 00. Local maximum at (-2, 8), local minimum at (2, -4). 0. Absolute maximum value is at x = Absolute minimum value is - 4 at x = -. Absolute maximum value is 4at x =. 03. None 04. Absolute maximum is 3 at x = 0; absolute minimum is - at x = Answers will vary. A general shape is indicated below: 06. (-, 0) , Local minimum at x = ; local maximum at x = -; concave up on (0, ); concave down on (-, 0) 09. Local minimum at x = 3 ; local maximum at x = -3 ; concave up on (0, ); concave down on (-, 0) 0. local minimum: (-,-0) no inflection points 28

29 Answer Key Testname: GENERAL CAL REVIEW. local minimum: (-2, -2) local maximum: (2, 2) inflection points: (0, 0), (-2 3, -2 3), (2 3, 2 3) 2. local minimum: (, -7) local maximum: (-2, 20) inflection point: - 2, local minimum: (0, 0) inflection points: - 6 3, 4, 6 3, 4 29

30 Answer Key Testname: GENERAL CAL REVIEW 4. Local minima: (0, 0), (2, 0) Local maximum:, No inflection points in in. 6.7 in.; in ft 5. ft 2.5 ft in. 20 in. 5 in. width 8. height = in ft $5 by 26.6 $ units 23.,250 gadgets at $37.50 each 24. L(x) = 35x L(x) = x L(x) = 8 9 x cm in. 29. No 30. Yes b

31 Answer Key Testname: GENERAL CAL REVIEW k xk x , , x , x , x , x4-2x 2 + 5x + C x - 2 x + C sin - x - 5 ln x + C tan - x - 6 ln x + C 49. y = 5 2 x2-5 6 x3 + 8x y = 3 2 x3-2x 2 + 4x + 5 d 5. 2 s dt 2 = -g, s (0) = v 0, s(0) = s m

32 Answer Key Testname: GENERAL CAL REVIEW ; x x + 5 cos t 6 - sin 2 t 4x C 32

33 Answer Key Testname: GENERAL CAL REVIEW (7x 2 + 3)-4 + C x3 5/4 + C sin ln (8 + sin2 (x - 8)) 3/2 + C 33