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Math1325: Test 3 Review Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the derivative. 1) y = 5x2 e3x 1) A) 10ex3x(3x + 2) 2) y = B) 5xe 3x(2x + 3) C) 10xe3x(2x + 3) D) 5xe 3x(3x + 2) e-x + 1 ex A) 2) -ex + 2 e2x B) ex - 2 e2x C) -ex - 2 e2x D) ex + 2 e2x Find the derivative of the function. 3) y = ln (4x3 - x2 ) 12x - 2 A) 4x2 3) 12x - 2 B) 4x2 - x 4x - 2 C) 4x2 - x 12x - 2 D) 4x3 - x ex B) x ex(x ln x + 1) C) x ex(ln x + x) D) x Find the derivative. 4) f(x) = ex ln x, x > 0 4) A) ex ln x 5) f(x) = ln x ex A) ln x xex 5) B) 1 - x ln x ex C) 1 xex D) 1 - x ln x xex Find all points where the function is discontinuous. 6) A) None 6) B) x = 1 C) x = -2 D) x = -2, x = 1 Find the largest open interval where the function is changing as requested. 7) Increasing y = (x2 - 9)2 A) (-3, 0) 8) Decreasing f(x) = A) (-∞, 3) 7) B) (-∞, 0) C) (3, ∞) D) (-3, 3) B) (-3, ∞) C) (-∞, -3) D) (3, ∞) x+3 8) 1 Find the indicated absolute extremum as well as all values of x where it occurs on the specified domain. 9) f(x) = x3 - 3x2 ; [0, 4] Minimum A) 16 at x = 4 C) No absolute minimum 9) B) -4 at x = 2 D) 0 at x = 0 Determine the location of each local extremum of the function. 10) f(x) = x3 - 6x2 + 12x - 1 10) A) Local maximum at 2; local minimum at -2 B) Local maximum at 2 C) Local minimum at 2 D) No local extrema Use the first derivative test to determine the location of each local extremum and the value of the function at that extremum. x2 11) f(x) = 11) x2+ 5 A) No local extrema C) Local minimum at (5, 0.83333333) B) Local minimum at (0, 0) D) Local maximum at (0, 0) Evaluate f''(c) at the point. 12) f(x) = (x2 - 4)(x3 - 3), c = 1 A) f''(1) = 36 B) f''(1) = -4 13) f(x) = ln (3x2 - 2), c = -1 A) f''(-1) = -30 12) C) f''(1) = -10 D) f''(1) = 4 C) f''(-1) = -6 D) f''(-1) = -1 13) B) f''(-1) = 30 Find the coordinates of the points of inflection for the function. 14) f(x) = x2 + 8x + 18 A) (-5, 1) C) (-3, 3) 14) B) (-4, 2) D) There are no points of inflection. Find the largest open intervals where the function is concave upward. 15) f(x) = 4x3 - 45x2 + 150x 15 A) ,∞ 4 15 B) -∞, 4 15 C) ,∞ 4 15) 15 D) -∞, 4 Find the indicated absolute extremum as well as all values of x where it occurs on the specified domain. 16) f(x) = (x + 1)2 (x - 2); [-2, 1] Maximum A) -4 at x = -2 C) 0 at x = -1 B) -2 at x = 0 D) No absolute maximum 2 16) Find all critical numbers for the function. State whether it leads to a local maximum, a local minimum, or neither. 17) f(x) = -x3 - 3x2 + 24x - 4 17) A) Local maximum at 4; local minimum at -2 B) Local maximum at -4; local minimum at 2 C) Local maximum at -2; local minimum at 4 D) Local maximum at 2; local minimum at -4 Find the absolute extremum within the specified domain. 1 18) Minimum of f(x) = x3 - 2x2 + 3x - 4; [-2, 5] 3 A) 2, - 62 3 B) 2, - 18) 61 3 C) -2, - 61 3 D) -2, - 62 3 Solve the problem. 19) Find the dimensions that produce the maximum floor area for a one-story house that is rectangular in shape and has a perimeter of 163 ft. A) 13.58 ft x 40.75 ft B) 40.75 ft x 163 ft C) 40.75 ft x 40.75 ft D) 81.5 ft x 81.5 ft 19) 20) An architect needs to design a rectangular room with an area of 72 ft2 . What dimensions should he use in order to minimize the perimeter? A) 8.49 ft × 8.49 ft B) 14.4 ft × 72 ft C) 8.49 ft × 18 ft D) 18 ft × 18 ft 20) 21) A piece of molding 152 cm long is to be cut to form a rectangular picture frame. What dimensions will enclose the largest area? A) 38 cm × 38 cm B) 12.33 cm × 38 cm C) 30.4 cm × 30.4 cm D) 12.33 cm × 12.33 cm 21) 22) Of all numbers whose difference is 12, find the two that have the minimum product. A) 0 and 12 B) 1 and 13 C) 6 and -6 D) 24 and 12 22) 23) Find two numbers whose sum is 490 and whose product is as large as possible. A) 10 and 480 B) 1 and 489 C) 244 and 246 D) 245 and 245 23) 24) Find two numbers x and y such that their sum is 120 and x2y is maximized. A) x = 90, y = 30 B) x = 40, y = 80 C) x = 80, y = 40 24) D) x = 30, y = 90 Find the open interval(s) where the function is changing as requested. 25) Increasing; f(x) = x2 - 2x + 1 A) (-∞, 1) B) (1, ∞) C) (-∞, 0) 25) D) (0, ∞) Find the x-value of all points where the function has relative extrema. Find the value(s) of any relative extrema. 26) f(x) = 3x4 + 16x3 + 24x2 + 32 26) A) Relative minimum of 30 at -1. B) Relative maximum of 48 at -2; Relative minimum of 32 at 0. C) No relative extrema. D) Relative minimum of 32 at 0. Sketch the graph and show all extrema, inflection points, and asymptotes where applicable. 3 27) f(x) = 2x3 - 12x2 + 18x 27) y 24 12 -8 -4 4 8 x -12 -24 A) Rel min: 2,10 No inflection points B) Rel:max 1,8 ; Rel min: 3,0 Inflection point: 2,4 y -8 y 24 24 12 12 -4 4 8 x -8 -4 4 -12 -12 -24 -24 C) No extrema Inflection point: 0, 0 8 x 8 x D) Rel:max: 0, 0 Rel min: 1,-1 Inflection point: 0.5,-0.5 y 24 y 24 12 12 -8 -4 4 8 x -8 -4 4 -12 -12 -24 -24 4 28) f(x) = x2 2 x +2 28) y 1 -3 -2 -1 1 2 3 x -1 A) Rel min: 0, 1 2 B) Rel min: (0,0) Inflection points: - No inflection points 6 1 , , 3 4 6 1 , 3 4 y y 1 1 -3 -2 -1 1 2 3 x -3 -2 -1 1 2 3 x 1 2 3 x -1 -1 C) Rel min: (0,0) No inflection points D) Rel min: 0,- 1 2 No inflection points y y 1 1 -3 -2 -1 1 2 3 x -3 -2 -1 -1 -1 5 Solve the problem. 29) 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 120 in. What dimensions will give a box with a square end the largest possible volume? A) 20 in. × 20 in. × 100 in. C) 20 in. × 20 in. × 40 in. B) 20 in. × 40 in. × 40 in. D) 40 in. × 40 in. × 40 in. Find the open interval(s) where the function is changing as requested. x+2 30) Decreasing; f(x) = x-3 A) (-∞, -3) 29) B) (-∞, 3), (3, ∞) C) (-∞, 2), (2, ∞) 6 30) D) none Answer Key Testname: 1325-3 1) D 2) C 3) B 4) C 5) D 6) B 7) C 8) B 9) B 10) D 11) B 12) C 13) A 14) D 15) C 16) C 17) D 18) D 19) C 20) A 21) A 22) C 23) D 24) C 25) B 26) D 27) B 28) B 29) C 30) B 7