Math142 Midterm Exam Review
Math142 Midterm Exam Review
Math142 Midterm Exam Review
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Math 142<strong>Midterm</strong> <strong>Exam</strong> <strong>Review</strong> with AnswersLim fx3Lim fx3( x)______________( x)EXIST / NOT EXISTLim fx3(Circle One)x 2 x 4b. Graph f ( x)Label four points.7 2xx 4( x)______________Lim fx4Lim fx4Lim fx4( x)( x)( x)____________________________EXIST / NOT EXIST(Circle One)Page 4 of 11
Math 142<strong>Midterm</strong> <strong>Exam</strong> <strong>Review</strong> with Answers6. [Sec. 1.5] Find the limit or if it does not exist. If it is infinite indicate whether it is +∞or -∞a.Limxx24x322 x x42b.Limxx28x627 x x772x(x 2)c. Lim 2x 0 xd.xLimx 2 x4162x x 5e. Lim3x 1 2xxf.Limxx122xx53x7. [Sec 2.1] For the given function use the definition of the derivative to find f '(x),Show all the required steps.2a. f ( x)4x3x4b.f ( x)7 x28. [Sec. 2.2] Find the derivative, f '(x)4 23 2a. f ( x)5x3x6x3b. f ( x)x 2 x2 3c. f ( x)4xd.4 3x xf ( x)5x3 29. [Sec 2.3] Find the derivative of the function using the product rule of derivatives,f’(x), simplify.23a. f ( x)( x x)(3x1)b. f ( x)(4x3)( x 3)Page 5 of 11
Math 142<strong>Midterm</strong> <strong>Exam</strong> <strong>Review</strong> with Answers10. [Sec 2.3] Find the derivative, y’, simplify if possible.a.4 2x x 1y b.2x 1y2x2xx 1511. [Sec 2.4] Find the derivative, f’(x), simplify if possible.1a. f ( x)b.334x4x3f3 2( x)x 3x412. [Sec. 2.4] Evaluate the second derivative of the given function.a. f ''(1)for the functionf ( x)3x25xb. f ''(1) for the function f ( x)3x213. [Sec 2.5] Solve the following applied problems, interpret results3a. A steel mill finds that its cost function is C ( x)8000 x 600 x dollars,where x is the daily production of steel (in tons).i. Find the Marginal Cost Function.ii.iii.Find the marginal cost when 64 tons of steel are produced.Interpret the results.b. The manufacturer determines that the profit P (in dollars) derived from selling x3units is given by P(x)0.0004x7x.i. Find the Marginal Profit Function.ii.Find the marginal profit for a production level of 52 units.iii.Interpret the results.Page 6 of 11
Math 142<strong>Midterm</strong> <strong>Exam</strong> <strong>Review</strong> with Answers14. [Sec. 2.4] A company’s profit function is P ( x)13x1700a. Find the Relative Rate of Change of the Profit Function.b. Find the Marginal Profit Function.c. Evaluate the Relative Rate of Change of the Profit Function at x = 250.15. [Sec. 2.3] Solve the following applied problems, interpret results.d. The number of weeds (in hundreds) n in lawn x weeks after a weed killer wasapplied is given by n ( x)2x 2x3 . Find the rate of change in thenumber of weeds in the lawn at the end of the second week.e. The velocity of an object in meters per second is v ( t)12 19t, 0 t .3Find the acceleration of the subject when t = .15 sec.16. [Sec 3.1 & 3.2] For the following functions:3 2f. The function, f ( x)x 6x 9x1,4 3 2g. The function, f ( x)x 4x20x12i. Make a sign diagram of f x ' ( )ii. Identify the relative extreme points and inflection pointsiii. Sketch the graphPage 7 of 11
Math 142<strong>Midterm</strong> <strong>Exam</strong> <strong>Review</strong> with Answers<strong>Midterm</strong> <strong>Review</strong> Math 142 Answers:#1 a. i. y = 4x + 2 ii. y = $22 millioniii. @ the end of the 7 th yearb. i. y = .1x + 4.95 or y1 x104. 95 ii. y = $ 14.95iii. 32 minutes3#2 a. i. x xii. x-intercept: None & y-intercept:22iii. f (3) = 3iv. Graph curve with label on any of three points (2, 2); (1, -2); )b. i. { x| -3 x 3}ii. x-intercept:iii. f (3) = 2iv. Graph curve with label on any of three points: (0, -1); (3, 2); (-3,2)#3 a. i. P ( x)150x1500ii. 10 canoesiii. x 17 canoes2b. i. P ( x)x 230x12000 ii. @ price of $80 and $150 per unitiii. price of $115 per unit#4 a. i. f(-2) = -1; f((3) = 3 ; g(3) = 9ii. Domain f(x) {R}; g(x) {R} iii. f(g(x)) = 3 – 9x 2iv. g(f(x)) = 9 – 3x 2b. i. f(-2) = Undefined; f(0) = Undefined; g(3) = 51ii. Domain f(x) x ; x ; g(x){R} iii. f ( g(x))6x43iv. g ( f ( x))3x1 1Page 8 of 11
Math 142<strong>Midterm</strong> <strong>Exam</strong> <strong>Review</strong> with Answers#5 a. Piecewise lines with points (0, 8), (1, 7), (2, 6), (3, 5); stop at the point (3, 5)And line with points (3, 3), (4, 5), (5, 7), (6, 9); (3, 5) do not exist.f( x ) 5limx3limx3limx3f( x ) 3f( x)do not exist.b. Piecewise lines with points (0, 7), (1, 5), (2, 3), (3, 1), (4, -1); stop at the point(4, -1) and line with points (4, 2), (5, 3), (6, 4); (4, -1) do not exist.f( x ) 1limx4limx4limx4f( x ) 2f( x)do not exist.#6 a. 0 b. 43d. Does not exist e. 0c.#7 a. f’(x) = 8x – 3 b. f’(x) = -2x#8 a. f’(x) = 20x 3 – 6x + 6 b. f’(x) =332x1x8 9c. f’(x) =5 4x x- 4 d. f’(x) =310x3 5#9 a. f’(x) = 9x 2 + 8x + 1 b. f’(x) = 16x 3 – 36x 2 – 352x4x#10 a. y ’ =2 2( x 1)322x10xb. y ’ =2(2x5)7Page 9 of 11
Math 142<strong>Midterm</strong> <strong>Exam</strong> <strong>Review</strong> with Answers23(12x4)#11 a. f’(x) =34(4x4x3)b. f’(x) =3( x22x33x4)23#12 a. f”(1) =45b. f”(1) =94#13 a. i. MC(x) =4000x200x3 2ii. MC(64) = 487.5iii. The cost is increasing by $487.5 for every ton of steel when the mill isproducing 64 tons of steel.b. i. MP(x) = .0012x 2 + 7 ii. MP(52) = 10.24iii. The profit is increasing at $10.24 per unit when manufacturesproduces 52 units.#14 a. RRCP(x) =1313x1700b. MP (x) 13 c. RRCP(250) = 0.008#15. a. n’(2)=-2, the rate of change is decreasing 2 hundred units per week.b. a(.15) = -2.7, the object is decreasing in acceleration of 2.7 m/sec. 2 .#16. a. i) + - +ii)Extreme Points (1,3) Relative Max(3,-1) Relative Min.Inflection Point (2, 1)b.i) - + - +ii) Extreme Points (-5,-387) Rel Minimum(0,-12) Rel. Maximum(2, -44) Rel. Minimum.Inflection points (1.0817, -28.97), (-3.082, -228.8)Page 10 of 11
Math 142<strong>Midterm</strong> <strong>Exam</strong> <strong>Review</strong> with AnswersPage 11 of 11
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