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UTA - MATH 1316 - Study Guide - Final

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Math 1316 Test 3 Review The test will consist of multiple choice questions modeled after those on your homework. You need to be able to do the following: 1] Be able to evaluatDon't forget about the age old question of which of the following does not occur during mitosis

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e indefinite integrals. 2] If the marginal revenue is given be able to use it to find the total revenue function. 3] Be able to use u-substitution to evaluate indefinite integrals of polynomials raised to a power other than one including those involving lns and exponentials. 4] Given the Marginal Cost Function and either the fixed costs or the costs resulting from a given production level be able to find the total cost function C(x). 5] Given the marginal Cost function MC and the marginal revenue function MR be able to find the optimal level of production 6] Be able to compute definite integrals. 7] Be able to use definite integrals to find the exact value of area between the curve and the x-axis over a specified interval. 8] Be able to compute consumer’s and producer’s surplus. 9] Be able to use definite integrals to find the area between 2 curves over a given interval. 10) Be able to use integration by parts to attempt to solve indefinite integral problems that cannot be solved utilizing other techniques learned in class. 11) Be able to identify the correct value for du in a u-substitution integration problem. 12) Be able to do summation problems like those on summation handout. 13) Be able to solve application problems involving indefinite and definite integrals.Formulas Given on Test Product Rule: if f (x) = u(x)∙ v(x)then f ′(x) = u(x)∙ v′(x)+ v(x)∙u′(x) f x =then ( )( ) ( ) ( ) ( ) Quotient Rule: if ( )( ) f x∙ ′ − ∙ ′ u x v(x) ′= v x u x u x v x ( ( ))2 v x f x = u xthen f (x) n(u(x)) u (x) n −1 Chain (Power)Rule: if ( ) ( ( ))n ′= ∙ ′ y1 ′= Derivative of Natural Log: if y = ln (x) then x 1u x y′= ∙ ′= ��′(��) if y = ln u(x)then ( ) u x ��(��) Derivative of Exponential Function: if x ( ) y′= e; y = ethen x if u( x) y = ethen ( ) ′= ∙ ′ u x ( ) y e u x x n + 1 ∫kdx = kx+C ∫+ n x dx = n + 1 C if n ≠ -1 dx ln 1u x u′dx = u du = u +C = u x +C ∫ ∫ ∫ ∫= = + − x dx x C x − − 1 1 ( ) ln ln ( ) x x∫ ���������� =������ ∫e dx = e +C ′= =+ + n n ��+C u 1 1 u x ( ) if n≠ -1 where du = u′dx ∫ ∫1 n n+ u x u dx u du ( ) n + 1 + = C n + C ′= = + = + ∫ ∫ u x u u u x ( ) ( ) e u dx e du e C e C CS =∫ [��(��) − ��0]���� ��0 0PS = ∫ [��0 − ��(��)]���� ��0 0 Integration by Parts: ∫ u dv = uv - ∫ �� ���� where du = u′dxand v = ∫ ���� n 1 ( )( ) 1 ( ) n n n + + ∑= n n ∑n n i + = n ∑i 2 = 1 2 1 ∑ ��3 =��2(��+1)2 ����=1 i = 1 i = 1 2 i = 1 6 4 1316 Test 3 Practice Test 1) ∫− x dx = 6 8 _____________________________________________________________ ex 2)∫dx = 9________________________________________________________________ x 3 ⎜⎝⎛− + dx ⎞ 3)∫⎟ = 8 52______________________________________________________ e x x ⎠ 1 __________________________________________________________ ⎜⎝⎛− ∫e x dx x 4 ⎞ 4) ⎟ = 3 ⎠ 5) ∫(3x −5x + 2)dx = 2 _____________________________________________________ 6) ∫−8dx = ______________________________________________________________ 10= _______________________________________________________________ 7)∫dx x ⎜⎝⎛+ +2 ∫⎟⎠⎞ x ex 4= _____________________________________________________ 8) dx 5 x 8____________________________________________________________ 9) ∫= −dx 6x 2 ⎜⎝⎛++dx x 2 10] ∫⎟⎠⎞ 2= _______________________________________________________ 5 20 x x ∫4x − 2 8dx 3= __________________________________________________________ 11) ( ) 12) ∫e dx = 2x 9 _____________________________________________________________ 13) If R(x) is a revenue function then R(0) must equal what? _________________________14) Suppose the marginal revenue is MR = 64x. What is the revenue generated at a production level of 10 units? 15) If the rate of change of cost for an item is MC = 4x-2 and the total cost of producing 6 items is $700, find the total cost function. 16) Find the optimal level of production given the marginal cost function for an item is: MC=5x + 50 and the marginal revenue function is: MR = 210 – 3x. 17) ( − ) = ∫4 3 8 x dx __________________________________________________________ 1 10 18)∫ 2 3dx =_________________________________________________________________ 19) ∫2 1 8xdx =_______________________________________________________________ y = 8 xand above the x axis over the interval [1, 8] ? 20) What’s the area under the curve 3 21) What is the area under the curve 2 y = 6xand above the x axis over the interval [1, 3]? 22)∫(−3��2��)���� =__________________________________________________________ 23) ∫(5����3��2)���� =__________________________________________________________ 4 6 6 +dx x5 24)∫= ___________________________________________________________ 25] Find R(2) if Marginal Revenue = 8x + 3. 1__________________________________________________________ 26) ∫= +dx 3x 5 27) Given MR = 3x + 10 find R(4). − 3 1 x + ⎜⎝⎛+ + ⎞ ∫dx 28) ⎟ = + 3 2 ______________________________________________ ex3 6 2 x x ⎠ 2________________________________________________________ 2 ⎜⎝⎛e − x dx x 2 ⎞ 29) ∫⎟ = 9 9 ⎠ 30) Suppose the marginal revenue is 78x. Find R(100). 31) What is the area found under the curve y = 9 xand above the x axis over the interval [0, 1]? 32) Give that the fixed costs for producing an item are $270 and that MC = 18 x + 4; find the total cost function. 33) The demand function for a product is p =D(x) = 100 – 4x. If the equilibrium price is $40 per unit, what is the consumer surplus? 2 34) The supply function for a good is ( ) 4 2 2 p = S x = x + x +. If the equilibrium price is $422 per unit, what is the producer surplus? 35.Use integration by parts to evaluate the following integral: ∫(�� ln(2��))����. Be able to identify u, dv, v, and du as well as the solution to the integral. 36. Use integration by parts to evaluate the following integral:∫(10����2��)���� . Be able to identify u, dv, v, and du as well as the solution to the integral. 37. ∫ (−14 ��+2) ���� = _____________________________________ 38. ∫(4����−3��2+7)���� = ________________________________ 3 39. ∫ (5��2 +7��4) ���� = 2_______________________________________ 40. ∫ (8��−1 6 1___________________________________________ 3) ���� = 5 √5���� = _________________________________ 41. ∫ (2��√��2 − 3 3 42.∫ (6�� 0______________________________________��2+1) ���� = 43) Find the area between the x-axis and the function f(x) = exover the interval [0, 2]. Round your answer off to 3 decimal places. 44. Find the total cost function C(x) given that the marginal cost function is: ���� = ��′(��) =6 ��+1and that the fixed costs for the product are $18. 45. Find the area between the 2 curves: y = x2- 30 and y = 10 – 3x from x = 1 to x = 5. 46. Find the area between the 2 curves: y = x2and y = x3from x = 0 to x = 1/2. 47. Use integration by parts to evaluate the definite integral: ∫ (�� ln(��))���� 41 48.∑ (3��2 − 7��) =? 40 ��=1 20 49. ∑ (2��3 + 3��2 + 4�� + 3 =? ��=1 50. Suppose that you are shooting paint balls into the air in such a way that the rate of change for the height of each ball t seconds after you shoot it is v = ��ℎ ����= 208 – 32t, in feet per second, and that 3 seconds after you shoot the ball it is 380 feet high in the air. Use this information to derive the function h(t) that describes the height of the ball at any time t. 51. Given that the marginal revenue function for a product is MR = 1.1 x + 14 thousand dollars where x = 0 corresponds to the year 2000. Use this data to compute the revenue generated b the product over the 5 year period from 2008 to 2013. The revenue generated is __________________________ thousand dollars.8 Solutions 27] $64 7 1] − x +C 7 1 x+ 3 1 2 28] e x x C −ln 3 6 2 x+ + + + 6 2] e C 2 2 9 29] e x C x− + 3 9 27 5 83 x− + 3ln | | + 3]e x x C 3 15 x 4] C ex− + 3 5 5] ��3 −5��2 2+ 2�� + �� 6] – 8x + C 7] 10ln | x | +C 5 x x 8] e x C 30] $390,000 31] 6 square units 32] ( ) 12 ( 4) 174 3 C x = x + + 33] $450 34] $2766.67 35] u = ln 2�� dv = x dx du = 1������ �� =��22 ��22ln 2�� −14��2 + �� 5 4 + 5 + 2ln | | + 36] u =10 x dv = ��2��dx 9] ln 6x − 2 +C 3 1 2 10] ln 5x + 20x +C 10 1 11] ( x − ) +C du = 10dx v = ��2�� 2 5����2�� −52��2�� + �� 4 4 2 2 9 12] e C 37]−14 ln|�� + 2| + �� 2 2 13] 0 x+ 38] −2��−3��2+7 3+C = −2 3��3��2+7 + �� 14] 2 R(x) = 32x; R(10) = $3200 15] ( ) 2 2 640 2 C x = x − x + 16] x = 20 17] – 39.75 18] 24 19] 12 20] 90 square units 21] 52 square units 22] −3��2�� 2+ �� 23] 5��3��2 6+C 6 5 6 24] x + x +C 39] 1.039 40] 4.77 41] 66.907 42] 3 ln10 = 6.908 43] Area = 6..389 square units 44] C(x) = 6 ln|x + 1| + 18 45] Area = 82.667 square units 46] Area =.026 square units 25 25] $22 2 5 47] 7.34 48] 60,680 49] 97,710 50] h(t) = 208t – 16t2– 100; 51] $127.75 26] 3x + 5 +C 3

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