MAT 210

MAT 210 Exam 3 Review Questions Section 13.1 Indefinite Integral 1. Find the indefinite integrals. (a) ∫ 2π‘₯ βˆ’ 4π‘₯ + 5π‘₯ + 3 𝑑π‘₯ (b) ∫ + 𝑑π‘₯ (c) ∫ βˆ’ 5√π‘₯ 𝑑π‘₯ (d) ∫ 𝑒 βˆ’ π‘₯ . 𝑑π‘₯ (e) ∫(π‘₯ + 3)(π‘₯ βˆ’ 2)𝑑π‘₯ (f) ∫ 𝑑π‘₯ Answer: (a) π‘₯ + βˆ’ + 3π‘₯ + 𝐢 (b) 7ln|π‘₯| βˆ’ +𝐢 (c) βˆ’ βˆ’ (d) 𝑒 βˆ’ π‘₯ +𝐢 . . +𝐢 =𝑒 βˆ’ (e) ∫ π‘₯ + π‘₯ βˆ’ 6 𝑑π‘₯ = (f) ∫ π‘₯ + 5 βˆ’ 𝑑π‘₯ = π‘₯ + . +𝐢 βˆ’ 6π‘₯ + 𝐢 + 5π‘₯ βˆ’ 2 ln|π‘₯| + 𝐢 2. Find 𝑓(π‘₯) if 𝑓(0) = βˆ’1 and the derivative 𝑓 (π‘₯) = 9𝑒 + 9. Answer: 𝑓(π‘₯) = 9𝑒 + 9π‘₯ βˆ’ 10 3. The velocity of a particle moving in a straight line is 𝑣(𝑑) = 𝑑 + 6. Find the expression for the position, 𝑠(𝑑), of the particle at time 𝑑, if 𝑠(3) = 0. Answer: 𝑠(𝑑) = 𝑑 + 6𝑑 βˆ’ 27 1 4. Suppose the function 𝐢(π‘₯) gives the total cost (in dollars) of producing π‘₯ units of a certain product. The marginal cost of producing the π‘₯th unit is 𝐢′(π‘₯) = 0.5π‘₯ + . If the cost to produce the first unit is 5 dollars, find the cost function 𝐢(π‘₯). Answer: 𝐢(π‘₯) = 0.25π‘₯ + ln |π‘₯| + 4.75 dollars 1 0.5π‘₯ + 𝑑π‘₯ = 0.25π‘₯ + ln |π‘₯| + 𝐾 π‘₯ 𝐢(1) = 5, So 5 = 0.25 βˆ™ 1 + ln |1| + 𝐾. Then solve for constant K: 𝐾 = 5 βˆ’ 0.25 = 4.75. 𝐢(π‘₯) = 𝐢′(π‘₯) 𝑑π‘₯ = Section 13.2 Substitution 5. Use integration by substitution to find the integrals. (a) ∫ 16𝑒 𝑑π‘₯ (can also use short-cut formula) (b) ∫(5π‘₯ βˆ’ 2) 𝑑π‘₯ (can also use short-cut formula) (c) ∫ (d) (e) (f) (g) (can also use short-cut formula) 𝑑π‘₯ 𝑑π‘₯ ∫ 4π‘₯𝑒 ∫ π‘₯(π‘₯ + 1) 𝑑π‘₯ ∫ 15π‘₯βˆšβˆ’π‘₯ + 7 𝑑π‘₯ ∫(3π‘₯ + 1)(π‘₯ + π‘₯ βˆ’ 2) 𝑑π‘₯ Answer: (a) 16 βˆ™ (b) (c) ( ) βˆ™ +𝐢 = 𝑒 +𝐢 (5π‘₯ βˆ’ 2) + 𝐢 ln|2π‘₯ βˆ’ 5| + 𝐢 (d) 2𝑒 (e) +𝐢 =βˆ’ +𝐢 (π‘₯ + 1) +𝐢 (f) βˆ’5(βˆ’π‘₯ + 7) + 𝐢 (g) (π‘₯ + π‘₯ βˆ’ 2) + 𝐢 2 Section 13.3 Definite Integral; Left Riemann Sum 6. Evaluate the definite integrals. (a) ∫ (6π‘₯ + 15π‘₯ βˆ’ 9π‘₯ + 1) 𝑑π‘₯ (b) ∫ π‘₯+ (c) ∫ 𝑑π‘₯ 𝑑π‘₯ (d) ∫ 𝑒 𝑑π‘₯ (e) ∫ 5𝑒 𝑑π‘₯ (f) ∫ 𝑑π‘₯ (g) ∫ 𝑒 𝑑π‘₯ Answer: (a) 2 (b) + 5 ln (c) (d) βˆ’1 + 𝑒 (e) (𝑒 βˆ’ 𝑒 ) (f) 4 (g) 8 7. Assume that 𝑏 is a positive number, solve the following equation for 𝑏. ∫ 2π‘₯ βˆ’ 4 𝑑π‘₯ = 9 Answer: 𝑏 = 5 8. Calculate the left Riemann sum for the function 𝑓(π‘₯) = 3π‘₯ + 2π‘₯ βˆ’ 3 over the interval [1, 3], with 𝑛 = 5. Answer: 22.56 βˆ†π‘₯ = = = 0.4, π‘₯ = π‘Ž = 1, π‘₯ = π‘₯ + βˆ†π‘₯ = 1.4, π‘₯ = 1.8, π‘₯ = 2.2, π‘₯ = 2.6. LRS = βˆ†π‘₯ βˆ™ 𝑓(1) + 𝑓(1.4) + 𝑓(1.8) + 𝑓(2.2) + 𝑓(2.6) = 0.4(2 + 5.68 + 10.32 + 15.92 + 22.48) = 22.56 9. Use a left Riemann sum to estimate the definite integral with 𝑛 = 4 subintervals. 1 𝑑π‘₯ 1 + 2π‘₯ 3 Answer: 0.18 βˆ†π‘₯ = 0.25, LRS = 0.25 ( ) + ( . ) + ( . ) + ( . = 0.25(0.2 + 0.18 + 0.17 + 0.15) = 0.18 ) Section 13.4 Fundamental Theorem of Calculus; Applications of Definite Integrals 10. A particle moves in a straight line with velocity 𝑣(𝑑) = βˆ’π‘‘ + 8 meters per second, where 𝑑 is time in seconds. Find the displacement of the particle between 𝑑 = 2 and 𝑑 = 6 seconds. Answer: βˆ’37 meters Displacement = 𝑠(6) βˆ’ 𝑠(2) = ∫ 𝑣(𝑑) 𝑑𝑑 = ∫ βˆ’π‘‘ + 8 𝑑𝑑 = βˆ’ 11. The marginal revenue of the π‘₯th box of flash cards sold is 500𝑒 selling box 101 through 5,000. β‰ˆ βˆ’37 meters. . dollars. Find the revenue generated by Answer: 448,598 dollars Total revenue generated = 𝑅(5000) βˆ’ 𝑅(101) = ∫ 𝑀𝑅 𝑑π‘₯ = ∫ 500𝑒 . 𝑑π‘₯ β‰ˆ 448597.54 dollars 12. Since YouTube first became available to the public in mid-2005, the rate at which video has been uploaded to this site can be approximated by 𝑓(𝑑) = 1.1𝑑 βˆ’ 2.6𝑑 + 2.3 million hours of videos per year (0 ≀ 𝑑 ≀ 9), where 𝑑 is time in years since June 2005. Use a definite integral to estimate the total number of hours of video uploaded from June 2007 to June 2010. Answer: 23 million hours of video Total number of hours = ∫ 𝑓(𝑑) 𝑑𝑑 = ∫ 1.1𝑑 βˆ’ 2.6𝑑 + 2.3 𝑑𝑑 β‰ˆ 23 million hours of video 13. Calculate the area of the region bounded by 𝑦 = √π‘₯, the π‘₯-axis, and the lines π‘₯ = 0 and π‘₯ = 16. Answer: Area under curve = ∫ √π‘₯ 𝑑π‘₯ = 4 Section 14.1 Integration by Parts Integration by parts formula: ∫ 𝑒𝑑𝑣 = 𝑒𝑣 βˆ’ ∫ 𝑣𝑑𝑒 14. Use integration by parts to find the integrals. (a) ∫ 2π‘₯𝑒 𝑑π‘₯ (b) ∫(3π‘₯ + 4)𝑒 𝑑π‘₯ (c) ∫ ln π‘₯ 𝑑π‘₯ (d) ∫ π‘₯ ln π‘₯ 𝑑π‘₯ Answer: (a) Let 𝑒 = 2π‘₯, 𝑑𝑣 = 𝑒 𝑑π‘₯. Then 𝑑𝑒 = 2𝑑π‘₯ and 𝑣 = 𝑒 . Using the formula: ∫ 𝑒𝑑𝑣 = 𝑒𝑣 βˆ’ ∫ 𝑣𝑑𝑒 to get 2π‘₯𝑒 𝑑π‘₯ = 2π‘₯𝑒 βˆ’ (b) βˆ’ (3π‘₯ + 4)𝑒 βˆ’ 𝑒 (Let 𝑒 = 3π‘₯ + 4, 𝑑𝑣 = 𝑒 +𝐢 = βˆ’ π‘₯βˆ’ 𝑒 𝑒 2𝑑π‘₯ = 2π‘₯𝑒 βˆ’ 2𝑒 + 𝐢 +𝐢 𝑑π‘₯. Then 𝑑𝑒 = 3𝑑π‘₯ and 𝑣 = βˆ’ 𝑒 ) (c) π‘₯ ln π‘₯ βˆ’ π‘₯ + 𝐢 (Let 𝑒 = ln π‘₯ , 𝑑𝑣 = 𝑑π‘₯) (d) π‘₯ ln π‘₯ βˆ’ π‘₯ + 𝐢 (Let 𝑒 = ln π‘₯ , 𝑑𝑣 = π‘₯ 𝑑π‘₯) 5 Section 14.2 Area between Curves 15. Find the area of the region enclosed by the curves of 𝑦 = βˆ’π‘₯ + 6π‘₯ + 2 and 𝑦 = 2π‘₯ + 9π‘₯ βˆ’ 4. Answer: 13.5 Find the intersection points: βˆ’π‘₯ + 6π‘₯ + 2 = 2π‘₯ + 9π‘₯ βˆ’ 4 0 = 3π‘₯ + 3π‘₯ βˆ’ 6 0 = 3(π‘₯ + 2)(π‘₯ βˆ’ 1) So π‘₯ = βˆ’2 and π‘₯ = 1. The area enclosed by the curves from βˆ’2 to 1 is ∫ top βˆ’ bottom 𝑑π‘₯ = ∫ (βˆ’π‘₯ + 6π‘₯ + 2) βˆ’ (2π‘₯ + 9π‘₯ βˆ’ 4) 𝑑π‘₯ = ∫ βˆ’3π‘₯ βˆ’ 3π‘₯ + 6 𝑑π‘₯ = βˆ’π‘₯ βˆ’ π‘₯ + 6π‘₯| = βˆ’1 βˆ’ 1 + 6(1) βˆ’ βˆ’(βˆ’2) βˆ’ ((βˆ’2) + 6(βˆ’2) = 13.5 16. Find the area of the region enclosed by the curves of 𝑓(π‘₯) = π‘₯ βˆ’ π‘₯ + 5 and 𝑔(π‘₯) = π‘₯ + 8. Answer: 17. Find the area of the region between 𝑦 = π‘₯ and 𝑦 = βˆ’1 from π‘₯ = βˆ’1 and π‘₯ = 1. Answer: 18. Which of the following calculates the area of the region(s) between the curves 𝑦 = π‘₯ and 𝑦 = 1 from π‘₯ = βˆ’1 to π‘₯ = 2? A. ∫ π‘₯ βˆ’ 1𝑑π‘₯ B. ∫ 1 βˆ’ π‘₯ 𝑑π‘₯ C. ∫ 1 βˆ’ π‘₯ 𝑑π‘₯ + ∫ π‘₯ βˆ’ 1𝑑π‘₯ D. ∫ π‘₯ βˆ’ 1𝑑π‘₯ + ∫ 1 βˆ’ π‘₯ 𝑑π‘₯ E. None of the above. Answer: C 6 Section 14.3 Average Value 19. Find the average value of 𝑓(π‘₯) = 6𝑒 Answer: 3(𝑒 . . βˆ’π‘’ . over the interval [βˆ’1, 3]. ) The average value of a continuous function 𝑓(π‘₯) over interval [π‘Ž, 𝑏] is 1 3 βˆ’ (βˆ’1) 6𝑒 . 1 𝑒 . 𝑑π‘₯ = βˆ™ 6 βˆ™ | 4 0.5 = 3𝑒 ∫ 𝑓(π‘₯) 𝑑π‘₯. . | = 3(𝑒 . βˆ’π‘’ . ) 20. Find the average of the function 𝑓(π‘₯) = π‘₯ βˆ’ π‘₯ over the interval [0, 2]. Answer: 1 21. Find the average value of the function 𝑓(π‘₯) = 6π‘₯ βˆ’ 4π‘₯ + 7 over the interval [βˆ’2, 2]. Answer: 15 Section 14.4 Consumers’ Surplus and Producers’ Surplus 22. Your video store has the exponential demand equation 𝑝 = 15𝑒 . , where π‘ž represents daily sales of used DVD’s and 𝑝 represents daily price you charge per DVD. Calculate the daily Consumer’s Surplus if you sell used DVDs at $5 dollars each. Answer: $450.69 . Consumer Surplus = ∫ 𝐷(π‘ž) βˆ’ 𝑝̅ π‘‘π‘ž = ∫ 15𝑒 . βˆ’ 5 π‘‘π‘ž = 450.69 23. Calculate the Producer’s Surplus for the supply equation 𝑝 = 13 + 2π‘ž at the unit price 𝑝̅ = 29. Answer: $64 Producer Surplus = ∫ 𝑝̅ βˆ’ 𝑆(π‘ž)π‘‘π‘ž = ∫ 29 βˆ’ (13 + 2π‘ž)π‘‘π‘ž = 16π‘ž βˆ’ π‘ž | = 64 24. Calculate the Producer Surplus for the supply equation 𝑝 = 7 + 2π‘ž / at price 𝑝̅ = 14. Answer: $75 π‘ž = 3.5 = 42.875, Producer Surplus = ∫ . 14 βˆ’ (7 + 2π‘ž / )π‘‘π‘ž = 7π‘ž βˆ’ π‘ž / | . = 75 7 Section 14.5 Improper Integral 25. Determine whether each improper integral is convergent or divergent. If it is convergent, find its value. (a) ∫ 𝑑π‘₯ (b) ∫ 𝑒 (c) ∫ 𝑑π‘₯ 𝑑π‘₯ (d) ∫ π‘₯ 𝑑π‘₯ (e) ∫ 𝑒 𝑑π‘₯ Answer: (a) Converges to 8 8 𝑑π‘₯ = lim β†’ π‘₯ 8π‘₯ 8 𝑑π‘₯ = lim βˆ’ | β†’ π‘₯ 8 8 = lim βˆ’ + =0+8=8 β†’ 𝑑 1 (b) Converges to 𝑒 𝑑π‘₯ = lim β†’ 𝑒 1 𝑑π‘₯ = lim βˆ’ 𝑒 β†’ 2 | 1 = lim βˆ’ 𝑒 β†’ 2 1 + 𝑒 2 ( ) 1 =0+ 𝑒 2 = 1 2𝑒 (c) Diverges 1 𝑑π‘₯ = lim β†’ π‘₯ 1 𝑑π‘₯ = lim (ln 𝑑 βˆ’ ln 1) = ∞ βˆ’ 1 = ∞ β†’ π‘₯ (d) Diverges (e) Converges to 8 Uz instructions des Locko Bob Question Evaluate S 3×2 + 2x +\dx x O None of the above O 6x + 2 O xΒ² + xΒ² + c + Question 2 Find the definite integral S3 dx O 9 (ln(3) – In(7)) O None of the above O 9 In(7) O 9 (In(7) – In(3)) Canvas 7 pts Question 3 Suppose that the marginal cost to produce backpacks at a a production level of x backpacks is 5x + 2x + 3 dollars per backpack, and the cost of producing 1 backpack is $15. Find the cost function C (x) O C(x) = x + xΒ² + 3x + 10 O C(x) = x + x2 + 3x + 15 O None of the above O C(x) = x + x2 + 13 7 pts Question 4 Which of the following would be the best substitution to integrates longue dx ? D Ou = 7x None of the above Ou = lnde 7 pts In(x) = Canvas Δ€ Question 5 7 pts = The rate of U.S. per capita sales of bottle water for the period 2007-2014 could be approximated by s(t) 0.25 12 – 1 + 29 gallons per year, 0 st 57, where t = 0) is time in years since the start of 2007 Use a definite integral to estimate the total U.S. per capita sales of bottle water in gallons from the start of 2008 to the start of 2012. Which expression is the correct setup for solving this problem? You don’t need to compute the final calculation O None of the above O 0.25 12 – + 29 dt O 0.25 + – + + 29 dt O 0.25 12 – + + 29 dt 7 pts Question 6 D Evaluate the integral |-|(8x+ x) dx 00 08 05 7 pts Carwas GΔ€ D Question 7 Evaluate the integral 3 dx 0 3+ O None of the above OO O 3x + c 1 pts Question 8 Evaluate the integral fet dx O None of the above Oetti + c X+1 O ex + c 7 pts O ext] + Question 9 Book Pro 19 Canvas Γ€ Question 9 Evaluate S 6xΒ² (2x + 5) dx Hint: Use the substitution rule O (2xΒΊ+5) + 10 O 2x ” + ] (2x +5) 10 + c 0 2×3 + (2x + 5) 0 + c + O 2×3 (2×3 + 5).Β° + c 7 pts Question 10 Evaluate / x e’ dx Hint: Use Integration by parts ext1 X+1 + c C O O xe* texto 이λ₯Ό er et +C – x +C MacBook Pro px Canvas G8 OxeΒ² – et + c Question 11 15 pts Free Response Question: For this problem, beside answering the question as the other multiple choice questions, you need to show all your work on the scratch/draft paper to receive full/partial credit. Calculate the Left Riemann Sum of the function f(x) = 3×2 + 2x – 3 over 12. 7) with n = 5 None of the above O 300 0 295 0 290 15 pts Question 12 Free Response Question: For this problem, beside answering the question as the other multiple choice questions, you need to show all your work on the scratch/draft paper to receive full/partial credit D Let f (x) = x2 – 3, and g(x) = x – 1 Find the area of the region enclosed by f (x) and g(x) You need to show all steps algebraically. You need to find the points of intersection and find the integral steps. You can use your your answer not to just answer the question MacBook Pro Cams 39Δ€ Question 12 Free Response Question: For this problem, beside answering the question as the other multiple choice questions, you need to show all your work on the scratch/draft paper to receive full/partial credit Let f(x) = x – 3, and g(x) = x – L Find the area of the region enclosed by f(x) and g(x) You need to show all steps algebraically. You need to find the points of intersection and find the integral steps. You can use your calculator to check your answer not to just answer the question. O 4.5 03 O None of the above 08 Thank you for finishing the exam. Please follow the following steps Step 1: Show the scratch papers for Questions 11 and 12 to the camera, Step 2: After you submit this exam you have 15 minutes to scan your work for Questions 11 and 12 as a single pdf file. Then submit it on Canvas —> Assignments –> Exam 3 Scratch Papers. Pro
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