FIU MAC 2233 - Final Exam Review

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Final Exam MAC 2233 Review 1. limπ₯β15(β30)= A. 15 B. 30 C. -15 D. -450 E. None of the above 2. Which statements is/are true? I. π(π₯)=7π₯3β 5π₯2+ 3π₯ β5 is continuous at π₯=β1 II. limπ₯β 2(π₯β2)(π₯β2)(π₯+2) ππππ  πππ‘ ππ₯ππ π‘ III. The horizontal asymptote of π¦=π₯2π₯β1 is π₯=1. A. I and II B. I and III C. II only D. I only E. All of them 3. Differentiate π¦=3βπ₯ β 7π₯5+ 10π₯β 8. A. π¦β²=32βπ₯β 35π₯4+ 10 B. π¦β²=6βπ₯β 7π₯4+ 5π₯β 1 C. π¦β²=2βπ₯β 35π₯4β 8 D. π¦β²=32βπ₯β 7π₯4+ 10 E. None of the above 4. In 2007 in the United States, the cost per gallon for gas was \$2.85, while in 2014 it was \$3.43. Find the average rate of change of the cost per gallon for gas per year from 2007 to 2014. Round to the nearest hundredth. A. \$0.58 B. \$12.07 C. \$0.08 D. \$0.90 E. None of the above 5. Find the value of A that makes π(π₯) continuous on all real numbers. π(π₯)={2π₯2β 3π₯ + 1 π₯<β2π΄π₯β 7 π₯β₯β2 A. π΄=β2 B. π΄=15 C. π΄=β10 D. π΄=β11 E. None of the above 6. How long will it take for \$3,000 to turn into \$30,000 at an annual interest rate of 3.7% compounded continuously? Round to the nearest whole number. A. 10 years B. 62 years C. 37 years D. 6 years E. None of the above7. How much money is in the account when \$800 is invested at a 4.6% interest rate compounded monthly after 10 years? A. \$1,267.26 B. \$1,387.62 C. \$1,266.15 D. \$1,357.32 E. None of the above 8. Find the marginal average cost given the total cost is πΆ(π₯)=4π₯+ 8,000 when x units are produced. A. πΆξͺ§β²(π₯)=4 B. πΆξͺ§β²(π₯)=β8,000π₯2 C. πΆξͺ§β²(π₯)=0 D. πΆξͺ§β²(π₯)=4 β 8,000π₯2 E. None of the above 9. Differentiate π¦=4lnπ₯6 A. π¦β²=24lnπ₯5 B. π¦β²=24π₯5 C. π¦β²=4ln6π₯5 D. π¦β²=24π₯ E. None of the above 10. Differentiate π¦=π₯3π2π₯. Factor completely. A. π¦β²=6π₯2π2π₯ B. π¦β²=π₯2π2π₯(3 + π₯) C. π¦β²=π₯2(3π2π₯+ 2π2π₯β1) D. π¦β²=π₯2π2π₯(3 + 2π₯) E. None of the above 11. Which of the following is/are true? I. (πΉπ)β²=πΉβ²π β πΉπβ² II. π¦=ln3π₯ βπ¦β²=3π₯ III. Profit = Cost - Revenue A. I and II B. II and III C. II only D. I and III E. None of them 12. Differentiate π¦=ππ₯2β5π₯+4 . A. π¦β²=ππ₯2β5π₯+4 B. π¦β²=π2π₯β5 C. π¦β²=ππ₯(2π₯ β 5)(π₯2β 5π₯+ 4) D. π¦β²=(2π₯ β 5)ππ₯2β5π₯+4 E. None of the above 13. Differentiate π¦=3π₯(2π₯2β 5)3. Factorize Completely. A. π¦β²=3(2π₯2β 5)2(14π₯2β 5) B. π¦β²=36π₯(2π₯2β 5)2 C. π¦β²=3(2π₯2β 5)2(2π₯2+ 3π₯β 5) D. π¦β²=12π₯2(2π₯2β 5)(4π₯) E. None of the above14. Use the graph of π¦=π(π₯) to find the interval on which πβ²(π₯)>0. A. (ββ,β3)βͺ (0,2) βͺ (3.5,β) B. (ββ,β2) βͺ (1,3) C. (β2,1)βͺ (3,β) D. (β3,0) βͺ (2,3.5) E. None of the above 15. Find all inflection points of π¦=3π₯5β 10π₯4. A. (0,0) πππ (2,β64) B. (0,0) πππ (83,β101.1) C. (2,β64) D. (0,0) E. None of the above 16. The graph of π(π₯) is given below. Find the intervals where πβ²β²(π₯)<0. A. (ββ,β1.5) βͺ (β0.5,β) B. (ββ,β1) C. (β1.5,β1) βͺ (β1,β0.5) D. (β1,β) E. None of the above 17. πβ²(π₯)=0 ππ‘ π₯=1 πππ π₯=5 (the critical numbers). Use the second derivative test to find the x-coordinate of the local extrema of the function, given the second derivative: πβ²β²(π₯)=β2π₯β 1 β 2. A. Local max at π₯=1 and local min at π₯=5 B. Local max at π₯=1 and π₯=5 C. Local min at π₯=1 and π₯=5 D. Local max at π₯=5 and local min at π₯=1 E. None of the above 18. Find two numbers whose difference is 71 and whose product is a minimum. A. 35.5 πππ 35.5 B. 81 πππ 10 C. 40 πππ β 31 D. β15.5 πππ 55.5 E. None of the above19. Evaluate β«3π₯β 242ππ₯ A. 2 B. 6 C. 14 D. 18 E. None of the above 20. Evaluate: β«(ππβ 6π₯7β ln(π₯))341212 A. 12 B. 34 C. 0 D. 12 β π E. None of the above 21. Which of these is/are true? (Choose all that apply) If π πππ π are continuous on [π,π], and πΉ is an antiderivative of π, then I.β«π(π₯)ππ₯=πΉ(π)+ πΉ(π)ππ II. β«π(π₯)Β± π(π₯)ππ₯= β«π(π₯)ππ₯Β± β«π(π₯)ππ₯ III. β«1π₯ππ₯=ln|π₯|+ π A. I and II B. II and III C. I and III D. All of them E. None of them 22. Evaluate the following limits. SHOW WORK FOR EACH to receive full credit. a. limπ₯β72π₯π₯+1= b. limπ₯β25π₯β25βπ₯β5= c. limπ₯ββ3π₯2β7π₯+16π₯2+8= d. limπ₯βββ(β5π₯5+ 6π₯β 10)= e. limπ₯ββ4+π₯β3π₯+4=23. Use the graph to answer the following questions. a. limπ₯ββ6π(π₯)= b. limπ₯ββπ(π₯)= c. limπ₯ββ1βπ(π₯)= d. Is π(π₯) continuous at π₯=β2? Why or why not? e. Is π(π₯) continuous at π₯=1? Why or why not? 24. Find the horizontal and vertical asymptotes of π(π₯) using limits. Failure to use limits will result in a zero. State if there are any holes in the graph, and if yes, give the coordinates. π(π₯)=π₯β 5π₯2β 4π₯ β 5 25. Use the definition of the derivative (four-step process in book) to differentiate π(π₯)=5π₯. Any other method of finding the derivative will result in a zero. 26. Let π¦=2π₯3β 3π₯2β 12π₯ + 4. a. Find the equation of the tangent line in slope intercept form when π₯=β1. b. Find the x-values where the tangent line is horizontal. 27. Differentiate. a. π¦=βπ₯43β5π₯2+ 7π₯β 10π₯4 b. π¦=5π₯2β7π₯π₯ 28. A function π(π₯) is continuous at a value π₯=π if the following conditions are met: 1. π(π) is defined 2. limπ₯βππ(π₯) ππ₯ππ π‘π  3. limπ₯βππ(π₯)=π(π) Use the three conditions of continuity to decide if π(π₯) is continuous at π₯=3. π(π₯)={2π₯ β 1 π₯<3β5 π₯=3π₯2β 4 π₯>3 1.

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# FIU MAC 2233 - Final Exam Review

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