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Final Exam MAC 2233 Review 1. limπ‘₯β†’15(βˆ’30)= A. 15 B. 30 C. -15 D. -450 E. None of the above 2. Which statements is/are true? I. 𝑓(π‘₯)=7π‘₯3βˆ’ 5π‘₯2+ 3π‘₯ βˆ’5 is continuous at π‘₯=βˆ’1 II. limπ‘₯β†’ 2(π‘₯βˆ’2)(π‘₯βˆ’2)(π‘₯+2) π‘‘π‘œπ‘’π‘  π‘›π‘œπ‘‘ 𝑒π‘₯𝑖𝑠𝑑 III. The horizontal asymptote of 𝑦=π‘₯2π‘₯βˆ’1 is π‘₯=1. A. I and II B. I and III C. II only D. I only E. All of them 3. Differentiate 𝑦=3√π‘₯ βˆ’ 7π‘₯5+ 10π‘₯βˆ’ 8. A. 𝑦′=32√π‘₯βˆ’ 35π‘₯4+ 10 B. 𝑦′=6√π‘₯βˆ’ 7π‘₯4+ 5π‘₯βˆ’ 1 C. 𝑦′=2√π‘₯βˆ’ 35π‘₯4βˆ’ 8 D. 𝑦′=32√π‘₯βˆ’ 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 𝑓(π‘₯) continuous on all real numbers. 𝑓(π‘₯)={2π‘₯2βˆ’ 3π‘₯ + 1 π‘₯<βˆ’2𝐴π‘₯βˆ’ 7 π‘₯β‰₯βˆ’2 A. 𝐴=βˆ’2 B. 𝐴=15 C. 𝐴=βˆ’10 D. 𝐴=βˆ’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 𝐢(π‘₯)=4π‘₯+ 8,000 when x units are produced. A. 𝐢ξͺ§β€²(π‘₯)=4 B. 𝐢ξͺ§β€²(π‘₯)=βˆ’8,000π‘₯2 C. 𝐢ξͺ§β€²(π‘₯)=0 D. 𝐢ξͺ§β€²(π‘₯)=4 βˆ’ 8,000π‘₯2 E. None of the above 9. Differentiate 𝑦=4lnπ‘₯6 A. 𝑦′=24lnπ‘₯5 B. 𝑦′=24π‘₯5 C. 𝑦′=4ln6π‘₯5 D. 𝑦′=24π‘₯ E. None of the above 10. Differentiate 𝑦=π‘₯3𝑒2π‘₯. Factor completely. A. 𝑦′=6π‘₯2𝑒2π‘₯ B. 𝑦′=π‘₯2𝑒2π‘₯(3 + π‘₯) C. 𝑦′=π‘₯2(3𝑒2π‘₯+ 2𝑒2π‘₯βˆ’1) D. 𝑦′=π‘₯2𝑒2π‘₯(3 + 2π‘₯) E. None of the above 11. Which of the following is/are true? I. (𝐹𝑆)β€²=𝐹′𝑆 βˆ’ 𝐹𝑆′ II. 𝑦=ln3π‘₯ →𝑦′=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 𝑦=𝑒π‘₯2βˆ’5π‘₯+4 . A. 𝑦′=𝑒π‘₯2βˆ’5π‘₯+4 B. 𝑦′=𝑒2π‘₯βˆ’5 C. 𝑦′=𝑒π‘₯(2π‘₯ βˆ’ 5)(π‘₯2βˆ’ 5π‘₯+ 4) D. 𝑦′=(2π‘₯ βˆ’ 5)𝑒π‘₯2βˆ’5π‘₯+4 E. None of the above 13. Differentiate 𝑦=3π‘₯(2π‘₯2βˆ’ 5)3. Factorize Completely. A. 𝑦′=3(2π‘₯2βˆ’ 5)2(14π‘₯2βˆ’ 5) B. 𝑦′=36π‘₯(2π‘₯2βˆ’ 5)2 C. 𝑦′=3(2π‘₯2βˆ’ 5)2(2π‘₯2+ 3π‘₯βˆ’ 5) D. 𝑦′=12π‘₯2(2π‘₯2βˆ’ 5)(4π‘₯) E. None of the above14. Use the graph of 𝑦=𝑓(π‘₯) to find the interval on which 𝑓′(π‘₯)>0. A. (βˆ’βˆž,βˆ’3)βˆͺ (0,2) βˆͺ (3.5,∞) B. (βˆ’βˆž,βˆ’2) βˆͺ (1,3) C. (βˆ’2,1)βˆͺ (3,∞) D. (βˆ’3,0) βˆͺ (2,3.5) E. None of the above 15. Find all inflection points of 𝑦=3π‘₯5βˆ’ 10π‘₯4. A. (0,0) π‘Žπ‘›π‘‘ (2,βˆ’64) B. (0,0) π‘Žπ‘›π‘‘ (83,βˆ’101.1) C. (2,βˆ’64) D. (0,0) E. None of the above 16. The graph of 𝑓(π‘₯) is given below. Find the intervals where 𝑓′′(π‘₯)<0. A. (βˆ’βˆž,βˆ’1.5) βˆͺ (βˆ’0.5,∞) B. (βˆ’βˆž,βˆ’1) C. (βˆ’1.5,βˆ’1) βˆͺ (βˆ’1,βˆ’0.5) D. (βˆ’1,∞) E. None of the above 17. 𝑓′(π‘₯)=0 π‘Žπ‘‘ π‘₯=1 π‘Žπ‘›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: 𝑓′′(π‘₯)=√2π‘₯βˆ’ 1 βˆ’ 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‘Žπ‘›π‘‘ 35.5 B. 81 π‘Žπ‘›π‘‘ 10 C. 40 π‘Žπ‘›π‘‘ βˆ’ 31 D. βˆ’15.5 π‘Žπ‘›π‘‘ 55.5 E. None of the above19. Evaluate ∫3π‘₯βˆ’ 242𝑑π‘₯ A. 2 B. 6 C. 14 D. 18 E. None of the above 20. Evaluate: ∫(π‘’πœ‹βˆ’ 6π‘₯7βˆ’ ln(π‘₯))341212 A. 12 B. 34 C. 0 D. 12 βˆ’ πœ‹ E. None of the above 21. Which of these is/are true? (Choose all that apply) If 𝑓 π‘Žπ‘›π‘‘ 𝑔 are continuous on [π‘Ž,𝑏], and 𝐹 is an antiderivative of 𝑓, then I.βˆ«π‘“(π‘₯)𝑑π‘₯=𝐹(𝑏)+ 𝐹(π‘Ž)π‘π‘Ž II. βˆ«π‘“(π‘₯)Β± 𝑔(π‘₯)𝑑π‘₯= βˆ«π‘“(π‘₯)𝑑π‘₯Β± βˆ«π‘”(π‘₯)𝑑π‘₯ III. ∫1π‘₯𝑑π‘₯=ln|π‘₯|+ 𝑐 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‘₯βˆ’25√π‘₯βˆ’5= c. limπ‘₯β†’βˆž3π‘₯2βˆ’7π‘₯+16π‘₯2+8= d. limπ‘₯β†’βˆ’βˆž(βˆ’5π‘₯5+ 6π‘₯βˆ’ 10)= e. limπ‘₯β†’βˆ’4+π‘₯βˆ’3π‘₯+4=23. Use the graph to answer the following questions. a. limπ‘₯β†’βˆ’6𝑓(π‘₯)= b. limπ‘₯β†’βˆžπ‘“(π‘₯)= c. limπ‘₯β†’βˆ’1βˆ’π‘“(π‘₯)= d. Is 𝑓(π‘₯) continuous at π‘₯=βˆ’2? Why or why not? e. Is 𝑓(π‘₯) continuous at π‘₯=1? Why or why not? 24. Find the horizontal and vertical asymptotes of 𝑓(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. 𝑓(π‘₯)=π‘₯βˆ’ 5π‘₯2βˆ’ 4π‘₯ βˆ’ 5 25. Use the definition of the derivative (four-step process in book) to differentiate 𝑓(π‘₯)=5π‘₯. Any other method of finding the derivative will result in a zero. 26. Let 𝑦=2π‘₯3βˆ’ 3π‘₯2βˆ’ 12π‘₯ + 4. a. Find the equation of the tangent line in slope intercept form when π‘₯=βˆ’1. b. Find the x-values where the tangent line is horizontal. 27. Differentiate. a. 𝑦=√π‘₯43βˆ’5π‘₯2+ 7π‘₯βˆ’ 10π‘₯4 b. 𝑦=5π‘₯2βˆ’7π‘₯π‘₯ 28. A function 𝑓(π‘₯) is continuous at a value π‘₯=𝑐 if the following conditions are met: 1. 𝑓(𝑐) is defined 2. limπ‘₯→𝑐𝑓(π‘₯) 𝑒π‘₯𝑖𝑠𝑑𝑠 3. limπ‘₯→𝑐𝑓(π‘₯)=𝑓(𝑐) Use the three conditions of continuity to decide if 𝑓(π‘₯) is continuous at π‘₯=3. 𝑓(π‘₯)={2π‘₯ βˆ’ 1 π‘₯<3βˆ’5 π‘₯=3π‘₯2βˆ’ 4 π‘₯>3 1.

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FIU MAC 2233 - Final Exam Review

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