Final Exam MAC 2233 Review 1. lim𝑥→15(−30)= A. 15 B. 30 C. -15 D. -450 E. None of the above 2. Which statements is/are true? I. 𝑓(𝑥)=7𝑥3− 5𝑥2+ 3𝑥 −5 is continuous at 𝑥=−1 II. lim𝑥→ 2(𝑥−2)(𝑥−2)(𝑥+2) 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡 III. The horizontal asymptote of 𝑦=𝑥2𝑥−1 is 𝑥=1. A. I and II B. I and III C. II only D. I only E. All of them 3. Differentiate 𝑦=3√𝑥 − 7𝑥5+ 10𝑥− 8. A. 𝑦′=32√𝑥− 35𝑥4+ 10 B. 𝑦′=6√𝑥− 7𝑥4+ 5𝑥− 1 C. 𝑦′=2√𝑥− 35𝑥4− 8 D. 𝑦′=32√𝑥− 7𝑥4+ 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(𝑥) continuous on all real numbers. 𝑓(𝑥)={2𝑥2− 3𝑥 + 1 𝑥<−2𝐴𝑥− 7 𝑥≥−2 A. 𝐴=−2 B. 𝐴=15 C. 𝐴=−10 D. 𝐴=−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 𝐶(𝑥)=4𝑥+ 8,000 when x units are produced. A. 𝐶′(𝑥)=4 B. 𝐶′(𝑥)=−8,000𝑥2 C. 𝐶′(𝑥)=0 D. 𝐶′(𝑥)=4 − 8,000𝑥2 E. None of the above 9. Differentiate 𝑦=4ln𝑥6 A. 𝑦′=24ln𝑥5 B. 𝑦′=24𝑥5 C. 𝑦′=4ln6𝑥5 D. 𝑦′=24𝑥 E. None of the above 10. Differentiate 𝑦=𝑥3𝑒2𝑥. Factor completely. A. 𝑦′=6𝑥2𝑒2𝑥 B. 𝑦′=𝑥2𝑒2𝑥(3 + 𝑥) C. 𝑦′=𝑥2(3𝑒2𝑥+ 2𝑒2𝑥−1) D. 𝑦′=𝑥2𝑒2𝑥(3 + 2𝑥) E. None of the above 11. Which of the following is/are true? I. (𝐹𝑆)′=𝐹′𝑆 − 𝐹𝑆′ II. 𝑦=ln3𝑥 →𝑦′=3𝑥 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=𝑒𝑥2−5𝑥+4 . A. 𝑦′=𝑒𝑥2−5𝑥+4 B. 𝑦′=𝑒2𝑥−5 C. 𝑦′=𝑒𝑥(2𝑥 − 5)(𝑥2− 5𝑥+ 4) D. 𝑦′=(2𝑥 − 5)𝑒𝑥2−5𝑥+4 E. None of the above 13. Differentiate 𝑦=3𝑥(2𝑥2− 5)3. Factorize Completely. A. 𝑦′=3(2𝑥2− 5)2(14𝑥2− 5) B. 𝑦′=36𝑥(2𝑥2− 5)2 C. 𝑦′=3(2𝑥2− 5)2(2𝑥2+ 3𝑥− 5) D. 𝑦′=12𝑥2(2𝑥2− 5)(4𝑥) E. None of the above14. Use the graph of 𝑦=𝑓(𝑥) to find the interval on which 𝑓′(𝑥)>0. A. (−∞,−3)∪ (0,2) ∪ (3.5,∞) B. (−∞,−2) ∪ (1,3) C. (−2,1)∪ (3,∞) D. (−3,0) ∪ (2,3.5) E. None of the above 15. Find all inflection points of 𝑦=3𝑥5− 10𝑥4. A. (0,0) 𝑎𝑛𝑑 (2,−64) B. (0,0) 𝑎𝑛𝑑 (83,−101.1) C. (2,−64) D. (0,0) E. None of the above 16. The graph of 𝑓(𝑥) is given below. Find the intervals where 𝑓′′(𝑥)<0. A. (−∞,−1.5) ∪ (−0.5,∞) B. (−∞,−1) C. (−1.5,−1) ∪ (−1,−0.5) D. (−1,∞) E. None of the above 17. 𝑓′(𝑥)=0 𝑎𝑡 𝑥=1 𝑎𝑛𝑑 𝑥=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′′(𝑥)=√2𝑥− 1 − 2. A. Local max at 𝑥=1 and local min at 𝑥=5 B. Local max at 𝑥=1 and 𝑥=5 C. Local min at 𝑥=1 and 𝑥=5 D. Local max at 𝑥=5 and local min at 𝑥=1 E. None of the above 18. Find two numbers whose difference is 71 and whose product is a minimum. A. 35.5 𝑎𝑛𝑑 35.5 B. 81 𝑎𝑛𝑑 10 C. 40 𝑎𝑛𝑑 − 31 D. −15.5 𝑎𝑛𝑑 55.5 E. None of the above19. Evaluate ∫3𝑥− 242𝑑𝑥 A. 2 B. 6 C. 14 D. 18 E. None of the above 20. Evaluate: ∫(𝑒𝜋− 6𝑥7− ln(𝑥))341212 A. 12 B. 34 C. 0 D. 12 − 𝜋 E. None of the above 21. Which of these is/are true? (Choose all that apply) If 𝑓 𝑎𝑛𝑑 𝑔 are continuous on [𝑎,𝑏], and 𝐹 is an antiderivative of 𝑓, then I.∫𝑓(𝑥)𝑑𝑥=𝐹(𝑏)+ 𝐹(𝑎)𝑏𝑎 II. ∫𝑓(𝑥)± 𝑔(𝑥)𝑑𝑥= ∫𝑓(𝑥)𝑑𝑥± ∫𝑔(𝑥)𝑑𝑥 III. ∫1𝑥𝑑𝑥=ln|𝑥|+ 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. lim𝑥→72𝑥𝑥+1= b. lim𝑥→25𝑥−25√𝑥−5= c. lim𝑥→∞3𝑥2−7𝑥+16𝑥2+8= d. lim𝑥→−∞(−5𝑥5+ 6𝑥− 10)= e. lim𝑥→−4+𝑥−3𝑥+4=23. Use the graph to answer the following questions. a. lim𝑥→−6𝑓(𝑥)= b. lim𝑥→∞𝑓(𝑥)= c. lim𝑥→−1−𝑓(𝑥)= d. Is 𝑓(𝑥) continuous at 𝑥=−2? Why or why not? e. Is 𝑓(𝑥) continuous at 𝑥=1? Why or why not? 24. Find the horizontal and vertical asymptotes of 𝑓(𝑥) 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(𝑥)=𝑥− 5𝑥2− 4𝑥 − 5 25. Use the definition of the derivative (four-step process in book) to differentiate 𝑓(𝑥)=5𝑥. Any other method of finding the derivative will result in a zero. 26. Let 𝑦=2𝑥3− 3𝑥2− 12𝑥 + 4. a. Find the equation of the tangent line in slope intercept form when 𝑥=−1. b. Find the x-values where the tangent line is horizontal. 27. Differentiate. a. 𝑦=√𝑥43−5𝑥2+ 7𝑥− 10𝑥4 b. 𝑦=5𝑥2−7𝑥𝑥 28. A function 𝑓(𝑥) is continuous at a value 𝑥=𝑐 if the following conditions are met: 1. 𝑓(𝑐) is defined 2. lim𝑥→𝑐𝑓(𝑥) 𝑒𝑥𝑖𝑠𝑡𝑠 3. lim𝑥→𝑐𝑓(𝑥)=𝑓(𝑐) Use the three conditions of continuity to decide if 𝑓(𝑥) is continuous at 𝑥=3. 𝑓(𝑥)={2𝑥 − 1 𝑥<3−5 𝑥=3𝑥2− 4 𝑥>3 1.