Math 4R Final Review 2023-2024 PDF

Name:_________________________ Per_________ Math 4R Final Review HW #FR1: WS - Final Review - # 1 - 11 HW # FR2: WS - Final Review - # 12 - 21 HW # FR3: WS - Final Review -...

Name:_________________________ Per_________ Math 4R Final Review HW #FR1: WS - Final Review - # 1 - 11 HW # FR2: WS - Final Review - # 12 - 21 HW # FR3: WS - Final Review - # 22 – 32 Final Exam Dates – Monday, June 10th and Tuesday, June 11th in class!! MATH 4R Name________________________ Final Exam Review ln 𝑥𝑥 2 +2 𝑒𝑒 5 1. Simplify: (a) ln 𝑒𝑒 10 (b) 𝑒𝑒 (c) ln 10 2. Evaluate the given limits: 4 (a) lim √𝑥𝑥 2 − 9 𝑥𝑥→5 𝑥𝑥 2 −3𝑥𝑥+2 (b) lim 𝑥𝑥→2 𝑥𝑥 2 −4 3. Find f ′(x) for each function given: (a) 𝑓𝑓 (𝑥𝑥 ) = (1 − 2𝑥𝑥)(𝑥𝑥 − 𝑥𝑥 2 ) (b) 𝑓𝑓 (𝑥𝑥 ) = (2𝑥𝑥 3 + cos 𝑥𝑥 + 7)30 (1−2𝑥𝑥 2 ) (c) 𝑓𝑓(𝑥𝑥 ) = 3𝑥𝑥 3 4. For the function given below, find the average rate of change for [−3.0, 2.2]. State whether the function is increasing or decreasing. 5. Suppose $1000 is invested in an account paying a nominal annual interest rate of 5.5%. How much is in the account after 8 years if the interest is compounded: (a) Annually? (b) Quarterly? (c) Monthly? (d) Daily? (e) Continuously? 6. Given 𝑟𝑟 = 3 + 3 cos 𝜃𝜃, state the name of the classical curve, show all information necessary for graphing, and graph the curve. 𝑟𝑟 = 3 + 3 cos 𝜃𝜃 Name:______________ Symmetry: _______________ 7. Given the polynomial 𝑓𝑓 (𝑥𝑥 ) = −𝑥𝑥 3 + 3𝑥𝑥 2 + 2𝑥𝑥 − 3, (a) Determine the end behavior of 𝑓𝑓(𝑥𝑥). Left End: As 𝑥𝑥 →______, _____________ and Right End: As 𝑥𝑥 →______, _____________ (b) Find the coordinates of all relative and absolute extrema. You may round coordinates to the nearest hundredth, if necessary. Relative Maximum: ________________ Relative Minimum: ________________ Absolute Maximum: ________________ Absolute Minimum: ________________ 2𝑥𝑥 − 1, 𝑥𝑥 < 2 8. Let 𝑓𝑓 (𝑥𝑥 ) = 4, 𝑥𝑥 = 2. 2 𝑥𝑥 − 7, 𝑥𝑥 > 2 (a) Graph 𝑓𝑓 (𝑥𝑥 ) on the axes below. (b) Find: (i) lim 𝑓𝑓(𝑥𝑥) = __________ 𝑥𝑥→2− (ii) lim 𝑓𝑓(𝑥𝑥) = __________ 𝑥𝑥→2+ (iii) lim 𝑓𝑓(𝑥𝑥) = __________ 𝑥𝑥→2 (iv) 𝑓𝑓 (2) = __________ 9. (a) Solve for 𝑥𝑥 in the interval 0 ≤ 𝑥𝑥 ≤ 2𝜋𝜋: 3 cos 𝑥𝑥 + 3 = 2(1 − cos2 𝑥𝑥) (b) Express in simplest form: (1 − sin2 𝑥𝑥)(sec 𝑥𝑥) 10. Matt bought a new car at a cost of $25,000. The car depreciates approximately 15% of its value each year. (a) Complete the table to model this situation. (b) Write an equation to model this situation. (c) How much will the car be worth after 5 years? (d) Matt wants to sell the car when it is worth less than $10,000. After how many years should he sell his car? 11. Do a complete analysis and graph the given function. 2𝑥𝑥 − 6 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 2 − 3𝑥𝑥 Intercepts Asymptotes 𝑥𝑥-int.:____________ Vert.:_________________ 𝑦𝑦-int.:____________ Horiz.:________________ Hole:_________________ Extent: End Behavior: Domain:__________________________ Range:___________________________ Maximum(s) Minimum(s) 12. Identify each of the following functions as even, odd or neither. (a) 𝑓𝑓 (𝑥𝑥) = 𝑥𝑥 2 − 4𝑥𝑥 + 3 (b) 𝑦𝑦 = 𝑥𝑥 4 − 5𝑥𝑥 2 + 4 (c) 𝑦𝑦 = 𝑥𝑥 3 − 7𝑥𝑥 2 + 14𝑥𝑥 − 7 13. For each rational function given below, state the equations of any vertical asymptotes or horizontal asymptotes and find the coordinates of any holes. 𝑥𝑥 2 −3𝑥𝑥+2 2𝑥𝑥 (a) 𝑓𝑓 (𝑥𝑥 ) = (b) 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 2 −4 𝑥𝑥 2 +𝑥𝑥−6 Vert. Asymp.: _________________ Vert. Asymp.: _________________ Holes: _________________ Holes: _________________ Horiz. Asymp.: _________________ Horiz. Asymp.: _________________ 14. Identify each equation: (a) 𝑟𝑟 = 5 cos 𝜃𝜃 + 5 (b) 𝑟𝑟 = 2 sin 𝜃𝜃 + 1 (c) 𝑟𝑟 = 3 sin 5𝜃𝜃 (d) 𝑟𝑟 = 3 sin 2𝜃𝜃 15. Solve the following exponential equations: (a) 3 + 8.5𝑥𝑥 = 2 (b) 7.5𝑥𝑥 + 2 = 23 16. Answer the following questions for the function given in the table below. (a) Find the average rate of change for [−0.4, 0.4]. Is the function increasing or decreasing? (b) Estimate the instantaneous rates of change at 𝑥𝑥 = 0.4. 17. Evaluate the following: (a) lim− 𝑔𝑔(𝑡𝑡) = 𝑡𝑡→4 (b) lim+ 𝑔𝑔(𝑡𝑡) = 𝑡𝑡→4 (c) lim 𝑔𝑔(𝑡𝑡) = 𝑡𝑡→−4 (d) 𝑔𝑔(−4) = 18. Mr. M. Bezzel is gradually withdrawing money from his grandmother’s safe which originally held $40,000. He takes out 10% of the balance each year and no other money is added or withdrawn. (a) Create a table showing the amount in the safe each year for the first 5 years. Time Amount 0 1 2 3 4 5 (b) Write the exponential equation that can be used to calculate directly the amount remaining in any year. (c) Use your equation to compute the amount in the safe at the end of 20 years. (d) In approximately how many years will the amount of money in the safe be reduced to $25000? Solve either graphically or algebraically and show all work. 19. Answer the following questions for the function 𝑓𝑓 (𝑥𝑥 ) = 4𝑥𝑥 2 − 3𝑥𝑥. (a) Use the definition of derivative to find 𝑓𝑓′(𝑥𝑥). (b) Use your answer to (a) to find the slope of the curve at 𝑥𝑥 = 2. (c) Write an equation of the tangent line at 𝑥𝑥 = 2. 20. Given the polynomial 𝑓𝑓 (𝑥𝑥 ) = 2𝑥𝑥 2 + 3𝑥𝑥 − 5, (a) Determine the end behavior of 𝑓𝑓(𝑥𝑥). Left End: As 𝑥𝑥 →______, _____________ and Right End: As 𝑥𝑥 →______, _____________ (b) Find the coordinates of all relative and absolute extrema. You may round coordinates to the nearest hundredth, if necessary. Relative Maximum: ________________ Relative Minimum: ________________ Absolute Maximum: ________________ Absolute Minimum: ________________ 21. (a) Solve for 𝑥𝑥 in the interval 0 ≤ 𝑥𝑥 ≤ 2𝜋𝜋: 2(1 − cos2 𝑥𝑥) = 1 − cos 𝑥𝑥 (b) Express in simplest form: 4 − 4 cos 2 𝑥𝑥 4 sin 𝑥𝑥 22. Identify each of the following functions as even, odd or neither. (a) 𝑓𝑓 (𝑥𝑥) = 4𝑥𝑥 3 − 3𝑥𝑥 (b) 𝑦𝑦 = 2𝑥𝑥 2 + 5𝑥𝑥 23. Solve the following equations for all values in the interval from 0 ≤ 𝑥𝑥 ≤ 2𝜋𝜋. (a) 2 cos 𝑥𝑥 + √3 = 0 (b) 3 tan 𝑥𝑥 − √3 = 0 24. Luigi invested $7,200 for 20 years at 7% compounded annually and his wife Izabella invested $5,500 at 9% compounded continuously. Support your answers by computing the two balances. (a) Who had the most money in 5 years? Luigi:_____________ Izabella: _____________ Answer: _____________ (b) Estimate, to the nearest year, the year the amounts were equal. 25. Evaluate the given limits: (a) lim (𝑥𝑥 3 + 3𝑥𝑥 2 − 2𝑥𝑥 − 17) = 𝑥𝑥→1 𝑥𝑥 2 −1 (b) lim = 𝑥𝑥→−1 3𝑥𝑥 2 −2𝑥𝑥−15 𝑥𝑥 2 −3𝑥𝑥−18 (c) lim = 𝑥𝑥→−3 𝑥𝑥 2 +𝑥𝑥−6 26. Answer the questions below, based on the given graph. (a) Name all points for which the function is negative. (b) Name all points for which the derivative is negative. (c) Name the point for which the derivative is the greatest positive. (d) Name all points for which 𝒚𝒚 is positive and the derivative is negative. (e) Name all points for which the function is positive. (f) Name all points for which the derivative is zero and is a maximum. 27. Find f ′(x) for each function given: (a) 𝑓𝑓 (𝑥𝑥 ) = (𝑥𝑥 2 − 2𝑥𝑥)(5𝑥𝑥 + 1) (b) 𝑓𝑓 (𝑥𝑥 ) = (𝑥𝑥 2 + sin 𝑥𝑥 − 3)60 (𝑥𝑥 2 +3𝑥𝑥+1) (c) 𝑓𝑓(𝑥𝑥 ) = 𝑥𝑥 3 28. Given the polynomial 𝑓𝑓 (𝑥𝑥 ) = 2𝑥𝑥 2 − 6𝑥𝑥 − 8, (a) Determine the end behavior of 𝑓𝑓(𝑥𝑥). Left End: As 𝑥𝑥 →______, _____________ and Right End: As 𝑥𝑥 →______, _____________ (b) Find the coordinates of all relative and absolute extrema. You may round coordinates to the nearest hundredth, if necessary. Relative Maximum: ________________ Relative Minimum: ________________ Absolute Maximum: ________________ Absolute Minimum: ________________ 29. Given 𝑟𝑟 = 2 + 5 sin 𝜃𝜃, state the name of the classical curve, show all information necessary for graphing, and graph the curve. 𝑟𝑟 = 2 + 5 sin 𝜃𝜃 Name:______________ Symmetry: _______________ 30. Do a complete analysis and graph the given function. 2𝑥𝑥 3 − 2𝑥𝑥 2 𝑓𝑓(𝑥𝑥) = 3 𝑥𝑥 − 9𝑥𝑥 Intercepts Asymptotes 𝑥𝑥-int.:____________ Vert.:_________________ 𝑦𝑦-int.:____________ Horiz.:________________ Hole:_________________ Extent: End Behavior: Domain:__________________________ Range:___________________________ Maximum(s) Minimum(s) 31. Use the graph below to find the values of each limit. (a) lim 𝑓𝑓(𝑥𝑥) = (b) lim 𝑓𝑓(𝑥𝑥) = (c) lim 𝑓𝑓(𝑥𝑥) = 𝑥𝑥→1+ 𝑥𝑥→1− 𝑥𝑥→1 (d) lim 𝑓𝑓(𝑥𝑥) = (e) lim 𝑓𝑓(𝑥𝑥) = (f) lim 𝑓𝑓(𝑥𝑥) = 𝑥𝑥→−2− 𝑥𝑥→−2+ 𝑥𝑥→−2 (g) lim 𝑓𝑓(𝑥𝑥) = (h) lim 𝑓𝑓(𝑥𝑥) = (i) lim 𝑓𝑓(𝑥𝑥) = 𝑥𝑥→−4+ 𝑥𝑥→−4− 𝑥𝑥→−4 (j) lim 𝑓𝑓(𝑥𝑥) = (k) lim 𝑓𝑓(𝑥𝑥) = (l) lim 𝑓𝑓(𝑥𝑥) = 𝑥𝑥→3+ 𝑥𝑥→3− 𝑥𝑥→3 (m) lim 𝑓𝑓(𝑥𝑥) = (n) lim 𝑓𝑓(𝑥𝑥) = 𝑥𝑥→+∞ 𝑥𝑥→−∞ 32. Answer the following questions for the function 𝑓𝑓 (𝑥𝑥 ) = 2𝑥𝑥 2 + 5𝑥𝑥. (a) Use the definition of derivative to find 𝑓𝑓′(𝑥𝑥). (b) Use your answer to (a) to find the slope of the curve at 𝑥𝑥 = 1. (c) Write an equation of the tangent line at 𝑥𝑥 = 1.