Presentation on theme: "Welcome to the Webinar “Live” Review for Final Exam (MAC 1105)"— Presentation transcript:
1Welcome to the Webinar “Live” Review for Final Exam (MAC 1105) We will start promptly at 8:00 pmEveryone will be placed on muteI am happy that you are able to join us
2Important NoteThe following slides present sample problems for Chapter 6 only.Use the slides from Tests1-3 to review the previous material.To be prepared for the test, you must review the material from all the assigned sections throughout the term.
3Which of the following is the graph of f(x) = (1/3)x ? (Each tic-mark is one unit.) a b.For f(x) = bxb > 1, function is increasing; 0 < b < 1, function is decreasing(Review definition of exponential function in section 6.1)Answer: b
4The range of the function f(x) = 3x is: a. x can be any real numberb. y can be any real numberc. x > 0d. y > 0Domain: Possible values of the input.Range: Possible values of the output.Base b for the exponential function: b > 0 and b 1So, we know that bx > 0.Answer is d Confirm with graph:
5State any transformations for f(x) = 2x+2 – 1, the horizontal asymptote, and the y-intercept. Horizontal Translation: left 2 unitsVertical Translation: down 1 unitHorizontal Asymptote: Since graph was shifted 1 unit down,the horizontal asymptote is y = -1.y-intercept let x = 0 and solve for yf(0) = 2(0+2) – 1 = 22 – 1 = 3
6Without graphing nor performing calculations, determine if the function p(t) = 8(1.7)t is increasing or decreasing, and find its vertical intercept.f(x) = cbxb = 1.7Since b > 1, this function is increasingThe y-intercept of f(x) = cbx is (0, c).Since c = 8, the vertical intercept is (0, 8)
7[Make sure to distribute correctly: 2(2x + 3) = 4x + 6] Solve 42x+3 = 8, by equating the bases.42x+3 = 8(22)2x+3 = 2324x+6 = 23[Make sure to distribute correctly: 2(2x + 3) = 4x + 6]Solving the equation:4x + 6 = 3x = -3/4
8logk r = t is equivalent to kr = t True b. FalseHint: Start at the base and “move counterclockwise”logk r = tlogk r = t is equivalent to kt = rAnswer: FalseNumeric verificationlog5 25 = 2 is equivalent to 52 = 25
9Solve algebraically for x : logx 4 = 1/2 In exponential form: x1/2 = 4We know x1/2 is equivalent toSo, x = 16 because (16)1/2 = 4
10The x-intercept implies that y = 0. So, we have logb x = 0 If y = logb x, for any base b (b > 0, b 1) the x-intercept is (1, 0).a. True b. FalseThe x-intercept implies that y = 0.So, we have logb x = 0In exponential form: b0 = xBy definition, for any number a not equal to 0, a0 = 1.Therefore, if b0 = x, then x = 1So, we have (1, 0)Answer: True(Review definition of Logarithmic Function in Section 6.2.)
11Simplify without a calculator: log3 3 a. 0 b. 1 c. 3 d. UndefinedApplying the basic property logb b = 1, we know the answer is b.Verifying: log3 3 = y means that 3y = 3, thus y must equal 1.
12Your time to work! Poll Question log mn is equivalent to m log n (log m)(log n)(log m)/(log n)Your time to work!Poll Question
13log mn is equivalent tom log nn log m(log m)(log n)(log m)/(log n)Answer: b (power property)
14True or False: The following is the graph of y = log (x – 4) (Each tic-mark is one unit.) y = log (x – 4) implies a horizontal translation 4 units right; not a vertical translation.Answer: FalseGraph of y = log(x) isAnswer should have been
15True or False:The function y = log8x is the inverse function of y = x8 .Answer: FalseThe inverse function of y = log8x is y = 8x .y = log8xx = log8ylog8y = x8x = yObserve that in y = x8 the exponent is not a variable!
16Hint: What formula is helpful here? Use your graphing calculator to evaluate log864. Round to 3 decimal places where needed.0.50.9031.6312Your time to work!Poll QuestionHint: What formula is helpful here?
17Use your graphing calculator to calculate log864 Use your graphing calculator to calculate log864. Round to 3 decimal places where needed.0.50.9031.6312Answer: 2log864 = log(64)log (8)
18ln e is equivalent to logee = 1 Solve algebraically for x: 3e2x + 3 = 123e2x + 3 = 12e2x = 4ln e2x + 3 = ln 4(2x + 3) ln e = ln 4ln e is equivalent to logee = 12x + 3 = ln 4x = ln 4 – 32
19True or False:ln (1/e-q) = q1/e-q is equivalent to eqTherefore,= ln (eq)= q ln (e)ln (e) is equivalent to logee = 1= q(1)= q True
20Solve the equation ln (2x – 1) = 3 and approximate your answer to 4 decimal places.b c dln (2x – 1) = 3This is equivalent to loge(2x – 1) = 3In exponential form, e3 = 2x – 12x = e3 + 12x =x = Answer is c.
21Rewrite as a single logarithm: logb (m) + 5 logb (n) - logb (r) = logb (mn5) - logb (r)= logb
23Suki bought a computer for $500 Suki bought a computer for $500. She estimates that the computer will depreciate so that each year its value will be ¾ of its worth for the previous year.Write a function V(t) that models this situation, where t represents number of years.Find Suki’s computer value after 2 years.If the current value of this computer is $160, approximate its age in years.Exponential growth and decay: P(t) = P0btThe initial value, V0 = 500, decay factor = ¾,so V(t) = 500(3/4)tb. V(2) = 500(3/4)2 = $281.25
24c. If the current value of this computer is $160, approximate its age in years.We know V(t) = 500(3/4)tSolving Algebraically:500(3/4)t = 160(3/4)t = 160/500(3/4)t = 0.32ln(3/4)t = ln t ln(3/4) = ln 0.32t = ln 0.32ln(3/4)t = 3.96 4 years
25c. If the current value of this computer is $160, approximate its age in years.Solving by Graphing: (3/4)t = 160Y1 = 500(3/4)xY2 = 160t = 3.96 4 yearsCaution: You need to find an appropriate window!
26Jeremy invested $1,000 in an account paying 1. 5% compounded monthly Jeremy invested $1,000 in an account paying 1.5% compounded monthly. Find the accumulated amount after 5 years.A =where P = 1000, r = 0.015, n = 12, and t = 5.A $(This formula will be provided on the test ) Careful with parentheses!
27NOTE: It is important that you follow the instructions on the test. For example, if asked for interval notation, write your answer using interval notation.If you are not asked for interval notation, then just write the values separated by commas (,) or semicolons (;) as specified on the question.If asked to write log(x) with parentheses, do not just write log x.So, read and follow instructions!