MAT 21A Lecture Notes - Lecture 3: Wilshire Grand Center, Logarithmic Form, Logarithm

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MAT 21A Lecture Notes - Lecture 2: Farad, Rate Function, Trigonometric Functions

MAT 21A Lecture Notes - Lecture 3: Wilshire Grand Center, Logarithmic Form, Logarithm

MAT 21A Lecture 4: MAT 21A Lecture Notes 10-05-2018

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Mat 21a - lecture #3 - 2. 1 rates of change and tangent lines to curves. If f is a one to one function, then the inverse of f is the function formed by interchanging dependent and independent variables for f. Most common inverse is the logarithmic function, which is an inverse of the exponential function. Properties of logarithmic functions : if b, m, and n are positive real numbers with b . 1 and p and x are real numbers, then. Log b m = log b n if and only if m=n. Calculus analyzes functions in detail, making it very important. Let f(t) be the distance traveled at time t. then, the average speed over [t 1 ,t 2 ]= Example: if you drove 30 miles in 45 minutes, to find the average rate of change in miles per hour, you would divide 30 miles by 0. 75 minutes to get an answer of 40 miles an hour.

Related Questions

Q1. Determine, without graphing, whether the quadratic function f(x) = x2 + 2x - 6 has a maximum value or a minimum value and then find that value. a. minimum; -7 b. minimum; -1 c. maximum; -1 d. maximum; -7 Q2. Find (f - g)(4) when f(x) = 5x2 + 6 and g(x) = x + 2. a. 80 b. 88 c. 84 d. -90 Q3. The graph of a function is given. Decide whether it is even, odd, or neither. a. even b. odd c. neither Q4. Find the average rate of change for the function f(x) = -3x2 - x from the values 5 to 6. a. 1/2 b. -2 c. -34 d. -1/6 Q5. Find the vertex and axis of symmetry of the graph of the function f(x) = 3x2 + 36x. a. (-6, -108); x = -6 b. (-6, 0); x = -6 c. (6, -108); x =6 d. (6, 0); x = 6 Q6. The graph of a function is given. Decide whether it is even, odd, or neither. a. even b. odd c. neither Q7. Use the graph to find the intervals on which it is increasing, decreasing, or constant. a. Increasing on (-3, -2) and (2, 4); decreasing on (-1, 1); constant on (-2, -1) and (1, 2) b. Decreasing on (-3, -2) and (2, 4); increasing on (-1, 1); constant on (-2, -1) and (1, 2) c. Decreasing on (-3, -2) and (2, 4); increasing on (-1, 1) d. Decreasing on (-3, -1) and (1, 4); increasing on (-2, 1) Q8. Graph the function F(x) = -5. State whether it is increasing, decreasing, or constant. a. b. c. d. Q9. Determine algebraically whether f(x) = -2x3 is even, odd, or neither. a. even b. odd c. neither Q10. Find the average rate of change for the function f(x) = 3/(x - 2) from the values 4 to 7. a. 7 b. 1/3 c. -3/10 d. 2 Q11. For the graph of the function y = f(x), find the absolute maximum and the absolute minimum, if it exists. a. Absolute maximum: f(4) = 7; Absolute minimum: f(1) = 2 b. Absolute maximum: f(4) = 7; Absolute minimum: none c. Absolute maximum: none; Absolute minimum: none d. Absolute maximum: none; Absolute minimum: f(1) = 2 Q12. The graph of a function f is given. Find the numbers, if any, at which f has a local minimum. What are the local minima? a. f has a local minimum at x = -2; the local minimum is 0 b. f has a local minimum at x = 0; the local minimum is 3 c. f has a local minimum at x = -2 and 2; the local minimum is 0 d. f has no local minimum Q13. Graph the function h(x) = -2x + 3. State whether it is increasing, decreasing, or constant. a. b. c. d. Q14. Find -f(x) when f(x) = -2x2 + 5x + 2. a. 2x2 - 5x + 2 b. -2x2 - 5x - 2 c. -2x2 - 5x + 2 d. 2x2 - 5x - 2 Q15. Determine algebraically whether f(x) = 1/x2 is even, odd, or neither. a. even b. odd c. neither Q16. Match the graph to one of the listed functions. a. f(x) = x2 - 8 b. f(x) = -x2 - 8x c. f(x) = x2 - 8x d. f(x) = -x2 - 8 Q17. In a certain city, the cost of a taxi ride is computed as follows: There is a fixed charge of $2.90 as soon as you get in the taxi, to which a charge of $1.70 per mile is added. Find an equation that can be used to determine the cost, C(x), of an x-mile taxi ride. a. C(x) = 1.70 + 2.90x b. C(x) = 3.10x c. C(x) = 4.60x d. C(x) = 2.90 + 1.70x Q18. Graph the function f(x) = -x2 + 3 by starting with the graph of y = x2 and using transformations (shifting, compressing, stretching, and/or reflection). a. b. c. d. Q19. Graph the function f(x) = x2 + 8x + 7 using its vertex, axis of symmetry, and intercepts. a. b. c. d. Q20. Determine where the function f(x) = -x2 + 8x - 7 is increasing and where it is decreasing. a. increasing on (4, â) and decreasing on (-â, 4) b. increasing on (-â, 4) and decreasing on (4, â) c. increasing on (9, â) and decreasing on (-â, 9) d. increasing on (-â, 9) and decreasing on (9, â) Q21. Find the domain of the rational function f(x) = (x + 9)/(x2 - 4x). a. {x|x â -2, x â 2} b. {x|x â -2, x â 2, x â -9} c. all real numbers d. {x|x â 0, x â 4} Q22. For the polynomial f(x) = 4(x - 5)(x - 6)3, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x-intercept. a. 5, multiplicity 1, touches x-axis; 6, multiplicity 3 b. -5, multiplicity 1, touches x-axis; -6, multiplicity 3 c. 5, multiplicity 1, crosses x-axis; 6, multiplicity 3, crosses x-axis d. -5, multiplicity 1, crosses x-axis; -6, multiplicity 3, crosses x-axis Q23. Find all zeros of the function f(x) = 2x4 + 3x3 + 16x2 + 27x - 18 and write the polynomial as a product of linear factors. a. f(x) = (2x - 1)(x + 2)(x + 3)(x - 3) b. f(x) = (2x + 1)(x - 2)(x + 3)(x - 3) c. f(x) = (2x - 1)(x + 2)(x + 3i)(x - 3i) d. f(x) = (2x + 1)(x - 2)(x + 3i)(x - 3i) Q24. For the polynomial f(x) = (1/5)x(x2 - 5), list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x-intercept. a. 0, multiplicity 1, touches x-axis; â5, multiplicity 1, touches x-axis; -â5, multiplicity 1, touches x-axis b. 0, multiplicity 1, crosses x-axis; â5, multiplicity 1, crosses x-axis; -â5, multiplicity 1, crosses x-axis c. â5, multiplicity 1, touches x-axis; -â5, multiplicity 1, touches x-axis d. 0, multiplicity 1 Q25. List the potential rational zeros of the polynomial function f(x) = x5 - 6x2 + 5x + 15. Do not find the zeros. a. Â± 1, Â± 1/5, Â± 1/3 Â± 1/15 b. Â± 1, Â± 5, Â± 3, Â± 15 c. Â± 1, Â± 5, Â± 3 d. Â± 1, Â± 1/5, Â± 1/3, Â± 1/15, Â± 5, Â± 3,Â± 15 Q26. The function f(x) = x4 - 5x2 - 36 has the zero -2i. Find the remaining zeros of the function. a. 2i, 6, -6 b. 2i, 3i, -3i c. 2i, 3, -3 d. 2i, 6i, -6i Q27. Use transformations of the graph of y = x4 to graph the function f(x) = 5x4. a. b. c. d. Q28. State whether the function f(x) = x(x - 9) is a polynomial function or not. If it is, give its degree. If it is not, tell why not. a. Yes; degree 2 b. No; it is a product c. Yes; degree 0 d. Yes; degree 1 Q29. Solve the inequality algebraically. Express the solution in interval notation. (x - 2)2(x + 9) < 0 a. (-â, -9) or (9, â) b. (-â, -9] c. (-â, -9) d. (-9, â) Q30. Use the Intermediate Value Theorem to determine whether the polynomial function f(x) = -4x4 - 9x2 + 4; has a zero in the interval [-1, 0]. a. f(-1) = -9 and f(0) = -4; no b. f(-1) = 9 and f(0) = 5; no c. f(-1) = 9 and f(0) = -4; yes d. f(-1) = -9 and f(0) = 4; yes Q31. Use the Factor Theorem to determine whether x + 5 is a factor of f(x) = 3x3 + 13x2 - 9x + 5. a. Yes b. No Q32. Find a bound on the real zeros of the polynomial function f(x) = x4 - 8x2 - 9. a. -17 and 17 b. -9 and 9 c. -18 and 18 d. -10 and 10 Q33. Find the x-intercepts of the graph of the function f(x) = (x - 3)(2x + 7)/(x2 + 9x - 8). a. (3, 0), (-7, 0) b. (-3, 0), (7/2, 0) c. (3, 0), (-7/2, 0) d. none Q34. Solve the inequality algebraically. Express the solution in interval notation. (9x - 5)/(x + 2) â¤ 8 a. (-2, 13] b. (-2, 21) c. (-2, 21] d. (-2, 13) Q35. A polynomial f(x) of degree 3 whose coefficients are real numbers has the zeros -4 and 4 - 5i. Find the remaining zeros of f. a. 4, -4 + 5i b. 4, 4 + 5i c. 4 + 5i d. -4 + 5i Q36. Find the intercepts of the function f(x) = x3 + 3x2 - 4x - 12. a. x-intercepts: -2, 2, 3; y-intercept: -12 b. x-intercepts: -3, -2, 2; y-intercept: -12 c. x-intercept: -3; y-intercept: -12 d. x-intercept: -2; y-intercept: -12 Q37. Find the real solutions of the equation 3x3 - x2 + 3x - 1 = 0. a. {-3, 1/3, -1} b. {1/3} c. {1/3, -1} d. {-3, -1/3, -1} Q38. Solve the inequality algebraically. Express the solution in interval notation. x3 â¥ 27 a. (-â, 3] b. (-â, -3] âª [3, â) c. [-3, 3] d. [3, â) Q39. Find the intercepts of the function f(x) = x2(x - 1)(x - 6). a. x-intercepts: 0, -1, -6; y-intercept: 0 b. x-intercepts: 0, 1, 6; y-intercept: 6 c. x-intercepts: 0, 1, 6; y-intercept: 0 d. x-intercepts: 0, -1, -6; y-intercept: 6 Q40. Find the domain of the rational function R(x) = (-3x2)/(x2 + 2x - 15). a. {x|x â 5, 3} b. {x|x â 5, -3} c. {x|x â - 15, 1} d. {x|x â -5, 3}

lemongiraffe898

PLEASE ANSWER ALL QUESTIONS OR DO NOT ANSWER.

1. The formula

A = P[1 + (r/n)]^nt

describes the accumulated value A of a sum of money P, the principal, after t years at annual percentage rate r (in decimal form) compounded n times a year. How long will it take $25,000 to grow to $500,000 at 9% annual interest compounded monthly

Also:

Solve the logarithmic equation algebraically: Ln (X+5) = Ln (Xâ1) â Ln (X+1)

2. What is the first derivative of the following:

f(x) = ln (2x / x+1)

it is Natural Log of 2x over x+1

3. Please read the article from Money Magazine, posted in our week 8 Module, (one of my favorite financial magazines) that talks about small drop in t-bills will result in a major return in your investments (second paragraph). Why do you think that is? The article is a bit old but the question here is quite relevant about what we cover this week.

Hint: It has to do with the exponential graphs.

ARTICLE: Photo provided

THE ECONOMY: "Though growth averaged 2.5% stock market. says Leuthold, who recently completed a study of the stock market's daily ns over the past 100 years. In fact the stock market's daily swings from 1992 through '95 were less than half the historical average. Recently, however, daily fluctuations BY MICHAEL SIVY prices have cruised frm one high tothe next the stock market's daily swings from 1992 MosT OF THE PAST FOUR YEARS. STOCK ups and dow from one high to the next h, silent ride of a Rolls-Royce. But since March, the market has been lurching at an annual rate invest additional cash in stocks when they're low. If volatili in the first half of oyour Donailene tem dieapsirm ke a broken-down jalopy If you're hesitant to in stock prices have doubled (see the chart be- n't worry. Consider putting some be a sign that we're getting close to a market so jumpy. do of your money in long-term bonds instead. close to a mak t 1996, it will slow likely to beat stocks over the This short-term forecast doesn't change presi- the fact that the five-year outlook for stocks is terrific-thanks to the enormous underlying JTTERY AGAIN Percentage of trading days strength of the U.S. econ- the Dow moved more than 1% omy and the fact that aging -Bonds are next 12 months, says Steve Leuthold, dent of the Leuthold Group. STOCK PRICES ARE GETTING later this year.â nnpolis cuipd AGAIN advisory firm. We couldn't agree more. In fact, the out look for long-term bonds is the best it's been for more omy and the fact that agi baby boomers are continu- ing to sock away more than S20 billion a month in "The Dow couldthan a year. reach 6000 by fall but wil Here's our analysis: We see the economy slowing from its current 2.5% growth rate to only 2% or so late this year. Furthermore, stock mutual funds. Nonetheless, this is atime to restrict any new invest- ments in stocks to conserva- it makes sense to favor issues that are likely to boost their dividends, such as the shares decline ofas much as 15% starting beforea result, long-term-bond year-end." yields could sink from to day's 69% to as low as 6% by year-end. That rate decline would produce total re turns the Wall Street Newsletter n page 58. Stocks with solid around less than growth on bonds and other Income investments could return uping to fall s to 20% in the coming year. Moreover, as you sell By contrast, stocks increasingly look as some of your past year's winners, don't feel hough they are poised for a temporary 15% you have to reinvest all your profits in stocks. Consider putting some money into are likely to 30-year Treasuries (and mutual funds that start worrying that corporate profits are go- hold such long-term. high-quality issues). In hort of analysts' estimates and short, fortify your asset mix with bigger mark down share prices. "Slower earnings holdings of long-term bonds and other in- gains could turn the stock market's ail come investments such as high-yield stocks. portfo- drop in the second half of 1996. Reason: As the economy siows, investors winds into head winds." says economist That will help you keep your overall Michael SivyKathryn J. Lunstrum at Duff & Phelps in lio on track to a double-digit return even if the stock market takes a nasty 15% tumble in The tip-off that a stock market pullback the coming year. may be close is the increasing volatility o share prices over the past three months. Wall Street editor Michad Sivy is a chartered fi Volatility tends to be low in a strong, rising analyst and a former Wall Street researth director. nancia

MAT 21A Lecture Notes - Lecture 3: Wilshire Grand Center, Logarithmic Form, Logarithm

MAT 21A Lecture Notes - Lecture 2: Farad, Rate Function, Trigonometric Functions

MAT 21A Lecture Notes - Lecture 3: Wilshire Grand Center, Logarithmic Form, Logarithm

MAT 21A Lecture 4: MAT 21A Lecture Notes 10-05-2018

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