Math 10A Final Review OutlinePrepared By Will GarnerCompiled on 7-28-2008Math 10A Final Exam Review Outline 1Chapter 1: Library of FunctionsSection 1.1: Functions and Change• Know the formula for a linear function: y = mx +b, where m is the slope and b is y-intercept.• Given two points, know how to compute m, the slope.• Know how to tell the difference between different lines (look at slopes/y-intercepts)• Given a function, interpret its meaning.– If P (x) is the price of x units, what is the meaning of P−1(200)?Section 1.2: Exponential Functions• Know the general exponential function: P = P0at, where P0is the initial quantity, and a isthe factor by which P changes when t increases by 1.• Given two points on an exponential curve, know how to find the equation– If f(1) = 12, f(3) = 108, find a formula for f (t) = Q0at.– The size of a bacteria colony grows exponentially as a function of time. If the size of thebacteria colony doubles every 3 hrs, how long will it take to triple?– The fraction of a lake’s surface covered by algae was initially 0.42 and was halved eachyear since the passage of anti-pollution laws. How long after the passage of the law wasonly 0.07 of the lake’s surface covered with algae?– In 1924, Granny invested $75 (the contents of her purse) at a fixed annual interest rate.In 1964, her investment was worth $528. How much is her investment worth today(2008)?– The number of people who have heard a rumor is 10 at 6:00am and from that pointdoubles every 20 minutes. When have 100 people heard the rumor?2 Prepared by Will GarnerSection 1.3: New Functions From OldKnow the different types of shifts and what they do to a graph1. f(x − h) + k translates h units to the right, k units vertically2. −f(x) reflects across x-axis3. f(−x) reflects across y-axis4. cf(x) dilates by factor of c vertically5. f(cx) dilates by factor of1chorizontally• Know some types of example questions using the above shifts– How are the graphs of y = (x + 2)2and y = x2related?– What would I need to do to a graph to reflect it about the y-axis and shift it up 3 units?– The graph below was made from y = x2by reflecting it about the x-axis, shifting it tothe right by 2, and up by 1. Find its equation.xyH2,1L• Know what it means for a function to be even or odd• Know how the graph of f(x) and f−1(x) are related (reflection about line y = x)• Know how to find the domain of a function– Find the domain of f(x) =x2+1x2−4• Know the relationship between the domain/range of f(x) and f−1(x). That is, the domain(range) of f(x) is the range (domain) of f−1(x)– Find the domain of f−1(x) by considering the range of f (x) =√x − 1• Know what f(g(x)) means, (that is, plug g(x) into f(x)) and how to do this.– If f(x) =xx−1, g(x) = 3x + 1, what is f(g(x))? f(g(3))? f (f(2))? f(g−1(1))?Math 10A Final Exam Review Outline 3Section 1.4: Logarithmic Functions• Know how to use logarithms to solve exponential problems• Know the properties of logarithms and be careful of the false properties:True Falseln AB = ln A + ln B ln A + B = ln A + ln Bln A/B = ln A − ln B lnA/lnB = lnA − lnBln Ap= p ln ASection 1.5: Trigonometric Functions• Know the graphs of y = A sin(Bt) + C and y = A cos(Bt) + C• Know how B and the period are related (Period = 2π ×1B)• Know how to find the amplitude, |A| (|A| =max−min2)• Know how to find the vertical shift, C (C = max − |A|)• Know the basic values of sin(t) and cos(t). What is sin(0)? What is cos(0)? etc.• Know how to tell the differences between various Sine/Cosine graphs.Section 1.6: Powers, Polynomials, and Rational Functions• Know what it means for f(x) to be a rational function. (f(x) =p(x)q(x)).• Know how to find horizontal asymptotes of rational functions– Find the horizontal asymptote of y =4x2+1x3−1and of y =105+e−t.• Know where a rational function is undefined (where it has vertical asymptotes).– Where is the function f(x) =x2+4x+3x+1undefined?• Know how to find asymptotes and domain/range of a function– Find the domain of f (x) =x+12x+1undefined. Find the horizontal/vertical asymptotes off(x). Find f−1(x) and its domain.4 Prepared by Will GarnerSection 1.7: Introduction to Continuity• Know what it means for a function to be continuous (the graph has no breaks/holes).• Know how to make a rational function continuous at every point– Find k such that f(x) =(x2+4x+3x+1x 6= −1,k x = −1is continuous on any interval.Section 1.8: Limits• Know when a limit exists or does not exist and how to evaluate limits algebraically– Evaluate limh→041+h− 4halgebraically.Chapter 2: Key Concept: The DerivativeSection 2.1: How Do We Measure Speed?• Know how to use average rates of change (i.e. slope) to estimate the derivative at a point(Think the difference quotient, which is just another equation for slope)– If g(x) = x2+ 1, estimate g0(2) using average rates of change. (Hint: Use two pointsnear 2, for example 2 and 2.01.).– Suppose a particle’s distance from the origin is given by1(3+t). Find the average velocityof the particle between t = 0 and t = 2 seconds. What is the instantaneous velocity att = 2 seconds?Section 2.2: The Derivative at a Point• Know the definition of the derivative (at a point). The formula is f0(a) = limh→0f(a + h) − f(a)h=limx→af(x) − f(a)x − aMath 10A Final Exam Review Outline 5– If f(x) = (x + 1)2− 2, find f0(0) using the definition of the derivative.• Know that the derivative at a point is the slope of the tangent line to the curve at that point.– The slope, m, of the tangent line of y = x2at x = 1 is 2. Visually, we have the picturebelow.-11x-11ym = 2• Know how to order various slopes– List quantities in increasing order: f0(0), f0(1), f0(2),f(1) − f(0)1 − 0,f(2) − f(1)2 − 1in thepicture below.-112xSection 2.3: The Derivative Function• Know the definition of the derivative (as a function). The formula is f0(x) = limh→0f(x + h) − f(x)h6 Prepared by Will Garner• Know what the derivative tells us graphically:? If f0(x) > 0 on an interval, then f(x) is increasing over that interval.? If f0(x) < 0 on an interval, then f(x) is decreasing over that interval.• Note: Just because f (x) is increasing does not mean that f0(x) is increasing. It only meansthat f0(x) > 0. It is easy to confuse the two.• Know the power rule for taking derivatives:ddx(axn) = anxn−1, where a is a constant.• As a special