AP Calculus AB Class Notes 1.1 Lines pp 3-11 Objectives: Find the x and y increments between two points. Find the slope of a line given two points, or, find the location of a point given the slope and another x or y value. Find equations of lines that are perpendicular or parallel to other lines. Find the equation of a line given two points or given one point and a slope. Identify and convert between point-slope form, slope intercept form, and general form for an equation of a line. Omissions: In AP calculus, we do not do problems involving regression analysis. Omit this material, pp 7-8. Increments: An increment is the displacement from one place to the other, along the x and y directions. Increments are found by subtracting the initial position from the final position. Our book writes this as Definition: Increments If a particle moves from the point (x₁, y₁) to the point (𝑥2 , 𝑦2 ), the increments in its coordinates are: ∆𝑥 = 𝑥2 − 𝑥1 and ∆𝑦 = 𝑦2 − 𝑦1 I prefer the notation x x f xo and y y f yo which is spoken as, “delta-x equals x-final minus x-naught.” Naught means initial. Slope of a Line: This is often defined as rise over run, and has the formula and definition Definition: Slope Let 𝑃1 (𝑥1 , 𝑦1 ) and 𝑃2 (𝑥2 , 𝑦2 ) be points on a nonvertical line, L. The slope of L is 𝑟𝑖𝑠𝑒 ∆𝑦 𝑦2 − 𝑦1 𝑚= = = 𝑟𝑢𝑛 ∆𝑥 𝑥2 − 𝑥1 Class Notes 1.1 Page 1 of 27 Finding Slope: Find the slope of the line passing through the given points. Then tell whether the line rises, falls, is horizontal, or is vertical. 1. (3,2), (-4, 3) 2. (1,-4), (2,6) 3. (14, -3), (4,11) Equations of Lines: Lines can be written in many forms, and there are three forms that we codify. They are Point-Slope form, Slope-Intercept form, and General form. Slope Intercept form: This formula is most useful if we know the slope and the y-intercept. Point Slope form: This is the easiest way to find the equation for a line when you’re given the slope and one point on the line. This is the most important form and the one we’ll use more than the others. It is used more because often in calculus, often we will be finding the equation for a line when we know its slope and one point through which it passes. It’s also worth mentioning that we can, and should, always leave our answers in point slope form. Definition: Point-Slope Equation The equation 𝑦 = 𝑚(𝑥 − 𝑥1 ) + 𝑦1 is the point-slope equation of the line that passes through the point (𝑥1 , 𝑦1 ) with slope m. Write an equation of the line that passes through the given points. 1. (2,-1), (3,8) 2. (-5,3), (2,2) Page 2 of 27 3. (1,-6), (1,-2) Class Notes 1.1 Parallel and Perpendicular Lines: Parallel lines have identical slopes. Perpendicular lines have slopes that are opposite reciprocals. This can be written 1 m m The only exception is with a horizontal line. Since a horizontal line has a slope of zero, its perpendicular is a vertical line. However, since we can never have a zero in the denominator, we say that the vertical line has a slope that is undefined. The normal: Normal is a synonym for perpendicular. Write an equation of the line that passes through (6, 10) and is perpendicular to the line that passes through (4, 6) and (3, 4). Write an equation of the line that passes through (2, 7) and is parallel to the line x = 5. Write an equation of the line that passes through (4, 6) and is parallel to the line that passes through (6, 6) and (10, -4). Equations of Vertical and Horizontal Lines: Example What is the equation for the horizontal and vertical lines that pass through the point (3, 2)? Draw out this example. General Linear Equation: The general form is useful if you’re wishing to convey the intercepts of line. It does not show up much otherwise. Definition: General Linear Equation The equation Ax + By = C (A and B both not 0) is a general linear equation in x and y. Example: Analyzing and Graphing a General Linear Equation Find the slope and y-intercept of the line 8x + 5y = 20. Graph the line. Class Notes 1.1 Page 3 of 27 Twelve Fundamental Functions You’ll be working with most of the functions that you’ve studied through precalculus, as shown below. The logistics function is not seen much in AP Calculus AB, but it is a part of most precalc and calc. Page 4 of 27 Class Notes 1.1 Graphical Transformations Recall that transformations come in two forms; rigid transformations do not distort the shape and size of the graph. These include horizontal and vertical shifts and reflections. Non-rigid transformations distort the size and shape and include horizontal and vertical stretches and shrinks. Rigid Transformations vertical shift horizontal shift x-axis reflection y-axis reflection y f ( x) b y f ( x n) y f (x) y f ( x) Non Rigid Transformations vertical stretch/shrink y cf (x) horizontal stretch/shrink x y f c shifts the graph up b units shifts the graphs n units to the right reflects the graph in the x-axis reflects the graph in the y-axis functions stretches by a factor of c in the vertical direction if c 1 ; shrinks if 0 c 1 . functions stretches by a factor of c in the horizontal direction if c 1 ; shrinks if 0 c 1 . Example 1: Example 2: 𝑦 = 𝑥 2 + 5.2 𝑦 = (𝑥 − 3)2 Class Notes 1.1 Page 5 of 27 Example 3: 𝑦 = √𝑥 − 5 Example 4: 𝑦 = √3 − 𝑥 Page 6 of 27 Class Notes 1.1 Example 5: 𝑓(𝑥) = 𝑥 3 − 2 𝑔(𝑥) = (𝑥 + 4)3 − 1 𝑔(𝑥) = 2(𝑥 − 1)3 𝑓(𝑥) = −2|𝑥| − 3 𝑔(𝑥) = 3|𝑥 + 5| + 4 ℎ(𝑥) = |3𝑥| Class Notes 1.1 Page 7 of 27 AP Calculus AB Class Notes 1.2 Functions and Graphs pp 12-21 Objectives: Find the domain and range of a function. State the definition of an even and odd function, and classify a function as even, odd, or neither. Graph piecewise functions. Given the graph of a piecewise function, write the function. Graph piecewise functions on your calculator. Graph the absolute value function. Transform functions. Create composite functions. Omissions: none Functions are mathematical machines that spit out an output for each input. The input is called the domain and is often the x-value. The output is called the range, and is often the y-value. Domains and Ranges: The domain is the set of all allowable inputs or x-values for a function. If the domain is not explicitly stated, it is assumed to be all real numbers that are available for the function. For example, the function f ( x) x has the domain x 0 because we cannot take the square root of a negative number. The range of a function is the set of all outputs or y-values. Recall that since the graph of the sine function is The range of the sine function is 1 y 1. Example 1: Identifying Domain and Range of a function Identify the domain and range, and then sketch a graph of the function. 1 (a) 𝑦 = 𝑥 (b) 𝑦 = √𝑥 Page 8 of 27 Class Notes 1.2 Example 2: Identifying Domain and Range of a Function Use a grapher to identify the domain and range, and then draw a graph of the function. 2 (a) 𝑦 = √4 − 𝑥 2 (b) 𝑦 = 𝑥 ⁄3 Even Function and Odd Functions – Symmetry: Functions that are symmetric with respect to either the y-axis or the origin are said to be even or odd, respectively. There are really three ways to determine if a function is odd, even, or neither. Polynomials are even or odd if all of their exponents are either even or odd respectively. Odd functions have graphs that display origin-symmetry. The graphs of even functions have yaxis symmetry. Finally, one can determine if the function is even, odd, or neither by the following definition: Definitions: Even Function, Odd Function A function 𝑦 = 𝑓(𝑥)is an even function of x if 𝑓(−𝑥) = 𝑓(𝑥), odd function of x if 𝑓(−𝑥) = −𝑓(𝑥), for every x in the function’s domain. Some rules about odd and even functions: The only function which is both even and odd is the constant function which is equal to zero (i.e., f(x) = 0 for all x). The sum of an even and odd function is neither even nor odd, unless one of the functions is equal to zero over the given domain. The sum of two even functions is even, and any constant multiple of an even function is even. The sum of two odd functions is odd, and any constant multiple of an odd function is odd. The product of two even functions is an even function. The product of two odd functions is an even function. The product of an even function and an odd function is an odd function. The quotient of two even functions is an even function. The quotient of two odd functions is an even function. The quotient of an even function and an odd function is an odd function. Class Notes 1.2 Page 9 of 27 State whether the function is odd, even, or neither. Support graphically and confirm algebraically. 1. 𝑓(𝑥) = 2𝑥 4 3. 𝑓(𝑥) = √𝑥 2 + 2 2. 𝑔(𝑥) = 𝑥 3 4. 𝑔(𝑥) = 3 1+ 𝑥 2 End of part one. Piecewise Functions The so-called piecewise functions are functions are defined on a particular interval. Example: Graphing Piecewise-Defined Functions −𝑥, 𝑥 < 0 Graph 𝑦 = 𝑓(𝑥) = {𝑥 2 , 0 ≤ 𝑥 ≤ 1 1, 𝑥 > 1 Page 10 of 27 Class Notes 1.2 Example: Graph the function 𝑓(𝑥). Let f(x) be defined as follows: 2 𝑖𝑓 𝑥 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑓(𝑥) = { 1 𝑖𝑓 𝑥 𝑖𝑠 𝑛𝑜𝑡 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 a) Find each of the following: f(0), f(1/3), f(1/2), f(√2), f(2), f(π) b) Sketch the graph of 𝑦 = 𝑓(𝑥) Suppose m and k are numbers and let g(x) be defined piecewise as follows: 1 𝑥 + 𝑘 𝑖𝑓 𝑥 < 3 𝑔(𝑥) = { 2 𝑚𝑥 + 2 𝑖𝑓 𝑥 ≥ 3 a) Suppose m = 3/2. What must the value of k be in order for the graph of y = g(x) to be connected? b) Suppose k = 4. What must the value of m be in order for the graph to be connected? Class Notes 1.2 Page 11 of 27 Example: Writing Formulas for Piecewise Functions Write a formula for the function y = f(x) whose graph consists of the two line segments in the figure below. Composite Functions When the output of one function becomes the input of a second function, the resulting function is called a composite function. This concept is used quite frequently in calculus. For example, suppose we have the following functions: g ( x) x h and f ( x) x 2 3 x To form the composite function f ( g ( x)) , we replace every occurrence of x in f(x) with g(x). Similarly, to form the composite function g ( f ( x)) , we replace every occurrence of x in g(x) with f(x). Finally, on the subject of composite functions, there is an alternate notation: f ( g ( x)) can also be written as f g g ( f ( x)) can be written g f . Domain of a Composite Function Finding the domain of a composite function consists of two steps: Step 1. Find the domain of the "inside" (input) function. Step 2. Construct the composite function. Find the domain of this composite function. The domain of the composite is the intersection of these two domains. Page 12 of 27 Class Notes 1.2 For each of the following, find the composite functions f◦g, g◦f, and the domain of each composite. 1. 𝑓(𝑥) = 𝑥 + 3 3. 𝑓(𝑥) = √𝑥 + 3 𝑔(𝑥) = √9 − 𝑥 2 𝑔(𝑥) = 2𝑥 − 5 2. 𝑓(𝑥) = 𝑥 2 + 2 2 4. 𝑓(𝑥) = 𝑥−3 𝑔(𝑥) = √𝑥 − 5 5 𝑔(𝑥) = 𝑥+2 Composition of Functions Numerically Using the table below, find the values on the right. x f(x) g(x) h(x) Find: -3 0 3 -2 -2 1 1 -3 0 3 0 -2 1 -3 -2 0 3 -2 -3 1 ( f g )(3) (h g )( 2) ( g g )(1) ( g f )(1) ( f f )(3) ( f h)(1) ( g h)(3) ( g f h)(1) Class Notes 1.2 Page 13 of 27 AP Calculus AB Class Notes 1.3 Exponential Functions pp 22-29 Objectives: x x Graph and identify the domains and ranges for the functions e and e . Use the rules for exponential algebra to perform algebraic manipulation of exponential functions. Distinguish between exponential growth and decay. Solve exponential grown and decay problems. Omissions: none The exponential function is one of the common functions we use in calculus. It’s most familiar examples are compound interest (exponential growth) and radioactive decay (exponential decay). Year 1998 1999 2000 2001 2002 2003 Population (millions) 276.1 279.3 282.4 285.3 288.2 291.0 Ratios (’98-‘99) (’99-’00) (’00-’01) (’01-’02) (’02-’03) 279.3/276.1 ≈ 1.0116 282.4/279.3 ≈ 1.0111 285.3/282.4 ≈ 1.0102 288.2/285.3 ≈ 1.0102 291.0/288.2 ≈ 1.0097 To say that something grows exponentially is to say that over time, for equal time intervals, the ratio of the final to the initial value is constant. For example look at the data showing U.S. population from ‘98 – ’03 at the right. Here, over equal time intervals (one year), the ratio of the population at the end of the year to that at the beginning of the year is an approximate constant of 1.01. This means that the U.S. population grew at about 1% per year during this time. While that may not seem like a lot, as you can see by the table, that is an increase of about 15 million people over a 5 year time period. Since exponential functions are most often used to describe how the value of something changes over time, we’re most likely to use the variable t for time, when describing these functions. Let a be a positive real number. The general form for the exponential function is f (t ) a t The a is called the base of the exponent. Page 14 of 27 Class Notes 1.3 Example 1: Graphing an Exponential Function Graph the function 𝑦 = 2(3𝑥 ) − 4. State its domain and range. Example 2: Finding Zeros Find the zeros of 𝑓(𝑥) = 5 − 2.5𝑥 graphically. Investigating Graphs: Identify the y-intercept and the asymptote of the graph of the function. 𝑦 = 5𝑥 𝑦 = −2 ∙ 4𝑥 𝑦 = 4 ∙ 2𝑥 Algebraic Rules for the Manipulation of Exponents: **These must be memorized** Rules for Exponents If 𝑎 > 0 and 𝑏 > 0, the following hold for all real numbers x and y. 1. 𝑎𝑥 ∙ 𝑎𝑦 = 𝑎𝑥+𝑦 4. 𝑎𝑥 ∙ 𝑏𝑥 = (𝑎𝑏)𝑥 2. 𝑎𝑥 𝑎𝑦 𝑎 𝑥 = 𝑎𝑥−𝑦 𝑥 5. (𝑏) = 𝑎𝑥 𝑏𝑥 3. (𝑎𝑥 )𝑦 = (𝑎𝑦 ) = 𝑎𝑥𝑦 Later in the year, when we are solving differential equations, we’ll have to use these rules to simplify our functions. For example, Class Notes 1.3 Page 15 of 27 et c et e c Properties of Rational Exponents: Simplify the expression. 1. 35/3 ∙ 31/3 2⁄ 1⁄ 3) 2 2. (5 1⁄ 4 3. 4 ∙ 64 1⁄ 4 4. 1 −1 36 ⁄2 Solving Exponential Equations: Solve the equation. 10𝑥−3 = 1004𝑥−5 25𝑥−1 = 1254𝑥 3𝑥−7 = 272𝑥 End of Part I Definitions: Exponential Growth, Exponential Decay The function 𝑦 = 𝑦𝑜 ∙ 𝑎 𝑥 , 𝑘 > 0 is a model for exponential growth if 𝑎 > 1, and a model for exponential decay if 0 < 𝑎 < 1. Examples of Exponential Growth Writing Models: For the following two problems, write and solve an exponential growth model that describes the situation. Using the data on the first page of this section, predict the U.S. population in 2010. Show all work. You buy a commemorative coin for $110. Each year, the value of the coin increases by 4%. What is its value after 10 years? Page 16 of 27 Class Notes 1.3 Exponential Decay: When the ratio of final to initial values over equal time intervals of a quantity turns out to be a number between 0 and 1, we say that we have exponential decay. Writing Models: For the following two problems, write an exponential decay model that describes the situation. You buy a stereo system for $780. Each year t, the value V of the stereo system decreases by 5%. What is the value after 5 years? You drink a beverage with 120 milligrams of caffeine. Each hour h, the amount c of caffeine in your system decreases by about 12%. How much caffeine is in your system after 3 hours? Radioactive Decay and Half Life When the initial value of a substance is Ao and the substance decays with a half life of h, The amount of a substance, A(t ) after a time t is given by the function: A(t ) Ao 12 t h Example: Modeling Radioactive Decay Suppose the half-life a certain radioactive substance is 20 days and that there are 5 grams present initially. When will there be only 1 gram of the substance remaining? Class Notes 1.3 Page 17 of 27 Thallium-208 has a half-life of 3.053 min. How long will it take for 120.0 g to decay to 7.50 g? If 100.0 g of carbon-14 decays until only 25.0 g of carbon is left after 11 460 y, what is the half-life of carbon-14? Page 18 of 27 Class Notes 1.3 AP Calculus AB Class Notes 1.5 Functions and Logarithms pp 37-45 Objectives: Define one-to-one, and graphically determine whether a function is or is not one-to-one. Describe the relation of one-to-one and the horizontal line test. Draw the inverse of a function. Algebraically determine the inverse of a function. Explain the meaning of the logarithmic function. Graph a log function. Use the properties of log functions to manipulate equations. Omissions: parametric equations This section has two parts: inverse functions, and the logarithmic function. This is probably set up like this because the logarithmic function is often introduced and taught as the inverse to the most accessible exponential function. Definition: One-to-One Function A function f(x) is one-to-one on a domain D if 𝑓(𝑎) ≠ 𝑓(𝑏) whenever 𝑎 ≠ 𝑏. (A one-to-one function is also called monotonic; it is either always decreasing, or always increasing.) Graphically, one-to-one means that the graph passes the horizontal line test: no horizontal line crosses the function more than once. Graphing Inverse Functions: To graph the inverse of a function, exchange each points x and y values. This is the same as a reflection in the line y = x. Example: The inverse graphically. Draw the inverse of each of the following graphs. Class Notes 1.5 Page 19 of 27 The Inverse Numerically Don’t forget what an inverse is: An inverse is the function that undoes its parent function. For example, if the function f has the then for f 1 , we f f 1 following map x y know… x y 1 3 2 8 3 11 a b Using the table below, find the values on the right. Find: f 1 (3) x -3 -2 0 1 h 1 (3) g 1 (1) f 1 (3) f(x) g(x) h(x) f 1 (h(3)) ( f h 1 )(1) f 1 ( f 1 (3)) 0 3 -2 1 1 -3 3 0 -2 -3 -2 0 3 -2 -3 1 ( g 1 h)(2) Determining the Inverse Function Analytically: Writing 𝒇−𝟏 as a Function of x 1. Solve the equation 𝑦 = 𝑓(𝑥) for x in terms of y. 2. Interchange x and y. The resulting formula will be 𝑦 = 𝑓 −1 (𝑥). 3. Interchange the domain and range. This means, switch x with y and solve for y. Inverses of Power Functions: Find the inverse power function. 1. 𝑓(𝑥) = 𝑥 7 2. 𝑓(𝑥) = −𝑥 6 , 𝑥 ≥ 0 3. 𝑓(𝑥) = 3𝑥 4 , 𝑥 ≤ 0 Page 20 of 27 Class Notes 1.5 Inverses of Nonlinear Functions: Find the inverse function. 1 𝑓(𝑥) = 𝑥 3 + 2 𝑓(𝑥) = −2𝑥 5 + 3 𝑓(𝑥) = 2 − 2𝑥 2 , 𝑥 ≤ 0 End of Part One Logarithmic Functions Logarithmic functions are inverse functions of exponential functions. I think that the best way to think about a logarithmic function is to first look at the exponential. For example: 2 3 asks the question, “what number is the result of multiplying 2 by itself 3 times?” 23 8 Practice: problem log 5 125 translation log 2 8 is read log base 2 of 8. It asks the question, “how many times do I multiply 2 by itself to get 8?_____ answer log 6 36 log 8 1 log 1 / 4 log 4 1 4 1 2 log 4 4 0.38 log 64 64 log 1 Class Notes 1.5 Page 21 of 27 The case of e: e is called the natural base, and gives rise to the natural logarithm. log e x is written as ln x . Properties of Logarithms: Memorize the following Inverse Properties and the Algebraic Properties. Inverse Properties for 𝒂𝒙 and 𝒍𝒐𝒈𝒂 𝒙 1. Base a: 𝑎𝑙𝑜𝑔𝑎 𝑥 = 𝑥, 𝑙𝑜𝑔𝑎 𝑎𝑥 = 𝑥, 𝑎 > 1, 𝑥 > 0 2. Base e: 𝑒ln 𝑥 = 𝑥, ln 𝑒𝑥 = 𝑥, 𝑥 > 0 The function f ( x) ln x has a domain (0, ) and a range (, ) . Its graph is on the right. It’s important to note that one can never take the logarithm of a negative number or zero, and that the log does not converge. It grows without bounds, and it is one of the slowest growing functions. The x-intercept of all functions in the form y log b x is at x 1. y x Properties of Logarithms For any real numbers 𝑥 > 0 and 𝑦 > 0, 1. Product Rule: 𝑙𝑜𝑔𝑎 𝑥𝑦 = 𝑙𝑜𝑔𝑎 𝑥 + 𝑙𝑜𝑔𝑎 𝑦 𝑥 2. Quotient Rule: 𝑙𝑜𝑔𝑎 𝑦 = 𝑙𝑜𝑔𝑎 𝑥 − 𝑙𝑜𝑔𝑎 𝑦 3. Power Rule: 𝑙𝑜𝑔𝑎 𝑥 𝑦 = 𝑦𝑙𝑜𝑔𝑎 𝑥 Solving Logarithmic Functions: −5 + 2 ln 3𝑥 = 5 ln(𝑥 + 5) = ln(𝑥 − 1) − ln(𝑥 + 1) Page 22 of 27 log(5 − 3𝑥) = log(4𝑥 − 9) ln(5.6 − 𝑥) = ln(18.4 − 2.6𝑥) 6.5log 5 3𝑥 = 20 10 ∙ ln 100𝑥 − 3 = 117 Class Notes 1.5 AP Calculus AB Class Notes 1.6 Trigonometric Functions pp 46-55 Objectives: Graph the six trigonometric functions and their inverses. Find the values of each of the six trig functions along the unit circle. Use the fundamental identities to simplify trig expressions. Use the double angle formula to reduce the order of trig expressions. Omissions: regression problems. A good starting point for our review of trigonometric functions is the arc lengths along the unit circle. The following diagram shows that the distance traveled, along the circle of radius one, from the positive x-axis in a counterclockwise direction are show on the right. Graphs of Trigonometric Functions: Students must know the graphs of all trig functions as well as their domains and ranges. sine graph domain: range: cosine graph domain: y x tangent graph domain: range: Class Notes 1.6 range: y secant graph domain: y x range: y x x Page 23 of 27 cosecant graph domain: range: cotangent graph domain: y range: y x x Students also must know the sine and cosine values for all of the whole multiples of 6 .and . 4 and should therefore be able to determine the values of all of the six trig functions at these multiples. r 2 3 5 sin r cos r tan r sec r csc r cot r 6 4 3 2 3 4 6 Page 24 of 27 Class Notes 1.6 7 5 4 3 5 7 11 6 4 3 2 3 4 6 Fundamental Identities Fundamental identities are those that are based on the sine and cosine functions. cos x sin x tan x cot x cos x sin x 1 1 sec x csc x cos x sin x We use them to simplify other expressions. 1. cot 𝜃 sec 𝜃 2. sin ∅ (csc ∅ − sin ∅) cot 𝑥 3. csc 𝑥 4. 1−𝑠𝑖𝑛2 𝑥 𝑐𝑠𝑐 2 𝑥−1 sin 𝛼 5. sec 𝑎 ∙ tan 𝛼 End of part 1 Class Notes 1.6 Page 25 of 27 Double Angle Formulas You must memorize these: sin 2 x 2 sin x cos x cos 2 x sin 2 x cos 2 x 1 2 sin 2 x 2 cos 2 x 1 The Double Angle formulas are used often in calculus. One reason for this is to reduce the order of a problem. For example, we will not learn how to solve the integral: 2 sin xdx However, if we know the formula: cos 2 x 2 sin 2 x 1 , 2 We can write sin x as sin 2 x 12 (cos 2 x 1) . From this, sin 2 xdx 12 (cos 2 x 1)dx , and this second integral is one that we can do. Use the figure to find the exact value of the trigonometric function. 1. 2. 3. 4. sin𝜃 tan𝜃 cos2𝜃 sin2𝜃 Page 26 of 27 Class Notes 1.6 Inverse Trigonometric Functions The inverse trigonometric functions arcsin, arccos and arctan are part of the AP Calculus AB Curriculum, y cos 1 x or y arccos x y sin 1 x or y arcsin x y tan 1 x or y arctan x Class Notes 1.6 Page 27 of 27