Unit 1 A Library of Functions The building blocks for Calculus

Functions and Graphs  Function: A rule that takes a set of input numbers and assigns to each a definite output number.  The set of all input numbers is called the domain of the function and the set of all resulting output numbers is called the range of the function.  Function: A rule that takes a set of input numbers and assigns to each a definite output number.  The set of all input numbers is called the domain of the function and the set of all resulting output numbers is called the range of the function.

The Rule of Four (Multiple Representations)  Representing mathematical functions with:  tables  graphs  formulas  words  Representing mathematical functions with:  tables  graphs  formulas  words

Situation The value of a car decreases over time. Can this be represented using a: Graph? Table? Formula? Can this be represented using a: Graph? Table? Formula?

Linear Functions  Are fundamental to all mathematical patterns.  “ the value of the function changes by equal amounts over equal intervals ”.  In other words, as the independent variable changes by a fixed amount, a constant amount is added to the value of the function.  Are fundamental to all mathematical patterns.  “ the value of the function changes by equal amounts over equal intervals ”.  In other words, as the independent variable changes by a fixed amount, a constant amount is added to the value of the function.

A different look at the slope formula: This expression is called the difference quotient.

Exponential Functions  are characterized by a simple property: the value of the function changes by equal ratios over equal intervals.  In other words, as the independent variable changes by a fixed amount, the value of the function changes by a constant factor.  are characterized by a simple property: the value of the function changes by equal ratios over equal intervals.  In other words, as the independent variable changes by a fixed amount, the value of the function changes by a constant factor.

General Exponential Function where P o is the initial quantity when t = 0 and a is the factor by which P changes when t increases by 1.

 If the rate a is > 1, we have exponential growth.  If the rate a is < 1 we have exponential decay.  If the rate a is > 1, we have exponential growth.  If the rate a is < 1 we have exponential decay.

Population growth can often be modeled with an exponential function: Ratio: World Population: million The world population in any year is about times the previous year. in 2010: About 7.6 billion people. Nineteen years past 1991.

Radioactive decay can also be modeled with an exponential function: Suppose you start with 5 grams of a radioactive substance that has a half-life of 20 days. When will there be only one gram left? After 20 days: 40 days: t days: In Pre-Calc you solved this using logs.

Concavity  The graph of a function is concave up if it bends upward as we move left to right.  It is concave down if it bends downward.  A line is neither concave up nor concave down.  The graph of a function is concave up if it bends upward as we move left to right.  It is concave down if it bends downward.  A line is neither concave up nor concave down.

Exponential function with base e  When the growth or decay is “ continuous ” the function can be written: to represent continuous growth

Exponential function with base e  When the growth or decay is “ continuous ” the function can be written as: to represent continuous decay

New Functions from Old Understanding how certain parent functions can be transformed into new functions is essential in using functions to model real world situations.

Key Points oGraphical interpretation of linear combinations of functions, o Modeling interpretation of composition of functions, o Basic manipulations of graphs, including shifts, flips, stretches o Odd and even functions, o The concept of an inverse function. oGraphical interpretation of linear combinations of functions, o Modeling interpretation of composition of functions, o Basic manipulations of graphs, including shifts, flips, stretches o Odd and even functions, o The concept of an inverse function.

Inverse functions: Given an x value, we can find a y value. Switch x and y : (inverse of x) Inverse functions are reflections about y = x. Solve for x :

Logarithms Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2004 Golden Gate Bridge San Francisco, CA

Logarithmic Functions  The logarithm (log) is the inverse of an exponential function. Base e is just another number. By varying the constant k in e kt, any exponential can be expressed with base e, with an inverse of the natural logarithm ln.  The logarithm (log) is the inverse of an exponential function. Base e is just another number. By varying the constant k in e kt, any exponential can be expressed with base e, with an inverse of the natural logarithm ln.

Consider This is a one-to-one function, therefore it has an inverse. The inverse is called a logarithm function. Example: Two raised to what power is 16? The most commonly used bases for logs are 10: and e : is called the natural log function. is called the common log function.

is called the natural log function. is called the common log function. In calculus we will use natural logs exclusively. We have to use natural logs: Common logs will not work.

Properties of Logarithms Since logs and exponentiation are inverse functions, they “un-do” each other. Product rule: Quotient rule: Power rule: Change of base formula:

Example $1000 is invested at 5.25 % interest compounded annually. How long will it take to reach $2500? We use logs when we have an unknown exponent years In real life you would have to wait 18 years.

Trig Functions The Mean Streak, Cedar Point Amusement Park, Sandusky, OH

Trigonometric functions are used extensively in calculus. When you use trig functions in calculus, you must use radian measure for the angles.

Trigonometric Functions  Trigonometry is organized into 3 main categories: 1. Triangle trig (right triangle and non-right triangle relationships) 2. The Unit Circle and the graphs of trig functions. 3. Trig Identities and their properties  Trigonometry is organized into 3 main categories: 1. Triangle trig (right triangle and non-right triangle relationships) 2. The Unit Circle and the graphs of trig functions. 3. Trig Identities and their properties

Even and Odd Trig Functions: “Even” functions behave like polynomials with even exponents, in that when you change the sign of x, the y value doesn’t change. Cosine is an even function because: Secant is also an even function, because it is the reciprocal of cosine. Even functions are symmetric about the y - axis.

Even and Odd Trig Functions: “Odd” functions behave like polynomials with odd exponents, in that when you change the sign of x, the sign of the y value also changes. Sine is an odd function because: Cosecant, tangent and cotangent are also odd, because their formulas contain the sine function. Odd functions have origin symmetry.

The rules for shifting, stretching, shrinking, and reflecting the graph of a function apply to trigonometric functions. Vertical stretch or shrink; reflection about x -axis Horizontal stretch or shrink; reflection about y -axis Horizontal shift Vertical shift Positive c moves left. Positive d moves up. The horizontal changes happen in the opposite direction to what you might expect. is a stretch. is a shrink.

When we apply these rules to sine and cosine, we use some different terms. Horizontal shift Vertical shift is the amplitude. is the period. A B C D

Trig functions are not one-to-one. However, the domain can be restricted for trig functions to make them one-to-one. These restricted trig functions have inverses called “arc”. 

Power Functions

Polynomial Functions  Flexibility of polynomial graphs on a small scale.  Basic resemblance to power functions on a large scale.  Varying the coefficient to produce particular graphs.  Flexibility of polynomial graphs on a small scale.  Basic resemblance to power functions on a large scale.  Varying the coefficient to produce particular graphs.

Polynomials are the sums of power functions with nonnegative integer exponents n is a nonnegative integer called the degree of the polynomial.

Rational Functions  Identifying horizontal and vertical asymptotes of rational functions.  Rational Functions are ratios of polynomial functions  Identifying horizontal and vertical asymptotes of rational functions.  Rational Functions are ratios of polynomial functions

Rational Functions  Vertical asymptotes of a rational function occur at every root of the denominator that are not also roots of the numerator.  Holes can occur in a rational function when a root of the denominator is also a root of the numerator.  Vertical asymptotes of a rational function occur at every root of the denominator that are not also roots of the numerator.  Holes can occur in a rational function when a root of the denominator is also a root of the numerator.

3 cases to identify a horizontal asymptote  Case 1: The degree of the numerator and denominator are the same. is a horizontal asymptote: a is the leading coefficient of the numerator and b is the leading coefficient of the denominator.

 Case 2: The degree of the numerator is less than the degree of the denominator, the x-axis (y = 0) is a horizontal asymptote.  Case 3: The degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.  Case 2: The degree of the numerator is less than the degree of the denominator, the x-axis (y = 0) is a horizontal asymptote.  Case 3: The degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.