Calculus I Hughes-Hallett 6/8/2018 Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University

Function: (Data Point of View) 6/8/2018 Function: (Data Point of View) One quantity H, is a function of another, t, if each value of t has a unique value of H associated with it. In symbols: H = f(t). We say H is the value of the function or the dependent variable or output; and t is the argument or independent variable or input.

Working Definition of Function: H = f(t) 6/8/2018 Working Definition of Function: H = f(t) A function is a rule (equation) which assigns to each element of the domain (independent variable) one and only one element of the range (dependent variable).

Working definition of function continued: 6/8/2018 Working definition of function continued: Domain is the set of all possible values of the independent variable (t). Range is the corresponding set of values of the dependent variable (H).

6/8/2018 Questions? 11

General Types of Functions (Examples): 6/8/2018 General Types of Functions (Examples): Linear: y = m(x) + b; proportion: y = kx Polynomial: Quadratic: y =x2 ; Cubic: y= x3 ; etc Power Functions: y = kxp Trigonometric: y = sin x, y = Arctan x Exponential: y = aebx ; Logarithmic: y = ln x

6/8/2018 Graph of a Function: The graph of a function is all the points in the Cartesian plane whose coordinates make the rule (equation) of the function a true statement.

6/8/2018 Slope m - slope : b: y-intercept a: x-intercept .

5 Forms of the Linear Equation 6/8/2018 5 Forms of the Linear Equation Slope-intercept: y = f(x) = b + mx Slope-point: Two point: Two intercept: General Form: Ax + By = C

Exponential Functions: If a > 1, growth; a<1, decay 6/8/2018 Exponential Functions: If a > 1, growth; a<1, decay If r is the growth rate then a = 1 + r, and If r is the decay rate then a = 1 - r, and

Definitions and Rules of Exponentiation: 6/8/2018 Definitions and Rules of Exponentiation: D1: D2: R1: R2: R3:

6/8/2018 Inverse Functions: Two functions z = f(x) and z = g(x) are inverse functions if the following four statements are true: Domain of f equals the range of g. Range of f equals the domain of g. f(g(x)) = x for all x in the domain of g. g(f(y)) = y for all y in the domain of f.

A logarithm is an exponent. 6/8/2018 .

6/8/2018 General Rules of Logarithms: log(a•b) = log(a) + log(b) log(a/b) = log(a) - log(b)

e = 2.718281828459045... Any exponential function 6/8/2018 e = 2.718281828459045... Any exponential function can be written in terms of e by using the fact that So that

Making New Functions from Old 6/8/2018 Making New Functions from Old Given y = f(x): (y - b) =k f(x - a) stretches f(x) if |k| > 1 shrinks f(x) if |k| < 1 reverses y values if k is negative a moves graph right or left, a + or a - b moves graph up or down, b + or b - If f(-x) = f(x) then f is an “even” function. If f(-x) = -f(x) then f is an “odd” function.

6/8/2018 Polynomials: A polynomial of the nth degree has n roots if complex numbers a allowed. Zeros of the function are roots of the equation. The graph can have at most n - 1 bends. The leading coefficient determines the position of the graph for |x| very large.

6/8/2018 Rational Function: y = f(x) = p(x)/q(x) where p(x) and q(x) are polynomials. Any value of x that makes q(x) = 0 is called a vertical asymptote of f(x). If f(x) approaches a finite value a as x gets larger and larger in absolute value without stopping, then a is horizontal asymptote of f(x) and we write: An asymptote is a “line” that a curve approaches but never reaches.

Asymptote Tests y = h(x) =f(x)/g(x) Vertical Asymptotes: Solve: g(x) = 0 If y    as x  K, where g(K) = 0, then x = K is a vertical asymptote. Horizontal Asymptotes: If f(x)  L as x   then y = L is a vertical asymptote. Write h(x) as: , where n is the highest power of x in f(x) or g(x).

Basic Trig radian measure: q = s/r and thus s = r q, 6/8/2018 Basic Trig radian measure: q = s/r and thus s = r q, Know triangle and circle definitions of the trig functions. y = A sin B(x - j) + k A amplitude; B - period factor; period, p = 2p/B j - phase shift k (raise or lower graph factor)

Continuity of y = f(x) A function is said to be continuous if there are no “breaks” in its graph. A function is continuous at a point x = a if the value of f(x)  L, a number, as x  a for values of x either greater or less than a.

Intermediate Value Theorem Suppose f is continuous on a closed interval [a,b]. If k is any number between f(a) and f(b) then there is at least one number c in [a,b] such that f(x) = k.