Calculus Highlights

Chapter 1 Highlights

What is a Function?

• A function f is a rule that assigns to each element x in a set D exactly one element, called f(x), in a set E.
• D = All possible inputs
• f(x) = the value of f at x
• Range of x = All possible outputs
• Independent Variable = represents domain
• Dependent Variable = represents range
• (a) f(x) = sqrt(x+2)
• Domain = [-2, infinity)

The Vertical Line Test

• The Vertical Line Test = A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once

Piecewise Defined Functions

• Piecewise Defined Functions = When a function is defined by different formulas in different parts of their domain
• f(x) = {1-x if x f(-2), f(-1), and f(0).
• x^2 if x > -1}
• Since -2 f(2) = 1 - (-2) = 3
• Since -1 f(-1) = 1 - (-1) = 2
• Since 0 > -1, we have f(0) = 0^2 = 0

Absolute Value

• Absolute Value = the distance from a to 0 on the real number line
• |a| = a if a >= 0
• |a| = a if a
• Examples: |3| = 3
• |-3| = 3
• |0| = 0
• |sqrt(2)-1| = sqrt(2) - 1
• |3-pi| = pi - 3

Symmetry

• Even Function = If a function f satisfies f(-x) = f(x) for every number x in its domain
• The function f(x) = x^2 is even because
• f(-x) = (-x)^2 = x^2 = f(x)
• Odd Function = If a function f satisfies f(-x) = -f(x) for every number x in its domain
• The function f(x) = x^3 is odd because
• f(-x) = (-x)^3 = -x^3 = -f(x)

Polynomials

• The domain of any polynomial is (-infinity, infinity)
• ax^3 + bx^2 +cx +d
• The parabola opens upward if a > 0
• The parabola opens downward if a

Quadratic Formula

• x = -b+- sqrt((b)^2-4ac/2a

Functions

• Logarithmic Function
• The domain is (0, infinity) and the range is (-infinity, infinity)
• Exponential Function
• The domain is (-infinity, infinity) and the range is (0, infinity)
• Rational Function
• A ratio of two polynomials Example: f(x) = 2x^4-x^2+1 / x^2-4 Domain: {x | x ~= +- 2}
• Trigonometric Functions
• The domain of sine and cosine is (-infinity, infinity) and the range is [-1, 1]
• tan(x) = sin(x) / cos(x)

Vertical and Horizontal Shifts

• Suppose c > 0. To obtain the graph of
• y = f(x) + c, shift the graph of y = f(x) a distance c units upward
• y = f(x) - c, shift the graph of y = f(x) a distance c units downward
• y = f(x-c), shift the graph of y = f(x) a distance c units to the right
• y = f(x+c), shift the graph of y = f(x) a distance of c units to the left

Stretching and Reflecting

• Suppose c > 1. To obtain the graph of
• y = cf(x), stretch the graph of y = f(x) vertically by a factor of c
• y = (1/c)f(x), shrink the graph of y = f(x) vertically by a factor of c
• y = f(cx), shrink the graph of y = f(x) horizontally by a factor of c
• y = f(x/c), stretch the graph of y = f(x) horizontally by a factor of c
• y = -f(x), reflect the graph of y = f(x) about the x-axis
• y = f(-x), reflect the graph of y = f(x) about the y-axis