Calculus: Functions

Distance from Home vs. Time

• The graph models Mr. Passwater's distance from his home over time.
• Time (in hours) is the independent variable.
• Distance from home (in miles) is the dependent variable.
• Option (D) is appropriate:
  • Mr. Passwater leaves his house and drives to a restaurant at a constant rate.
  • He stops to eat breakfast for a period of time.
  • After eating, he drives to his school at a faster constant rate.

Water Pouring into a Container

• Water pours into an empty container at a constant rate until it's full.
• Initially, the depth of water increases at an increasing rate.
• Then, the depth increases at a decreasing rate.

Mr. Passwater Skydiving - Distance Fallen vs. Time

• Mr. Passwater skydives.
• Initially, the distance he falls increases at an increasing rate until he opens his parachute.
• After parachute opens, the distance he falls increases at a decreasing rate until he lands.

Mr. Passwater Skydiving - Distance from Ground vs. Time

• Mr. Passwater's skydive is analyzed with distance from the ground as the dependent variable.
• Initially, his distance from the ground decreases at an increasing rate.
• After the parachute opens, his distance from the ground decreases at a decreasing rate.

Conical Tank Filling with Water

• A conical tank is filling with water.
• The water pours in such a way that the depth of the water increases at a constant rate with respect to time.
• The graph depicts depth of water (independent) vs. volume of water (dependent).

Graph of $f$ Problems 6-10

• Figure shows the graph of a function $f$ for x < L.

Points of Inflection

• Determine the number of inflection points on the graph of $f$.

Statements about the Graph

• Analyze intervals to determine if $f$ is positive/negative and increasing/decreasing.

Rate of Change Analysis

• Determine if $f$ is increasing/decreasing at an increasing/decreasing rate.

Concavity Analysis

• Determine if $f$ is increasing/decreasing and if the graph is concave up/down.

Rate of Change Over Intervals

• Analyze if the rate of change of $f$ is positive/negative and increasing/decreasing.

Graph of f$ Problems 11-15

• Figure shows the graph of a function f$for$x < L.

Points of Inflection

• Determine the number of inflection points on the graph of f.

Statements about the Graph

• Analyze intervals to determine if f is positive/negative and increasing/decreasing.

Rate of Change Analysis

• Determine if f is increasing/decreasing at an increasing/decreasing rate.

Concavity Analysis

• Determine if f is increasing/decreasing and if the graph is concave up/down.

Rate of Change Over Intervals

• Analyze if the rate of change of f is positive/negative and increasing/decreasing.

Graph Analysis of f, Problems 16-23

• Figures show the graph of a function f$for$x < L.
• Determine if the function is increasing or decreasing at an increasing or decreasing rate.

Graph Analysis of f, Problems 24-31

• Figures show graphs of a function f$for$x < L.
• Determine if the rate of change of f is positive/negative and increasing/decreasing.

Graph Analysis of f, Problems 32-39

• Figures show graphs of a function f$for$x < L.
• Determine if the function f$is increasing/decreasing, and if the graph of$f is concave up/down.

Comparing Rates of Change for Functions, Problems 40-42

• Graphs of functions f$,$g$, and$h$are shown for$x < L.
• Determine for which function(s) the rate of change is increasing or negative, or identify when both the function and its rate of change are negative.

Analyzing Rate of Change and Concavity from a graph 43 - 46

• The figure shows the graph of a function. The endpoints and extrema are labeled, as well as the only point of inflection of the graph of f.
• Let A, B, C,$and$D$represent the$x-values at those points.
  • Determine intervals where the rate of change of f is increasing.
  • Determine intervals where f$is increasing and the graph of$f is concave down.
  • Determine intervals where the rate of change of f is positive.
  • Determine intervals where f is decreasing at an increasing rate.

Rate of change of graphs 47-49

• The figure shows the graph of a function f$for$x < L.
• Determine if the function is:
  • Increasing at an increasing rate
  • Increasing at a decreasing rate
  • Decreasing at an increasing rate
  • Decreasing at a decreasing rate

Analyzing the function's graph:

• The graph of f is positive

  • The graph f is increasing
  • The rate of change of f is positive
  • The rate of change of f is increasing

• The graph of f$is concave up, because the rate of change of$f is positive

  • The graph of f$is concave up, because the rate of change of$f is increasing
  • The graph of f$is concave down, because the rate of change of$f is negative
  • The graph of f$is concave down, because the rate of change of$f is decreasing

Rates of change 50-53.

• The figure shows the graph of a function f$with 3-4 labeled points. It is known that a relative is a relative maximum of$f$occurs at$x=B.

• Consider the rates of change of f$at the three labeled points. Which of the following orders the rates of change of$f in order from least to greatest?

  • f'(A) < f'(C) < f'(B)

• For which of the following pairs of points is the average rate of change of f the least?

• Of the following points, at which is the rate of change of f the least/greatest

Average rate of change problems

• Average rate of change problems will typically have a table or graph that you can use to calculate the average rate of change.
• AverageRateOfChange = {f(b) - f(a) } / {b-a}$, where$a$and$b define the closed interval!

Analyzing rates of change of polynomial functions from tables 64-71

• A table is provided of points from a polynomial function
• Based on the data, determine if the graph is:
  • Concave up
  • Concave down
  • Modeled by a linear function
  • Modeled by a quadratic function

Determining Concavity From Tables

• Positive average rates of change over consecutive equal-length input-value intervals indicate that the graph could be concave up
• Increasing average rates of change over consecutive equal-length input-value intervals indicate that the graph could be concave up.
• Constant rate of change over consecutive equal-length input-value intervals indicate the function can be modeled as a linear function
• Change in the average rates of change over consecutive equal-length input-value intervals is constant indicating that it is best modeled by a quadratic function.

Concavity Based on Tables 72-77

• The graph of the polynomial function f is concave up/down on the interval.
• Which of the following could be a table of values containing points on the function.
• Determine if the function is increasing or decreasing and if the graph is concave up or concave down.

Polynomial Inflection Points, Problems 78-79

• Figure shows the graph of a quartic/cubic polynomial function f.
• Determine how many points of inflection the graph has.

Identify Point of Inflection graph 80

• The figure shows the graph of a derivative f'(x)$. Which point on the graph of$f'(x)$equals the point of inflection on the graph of$f(x)

Graphing Polynomials Problems 81-83

• Match equations to their graphs.

Odd and Even Functions From Tables 84-86

• The table gives values of the polynomial function f$at selected values of$x.

• Determine if the function is odd or even, and whether f(-1) = -6$,$f(-1) = 6$,$f(-1) = 8$, or$f(-1) = -8.

• Odd functions are symmetric about the origin.

  • An odd function will have, f(-x) = -f(x).

• Even functions are symmetric about the y-axis.

  • An even function will have, f(-x) = f(x).

Increasing or Decreasing graphs 87-91

• Determine which interval is increasing or decreasing for various polynomial functions.

Comparing Polynomials92-94

• Determine intervals for which f(x) = g(x)$,$f(x) < g(x)$, and$f(x) > g(x).

Distinct Real Zeros 95-98

• Given a polynomial function, determine the number of distinct real zeros.

Least Possible Degree Tables 99-101

• Given a table of a polynomial function, find the least possible degree of the function.

Function Modeling Type

• Determine from table data if the function is Linear or Quadratic via average rate of change or the 2nd differences of the output values.

Least Possible Degree 104

• Given the graph of a polynomial function that has a zero, what is the least possible degree of f(x)
  *End Behavior

• Match graphs to their end behaviors.

• lim(x-> -inf) = +inf

• lim(x-> +inf) = +inf

• As x approaches negative infinity, f(x) approaches positive infinity

• As x approaches positive infinity, f(x) approaches positive infinity

Identifying End Behavior 110-113

• The sign of the leading term of f$is positive, and the degree of the leading term of$f is odd; therefore

  • lim x->+inf = +inf
  • lim x->-inf = -inf

• The sign of the leading term of f$is negative, and the degree of the leading term of$f is even; therefore,

  • lim x->+inf = -inf
  • lim x->-inf = -inf

• The sign of the leading term of f$is positive, and the degree of the leading term of$f is even; therefore,

  • lim x->+inf = +inf
  • lim x->-inf = +inf

• The sign of the leading term of f$is negative, and the degree of the leading term of$f is odd; therefore,

  • lim x->+inf = -inf
  • lim x->-inf = +inf

Determining Function Equations Based on End Behavior 114-117

• Determine function equations based on end behavior. If the end behavior of f is given by the statements
  • limx→−∞f(x) = ∞
  • limx→∞f(x) = ∞

Rational Functions End Behavior

• As the input values of x$increase without bound, the output values of$f(x)$get arbitrarily close to 0. The rational function will eventually approach$$y=0$

Rational function facts

• Given rational functions:
  • Determine matching end behaviors
  • State intervals on which f(x) > 0
  • State zero of the function
  • Identify an expression for $f(x)$

Rational Functions & Vertical Asymptotes

• A vertical asymptote occurs when the denominator of a rational function approaches 0. It is displayed as a dashed line on the graph of a function
  • In the xy-plane, the graph of a rational function ff has a vertical asymptote at x=5x=5

Limits for Rational Functions 128-131

• Given $$\lim{x->a^-}$f(x) and$\lim{x->a^+} f(x) determine what f(x) is approaching. Functions will approach +inf, -inf, and specific values. Graphing can be a useful tool.

Limits and Output Values 132-134

• The Input values of x sufficiently close to 4 correspond to output values arbitrarily close to 8.
  • It means, \lim_{x->4}f(x) = 8

Definition of Holes

• Holes are open circle discontinuities in a rational function when a rational function factors so that the numerator and denominator share a similar term.

  • Rational functions
    • State its domain
      *Vertical Asymptotes

• Vertical asymptotes can be determined by setting the denominator of a rational function to 0, and solving for x.

Rational equation to graphs 141-146

• Given some rational functions. Determine the zeros and the vertical/horizontal asymptotes.
• This can be tested by setting the numerator and denominator equal to 0, respectively.

Slant Asymptotes with Long Division

• If the degree of the numerator is one more than the degree of the denominator, the rational function will have what is called a slant asymptote. To find the equation for the slant asymptote, simply perform long division.

Properties of the Remainder

• Remainder Theorem

  • What is the remainder that results when the polynomial $f(x)$ is divided by the polynomial $g(x)$
    *Slant Asymptotes

• Equation relating polynomials to vertical and slant asymptotes.

Applying the Binomial Theorem

• The binomial theorem can be used to expand an expression of the form
  • $(x+y)^n$

Finding coefficients terms in a polynomial 153

• What is the coefficient of the $x^2$ term in the expanded polynomial?

Horizontal Asymptote Rules 154-155

• $y = 0$ if the degree of the numerator < degree of the denominator
• $y = a/b$ if the degree of the numerator = degree of the denominator
• No horizontal asymptote if the degree of the numerator > degree of the denominator

Describing Transformation of Functions

• The sequence of transformations to map the graph of $f$ to the graph of $g$ includes a horizontal dilation of the graph of f by a factor of +$1/4$ followed by a vertical translation of the graph of $f$ by 7 unit.

Graph Transformations and Effects on Domain 158-159

• What happens to the domain during graph transformations
  *Graph Transformations Effects on Range

• What happens to the Range during graph transformations

Transformations and Points 162-164

• The point (2,-4) is on the graph of,$f(x)$. Which of the following points is on the graph of $$g(x) = -f(1/2(x-4)) + 5?

Vertical and Horizontal Asymptotes and Transformations 165

• The graph of the function f$has a vertical asymptote at$$x = 3$ and a horizontal asymptote of $y =4$ . The graph of $g$ is the result of the transformation

• What Happens to the Horizonal Asymptotes.

• How does the Domain Effect the Graph
  *Transformations from Tables of Values

Transformations of values 166-169

• Given the selected values of $x$ and $y$, what is the coordinate pair for $$g(x)=2 * f(-x+3) - 1?

Graph for Transformed Functions

• The transformed graph for f(x) $= -2f(x-2)$

Describing Function Transformation

• Determine the sequence of transformations that maps the graph of f(x) to the graph g(x) in the $xy$ -plane

Horizontal and Vertical Translations, Problems 177-178

• The function $f$ is given by $f(x) = x^2$

• The graph of which of the following functions is the image of the graph of $f$ after a horizontal translation of the graph of $f$ units?

• Function Modeling With Tables

  • Given a table, find the function type (Linear, Quadratic, polynomial greater than degree 2, exponential

• Graph Features

  • Select a scenario based on the graph features

• Analyzing Data

  • Given cost to produce pez related data, when should money be spent.