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$.</li> </ul> <h4 id="pointsofinflection-1">Points of Inflection</h4> <ul> <li>Determine the number of inflection points on the graph of$f$.</li> </ul> <h4 id="statementsaboutthegraph-1">Statements about the Graph</h4> <ul> <li>Analyze intervals to determine if$f$is positive/negative and increasing/decreasing.</li> </ul> <h4 id="rateofchangeanalysis-1">Rate of Change Analysis</h4> <ul> <li>Determine if$f$is increasing/decreasing at an increasing/decreasing rate.</li> </ul> <h4 id="concavityanalysis-1">Concavity Analysis</h4> <ul> <li>Determine if$f$is increasing/decreasing and if the graph is concave up/down.</li> </ul> <h4 id="rateofchangeoverintervals-1">Rate of Change Over Intervals</h4> <ul> <li>Analyze if the rate of change of$f$is positive/negative and increasing/decreasing.</li> </ul> <h3 id="graphanalysisofddfddproblems1623">Graph Analysis of$f$, Problems 16-23</h3> <ul> <li>Figures show the graph of a function$f$for$x < L$.</li> <li>Determine if the function is increasing or decreasing at an increasing or decreasing rate.</li> </ul> <h3 id="graphanalysisofddfddproblems2431">Graph Analysis of$f$, Problems 24-31</h3> <ul> <li>Figures show graphs of a function$f$for$x < L$.</li> <li>Determine if the rate of change of$f$is positive/negative and increasing/decreasing.</li> </ul> <h3 id="graphanalysisofddfddproblems3239">Graph Analysis of$f$, Problems 32-39</h3> <ul> <li>Figures show graphs of a function$f$for$x < L$.</li> <li>Determine if the function$f$is increasing/decreasing, and if the graph of$f$is concave up/down.</li> </ul> <h3 id="comparingratesofchangeforfunctionsproblems4042">Comparing Rates of Change for Functions, Problems 40-42</h3> <ul> <li>Graphs of functions$f$,$g$, and$h$are shown for$x < L$.</li> <li>Determine for which function(s) the rate of change is increasing or negative, or identify when both the function and its rate of change are negative.</li> </ul> <h3 id="analyzingrateofchangeandconcavityfromagraph4346">Analyzing Rate of Change and Concavity from a graph 43 - 46</h3> <ul> <li>The figure shows the graph of a function. The endpoints and extrema are labeled, as well as the only point of inflection of the graph of$f$.</li> <li>Let$A, B, C,$and$D$represent the$x$-values at those points.<ul> <li>Determine intervals where the rate of change of$f$is increasing.</li> <li>Determine intervals where$f$is increasing and the graph of$f$is concave down.</li> <li>Determine intervals where the rate of change of$f$is positive.</li> <li>Determine intervals where$f$is decreasing at an increasing rate.</li></ul></li> </ul> <h3 id="rateofchangeofgraphs4749">Rate of change of graphs 47-49</h3> <ul> <li>The figure shows the graph of a function$f$for$x < L$.</li> <li>Determine if the function is:<ul> <li>Increasing at an increasing rate</li> <li>Increasing at a decreasing rate</li> <li>Decreasing at an increasing rate</li> <li>Decreasing at a decreasing rate</li></ul></li> </ul> <h3 id="analyzingthefunctionsgraph">Analyzing the function's graph:</h3> <ul> <li><p>The graph of$f$is positive</p> <ul> <li>The graph$f$is increasing</li> <li>The rate of change of$f$is positive</li> <li>The rate of change of$f$is increasing</li></ul></li> <li><p>The graph of$f$is concave up, because the rate of change of$f$is positive</p> <ul> <li>The graph of$f$is concave up, because the rate of change of$f$is increasing</li> <li>The graph of$f$is concave down, because the rate of change of$f$is negative</li> <li>The graph of$f$is concave down, because the rate of change of$f$is decreasing</li></ul></li> </ul> <h3 id="ratesofchange5053">Rates of change 50-53.</h3> <ul> <li><p>The figure shows the graph of a function$f$with 3-4 labeled points. It is known that a relative is a relative maximum of$f$occurs at$x=B$.</p></li> <li><p>Consider the rates of change of$f$at the three labeled points. Which of the following orders the rates of change of$f$in order from least to greatest?</p> <ul> <li>$f'(A) < f'(C) < f'(B)$</li></ul></li> <li><p>For which of the following pairs of points is the average rate of change of$f$the least?</p></li> <li><p>Of the following points, at which is the rate of change of$f$the least/greatest</p></li> </ul> <h3 id="averagerateofchangeproblems">Average rate of change problems</h3> <ul> <li>Average rate of change problems will typically have a table or graph that you can use to calculate the average rate of change.</li> <li>$AverageRateOfChange = {f(b) - f(a) } / {b-a}$, where$a$and$b$define the closed interval!</li> </ul> <h3 id="analyzingratesofchangeofpolynomialfunctionsfromtables6471">Analyzing rates of change of polynomial functions from tables 64-71</h3> <ul> <li>A table is provided of points from a polynomial function</li> <li>Based on the data, determine if the graph is:<ul> <li>Concave up</li> <li>Concave down</li> <li>Modeled by a linear function</li> <li>Modeled by a quadratic function</li></ul></li> </ul> <h3 id="determiningconcavityfromtables">Determining Concavity From Tables</h3> <ul> <li>Positive average rates of change over consecutive equal-length input-value intervals indicate that the graph could be concave up</li> <li>Increasing average rates of change over consecutive equal-length input-value intervals indicate that the graph could be concave up.</li> <li>Constant rate of change over consecutive equal-length input-value intervals indicate the function can be modeled as a linear function</li> <li>Change in the average rates of change over consecutive equal-length input-value intervals is constant indicating that it is best modeled by a quadratic function.</li> </ul> <h3 id="concavitybasedontables7277">Concavity Based on Tables 72-77</h3> <ul> <li>The graph of the polynomial function$f$is concave up/down on the interval.</li> <li>Which of the following could be a table of values containing points on the function.</li> <li>Determine if the function is increasing or decreasing and if the graph is concave up or concave down.</li> </ul> <h3 id="polynomialinflectionpointsproblems7879">Polynomial Inflection Points, Problems 78-79</h3> <ul> <li>Figure shows the graph of a quartic/cubic polynomial function$f$.</li> <li>Determine how many points of inflection the graph has.</li> </ul> <h3 id="identifypointofinflectiongraph80">Identify Point of Inflection graph 80</h3> <ul> <li>The figure shows the graph of a derivative$f'(x)$. Which point on the graph of$f'(x)$equals the point of inflection on the graph of$f(x)$</li> </ul> <h3 id="graphingpolynomialsproblems8183">Graphing Polynomials Problems 81-83</h3> <ul> <li>Match equations to their graphs.</li> </ul> <h3 id="oddandevenfunctionsfromtables8486">Odd and Even Functions From Tables 84-86</h3> <ul> <li><p>The table gives values of the polynomial function$f$at selected values of$x$.</p></li> <li><p>Determine if the function is odd or even, and whether$f(-1) = -6$,$f(-1) = 6$,$f(-1) = 8$, or$f(-1) = -8$.</p></li> <li><p>Odd functions are symmetric about the origin.</p> <ul> <li>An odd function will have,$f(-x) = -f(x)$.</li></ul></li> <li><p>Even functions are symmetric about the y-axis.</p> <ul> <li>An even function will have,$f(-x) = f(x)$.</li></ul></li> </ul> <h3 id="increasingordecreasinggraphs8791">Increasing or Decreasing graphs 87-91</h3> <ul> <li>Determine which interval is increasing or decreasing for various polynomial functions.</li> </ul> <h3 id="comparingpolynomials9294">Comparing Polynomials92-94</h3> <ul> <li>Determine intervals for which$f(x) = g(x)$,$f(x) < g(x)$, and$f(x) > g(x)$.</li> </ul> <h3 id="distinctrealzeros9598">Distinct Real Zeros 95-98</h3> <ul> <li>Given a polynomial function, determine the number of distinct real zeros.</li> </ul> <h3 id="leastpossibledegreetables99101">Least Possible Degree Tables 99-101</h3> <ul> <li>Given a table of a polynomial function, find the least possible degree of the function.</li> </ul> <h3 id="functionmodelingtype">Function Modeling Type</h3> <ul> <li>Determine from table data if the function is Linear or Quadratic via average rate of change or the 2nd differences of the output values.</li> </ul> <h3 id="leastpossibledegree104">Least Possible Degree 104</h3> <ul> <li><p>Given the graph of a polynomial function that has a zero, what is the least possible degree of f(x)<br /> *End Behavior</p></li> <li><p>Match graphs to their end behaviors.</p></li> <li><p>$lim(x-> -inf) = +inf$</p></li> <li><p>$lim(x-> +inf) = +inf$</p></li> <li><p>As x approaches negative infinity, f(x) approaches positive infinity</p></li> <li><p>As x approaches positive infinity, f(x) approaches positive infinity</p></li> </ul> <h3 id="identifyingendbehavior110113">Identifying End Behavior 110-113</h3> <ul> <li><p>The sign of the leading term of$f$is positive, and the degree of the leading term of$f$is odd; therefore</p> <ul> <li>$lim x->+inf = +inf$</li> <li>$lim x->-inf = -inf$</li></ul></li> <li><p>The sign of the leading term of$f$is negative, and the degree of the leading term of$f$is even; therefore,</p> <ul> <li>$lim x->+inf = -inf$</li> <li>$lim x->-inf = -inf$</li></ul></li> <li><p>The sign of the leading term of$f$is positive, and the degree of the leading term of$f$is even; therefore,</p> <ul> <li>$lim x->+inf = +inf$</li> <li>$lim x->-inf = +inf$</li></ul></li> <li><p>The sign of the leading term of$f$is negative, and the degree of the leading term of$f$is odd; therefore,</p> <ul> <li>$lim x->+inf = -inf$</li> <li>$lim x->-inf = +inf$</li></ul></li> </ul> <h3 id="determiningfunctionequationsbasedonendbehavior114117">Determining Function Equations Based on End Behavior 114-117</h3> <ul> <li>Determine function equations based on end behavior. If the end behavior of$f$is given by the statements<ul> <li>limx→−∞f(x) = ∞</li> <li>limx→∞f(x) = ∞</li></ul></li> </ul> <h3 id="rationalfunctionsendbehavior">Rational Functions End Behavior</h3> <ul> <li>As the input values of$x$increase without bound, the output values of$f(x)$get arbitrarily close to 0. The rational function will eventually approach$$y=0$