Section 2.3 Properties of Functions
|
|
- Silvia Harris
- 6 years ago
- Views:
Transcription
1 22 Section 2.3 Properties of Functions In this section, we will explore different properties of functions that will allow us to obtain the graph of the function more quickly. Objective #1 Determining Even and Odd Functions from a Graph. In mathematics, we use the words even and odd to describe the symmetry of the graph of a function. Let's begin with some definitions. Even and Odd Functions and Symmetry 1) A function is even if, for every ( x, y) (x, y) number x in the domain, the number x is also in the domain and f( x) = f(x) = y. This means that the function is symmetric with respect to the y-axis if and only if it is even. 2) A function is odd if, for every (x, y) number x in the domain, the number x is also in the domain and f( x) = f(x) = y. This means that the function is symmetric with respect to the origin if and only if it is odd. ( x, y) Determine if the following functions are even, odd, or neither: Ex. 1a Ex. 1b Ex. 1c
2 a) Since the graph is not symmetric with respect to the y-axis and not symmetric with respect to the origin, the function is neither even nor odd. b) Since the graph is symmetric with respect to the y-axis, the function is even. c) Since the graph is symmetric with respect to the origin, the function is odd. 23 Objective #2: Determining if a Function is Even, Odd, or Neither Algebraically. To determine if a function is even or odd algebraically, we replace x by x and simplify to see if we get f( x) = f(x) (even; original function) or f( x) = f(x) (odd; opposite of the original function). Determine if the following functions are even, odd, or neither: Ex. 2a f(x) = 4x 2 1 Ex.2b g(x) = x x Ex. 2c h(x) = x + x We will begin by plugging x in for x and then try to relate it back to the original function: a) f( x) = 4( x) 2 1 = ( x) 2 4x2 1 = f(x). x 2 Since f( x) = f(x), then f is an even function. b) g( x) = x + 1 = x + 1 = ( x 1) g(x) or g(x). Since g( x) g(x) and g( x) g(x), then g in neither an even or odd function. 3 c) h( x) = x + ( x) 3 = x x 3 = ( x + x ) = h(x) Since h( x) = h(x), then h is an odd function. If we were to use a graphing utility, we can see that the graph of f is symmetric with respect to the y-axis and the graph of h is symmetric with respect to the origin. The graph of g is neither symmetric to the origin nor to the y-axis.
3 24 f is symmetric with g is neither symmetric h is symmetric respect to the y-axis. with respect to the y-axis with respect to nor to the origin. the origin Objective #3 Determining when a Function is Increasing, Decreasing, or Neither. If we were to plot the miles per gallon a car was achieving versus its speed, we would find that the faster the car travels, the better gas mileage it would get until the car s speed was near mph. Once the car s speed exceeded that point, the gas mileage would start to decrease as the car started going faster. The optimal speed range would be between and 6 mph. That is where the car is most fuel-efficient. Let s take a closer look at the graph: 3 2 Fuel Efficiency vs. Speed Gas Mileage (MPG) Speed of the Car (MPH)
4 2 Notice that as the speed increases from mph to about 2 mph, the fuel efficiency increases. Thus, the function is increasing on the interval (, 2). When the speed is between 2 mph and about 7 mph, the fuel efficiency stays constant. Thus, the function is constant on the interval (2, 7). Finally, when the speed exceeds 7 mph, the fuel efficiency decreases. Hence, the function is decreasing on the interval (7, 8). Intervals over which a function is Increasing, Decreasing, or Constant Let I be an open interval in the domain of the function f. Then, 1) f is increasing on I if f(a) < f(b) for all a < b in I. (a function is increasing on an interval if it rises or goes uphill as you move from left to right. 2) f is decreasing on I if f(a) > f(b) for all a < b in I. (a function is decreasing on an interval if it falls or goes downhill as you move from left to right. 3) f is constant on I if f(a) = f(b) for all a and b in I. a function is constant on an interval if it stays level or flat as you move from left to right. Determine where the following function is a) increasing, b) decreasing, and c) constant: Ex. 3 Cost in Dollars Cost to Drive to Houston vs. Speed Speed of the Car (MPH)
5 a) The function rises as we move from about 8 mph to 8 mph, thus the function is increasing on (8, 8). b) The function falls as we move from mph to about 8 mph, thus the function is decreasing on (, 8). c) The function is constant nowhere. Ex. 4 Ex a) The function rises as we move from -1 to and 2 to, thus the function is increasing on ( 1, ) U (2, ). 4b) The function falls as we move from to 1 and to 2, thus the function is decreasing on (, 1) U (, 2). 4c) The function is constant nowhere. a) The function rises as we move from - to, thus the function is increasing on (, ). b) The function falls as we move from 3 to, thus the function is decreasing on (3, ). c) The function stays the same as we move from to 3, thus the function is constant on (, 3).
6 27 Objective #4: Finding Local Maximum & Minimum Values of a Function. If we look at the graph from example #4, we can see that the graph has two "valleys". The lowest points in these "valleys" occur at x = 1 and at x = 2. We can say that f( 1) and f(2) are local minimum values (relative minimum). Similarly, the graph has one "top of a hill" point. The "top of the hill" point occurs at x =. We can then say that f() is a local maximum value (relative maximum) "valleys" Definition of local extrema A function f has a local (relative) maximum at x = a if f(a) f(x) for all x in an interval c < x < d containing a. A function f has a local (relative) minimum at x = b if f(b) f(x) for all x in an interval c < x < d containing b. In other words, a function has a local maximum at x = a if y = f(a) is the "highest" y-value in a small neighborhood containing a and a function has a local minimum at x = b if y = f(b) is the "lowest" y-value in a small neighborhood containing b. Notice that with our definition, the local maxima (plural of maximum) and minima (plural of minimum) occur when the function is changing from increasing to decreasing or vice-versa. Find the local maxima and minima values of the following functions: Ex. 6a Ex. 6b "top of a hill"
7 28 a) There are two peaks at x = 4 and x 4. and one valley at x = 1. Thus, the function has two local maxima, one of f( 4) = 6 at x = 4 and another of f(4.) = 3 at x = 4.. The function has a local minimum of f(1) = 1 at x = 1. b) There is a peak at x 3. and one valley at x 3.. Thus, the function has a local maximum of f(3.) = 2 at x = 3. and a local minimum of f( 3.) = 2 at x = 3.. Objective # Finding Absolute Extrema of a Function. In example #4, the function had to local minima; one of 4 at x = 1 and the other of 32 at x = 2. The second local minimum of 32 at x = 2 was the lowest point of the entire graph. We call such a point the absolute minimum of the function. Since the function in example #4 increasing without bound as x get large both positively and negative, the graph has no absolute maximum. In example #6a, the function had to local maxima; one of 6 at x = 4 and the other of 3 at x = 4.. The first local maximum of 6 at x = 4 was the highest point of the entire graph. We call such a point the absolute maximum of the function. Since the function in example #6a decreasing without bound as x get large both positively and negative, the graph has no absolute minimum. Definition of absolute extrema A function f has an absolute maximum at x = a if f(a) f(x) for all x in the domain of f. A function f has an absolute minimum at x = b if f(b) f(x) for all x in domain of f. Find the absolute extrema of the following functions: Ex. 7a Ex. 7b
8 Ex. 8a a) The lowest point on the graph is ( 6, 2) so the function has an absolute minimum of 2 at x = 6. The highest point on the graph is ( 2, 8), so the function has an absolute maximum of 8 at x = 2. b) The lowest point on the graph is ( 3, 14) so the function has an absolute minimum of 14 at x = 3. Since the function increases without bound for large positive and negative x, there is no absolute maximum. Ex. 8b 29 a) The lowest point on the graph seams to be near (3, 1), but x = 3 is excluded from the domain since there is a hole at (3, 1). Hence there is no absolute minimum. Since the function increases without bound for large positive and negative x, there is no absolute maximum either. b) The lowest point on the graph is every point between ( 1, 1) and (2, 1) inclusively, so all of those points are absolute maxima. The highest point on the graph is ( 3, 9), so the function has an absolute maximum of 9 at x = 3. In Calculus, you will study an important theorem called the Extreme Value Theorem which applies to functions that are "continuous" (i.e., a function is continuous at a point c if you can draw the graph through c without lifting a pencil). The theorem states that if a function is "continuous" on a closed interval [a, b], then f has an absolute maximum and minimum on [a, b].
9 3 Objective #6: Finding the Absolute Extrema in an Application: Ex. 9 An elementary school plans to build a rectangular playground that is 4,9 square meters in area. The playground is to be surrounded by a fence. Express the length of the fencing as a function of the length of one of the sides of the playground, draw the graph, and find the dimensions of the playground requiring the least amount of fencing. Let L = the length of the playground & w = the width of the playground The perimeter of the playground is: P = 2L + 2w and the area is A = Lw = 49. Solving the area equation for L yields: L = 49. Substituting this answer into the perimeter equation yields: w P = 2L + 2w = 2( 49 w Thus, our function is P(w) = 98 w ) + 2w = 98 w + 2w. + 2w. The domain is w > As an aid to graphing the function, let s generate a table of values, plot the points, and draw a smooth curve through the points: w P(w) w P(w)
10 From the graph, it appears that the dimensions should be 7 feet by 7 feet in order to use the minimum amount of fencing. 31 Objective #7: Average Rate of Change. In order to calculate the slope of a line, we took the vertical change (y 2 y 1 ) divided by the horizontal change (x 2 x 1 ). Thus, the slope gives us the average rate of change in y-units compared to a unit of x. To illustrate how this works, consider the following example: Calculate the slope: Ex. 1 Leroy started his driving trip 8 miles north of Austin and headed north at 6 mph. After every hour, Leroy plotted the distance he was from Austin versus the time had been on the road. The graph he created is to the right. Calculate the slope and interpret what it represents Distance from Austin (miles) (3, 27) (2, 21) (1, 14) Hours Driving We first pick two distinct points on the graph: (2, 21) & (3, 27). Calculating the slope, we get: m = y 2 y 1 x 2 x 1 = = 6 1 = 6. Thus, the slope is the average rate of change of the distance travelled compared to one hour of time or more simply, the average speed. For any two distinct points on a curve, we can find the average rate of change from the first point to the second point by drawing a straight line (secant line) through those points and calculating the slope of that line. This will give us the average rate of change between those two points. Average Rate of Change The average rate of change in the function y = f(x) from x = a to x = b is average rate of change = change in y change in x f(b) f(a) = b a
11 Average rate of change is the slope of the secant line between the points (a, f(a)) and (b, f(b)) 32 f(x) f(b) Secant line (b, f(b)) f(a) (a, f(a)) a Determine the average rate of change of the function between the given values of the variable. Then find the equation of the secant line: Ex. 11a f(x) = 4x 2 x 3 ; x = 2, x = 4 Ex. 11b g(x) = x 2 ; x = 3, x = a) f(4) = 4(4) 2 (4) 3 = and f( 2) = 4( 2) 2 ( 2) 3 = 24 Average rate of change = b f(b) f(a) b a = f(4) f( 2) 4 ( 2) = = 24 6 = 4. This also is the slope of the secant line. Using m = 4 and (x 1,y 1 ) = (4, ), we can find the equation of the secant line: y y 1 = m(x x 1 ) y = 4(x 4) y = 4x + 16 b) g() = () 2 = 2 and g(3) = (3) 2 = 4 Average rate of change = g(b) g(a) b a = g() g(3) (3) = = 16 2 = 8. This also is the slope of the secant line. Using m = 8 and (x 1,y 1 ) = (3, 4), we can find the equation of the secant line: y y 1 = m(x x 1 ) y 4 = 8(x 3) y 4 = 8x 24 y = 8x 2
Notes: Piecewise Functions
Objective: Students will be able to write evaluate piecewise defined functions, graph piecewise defined functions, evaluate the domain and range for piecewise defined functions, and solve application problems.
More informationSection 1.3 Rates of Change and Behavior of Graphs
Section 1. Rates of Change and Behavior of Graphs 5 Section 1. Rates of Change and Behavior of Graphs Since functions represent how an output quantity varies with an input quantity, it is natural to ask
More informationAP Calculus AB. Chapter IV Lesson B. Curve Sketching
AP Calculus AB Chapter IV Lesson B Curve Sketching local maxima Absolute maximum F I A B E G C J Absolute H K minimum D local minima Summary of trip along curve critical points occur where the derivative
More informationPolynomial functions right- and left-hand behavior (end behavior):
Lesson 2.2 Polynomial Functions For each function: a.) Graph the function on your calculator Find an appropriate window. Draw a sketch of the graph on your paper and indicate your window. b.) Identify
More informationSection 3.1 Extreme Values
Math 132 Extreme Values Section 3.1 Section 3.1 Extreme Values Example 1: Given the following is the graph of f(x) Where is the maximum (x-value)? What is the maximum (y-value)? Where is the minimum (x-value)?
More informationName Date. Answers 1.
Name Date Honors Algebra 2 Summer Work Due at Meet the Teacher Night Show all work. You will be graded on accuracy and completion. Partial credit will be given on problems where work is not shown. 1. Plot
More informationA function is actually a simple concept; if it were not, history would have replaced it with a simpler one by now! Here is the definition:
1.2 Functions and Their Properties A function is actually a simple concept; if it were not, history would have replaced it with a simpler one by now! Here is the definition: Definition: Function, Domain,
More information2.4 Slope and Rate of Change
2.4 Slope and Rate of Change Learning Objectives Find positive and negative slopes. Recognize and find slopes for horizontal and vertical lines. Understand rates of change. Interpret graphs and compare
More informationThe Mean Value Theorem and its Applications
The Mean Value Theorem and its Applications Professor Richard Blecksmith richard@math.niu.edu Dept. of Mathematical Sciences Northern Illinois University http://math.niu.edu/ richard/math229 1. Extreme
More informationDetermine whether the formula determines y as a function of x. If not, explain. Is there a way to look at a graph and determine if it's a function?
1.2 Functions and Their Properties Name: Objectives: Students will be able to represent functions numerically, algebraically, and graphically, determine the domain and range for functions, and analyze
More informationLimits, Rates of Change, and Tangent Lines
Limits, Rates of Change, and Tangent Lines jensenrj July 2, 2018 Contents 1 What is Calculus? 1 2 Velocity 2 2.1 Average Velocity......................... 3 2.2 Instantaneous Velocity......................
More informationThe plot shows the graph of the function f (x). Determine the quantities.
MATH 211 SAMPLE EXAM 1 SOLUTIONS 6 4 2-2 2 4-2 1. The plot shows the graph of the function f (x). Determine the quantities. lim f (x) (a) x 3 + Solution: Look at the graph. Let x approach 3 from the right.
More informationChapter 2. Polynomial and Rational Functions. 2.3 Polynomial Functions and Their Graphs. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter Polynomial and Rational Functions.3 Polynomial Functions and Their Graphs Copyright 014, 010, 007 Pearson Education, Inc. 1 Objectives: Identify polynomial functions. Recognize characteristics
More informationChapter 3: The Derivative in Graphing and Applications
Chapter 3: The Derivative in Graphing and Applications Summary: The main purpose of this chapter is to use the derivative as a tool to assist in the graphing of functions and for solving optimization problems.
More informationApplications of Derivatives
Applications of Derivatives Extrema on an Interval Objective: Understand the definition of extrema of a function on an interval. Understand the definition of relative extrema of a function on an open interval.
More informationSecondary Math 3 Honors Unit 10: Functions Name:
Secondary Math 3 Honors Unit 10: Functions Name: Parent Functions As you continue to study mathematics, you will find that the following functions will come up again and again. Please use the following
More information1.2 Functions and Their Properties Name:
1.2 Functions and Their Properties Name: Objectives: Students will be able to represent functions numerically, algebraically, and graphically, determine the domain and range for functions, and analyze
More informationMath Practice Exam 2 - solutions
Math 181 - Practice Exam 2 - solutions Problem 1 A population of dinosaurs is modeled by P (t) = 0.3t 2 + 0.1t + 10 for times t in the interval [ 5, 0]. a) Find the rate of change of this population at
More informationTopic 25: Quadratic Functions (Part 1) A quadratic function is a function which can be written as 2. Properties of Quadratic Functions
Hartfield College Algebra (Version 015b - Thomas Hartfield) Unit FOUR Page 1 of 3 Topic 5: Quadratic Functions (Part 1) Definition: A quadratic function is a function which can be written as f x ax bx
More information3 Geometrical Use of The Rate of Change
Arkansas Tech University MATH 224: Business Calculus Dr. Marcel B. Finan Geometrical Use of The Rate of Change Functions given by tables of values have their limitations in that nearly always leave gaps.
More informationGraphs of Polynomial Functions
Graphs of Polynomial Functions By: OpenStaxCollege The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in [link]. Year 2006 2007 2008 2009 2010 2011 2012 2013
More information12 Rates of Change Average Rates of Change. Concepts: Average Rates of Change
12 Rates of Change Concepts: Average Rates of Change Calculating the Average Rate of Change of a Function on an Interval Secant Lines Difference Quotients Approximating Instantaneous Rates of Change (Section
More information5.1 Extreme Values of Functions
5.1 Extreme Values of Functions Lesson Objective To be able to find maximum and minimum values (extrema) of functions. To understand the definition of extrema on an interval. This is called optimization
More informationPolynomial and Rational Functions. Chapter 3
Polynomial and Rational Functions Chapter 3 Quadratic Functions and Models Section 3.1 Quadratic Functions Quadratic function: Function of the form f(x) = ax 2 + bx + c (a, b and c real numbers, a 0) -30
More informationSection 3.4 Library of Functions; Piecewise-Defined Functions
Section. Library of Functions; Piecewise-Defined Functions Objective #: Building the Library of Basic Functions. Graph the following: Ex. f(x) = b; constant function Since there is no variable x in the
More informationPrecalculus Lesson 4.1 Polynomial Functions and Models Mrs. Snow, Instructor
Precalculus Lesson 4.1 Polynomial Functions and Models Mrs. Snow, Instructor Let s review the definition of a polynomial. A polynomial function of degree n is a function of the form P(x) = a n x n + a
More informationDerivatives and Shapes of Curves
MATH 1170 Section 43 Worksheet NAME Derivatives and Shapes of Curves In Section 42 we discussed how to find the extreme values of a function using the derivative These results say, In Chapter 2, we discussed
More informationMATH 116, LECTURE 13, 14 & 15: Derivatives
MATH 116, LECTURE 13, 14 & 15: Derivatives 1 Formal Definition of the Derivative We have seen plenty of limits so far, but very few applications. In particular, we have seen very few functions for which
More informationThe Mean Value Theorem Rolle s Theorem
The Mean Value Theorem In this section, we will look at two more theorems that tell us about the way that derivatives affect the shapes of graphs: Rolle s Theorem and the Mean Value Theorem. Rolle s Theorem
More informationDefinition: For nonempty sets X and Y, a function, f, from X to Y is a relation that associates with each element of X exactly one element of Y.
Functions Definition: A relation is a correspondence between two sets. If x and y are two elements in these sets and if a relation exists between x and y, then we say that x corresponds to y or that y
More informationSection 1.2 DOMAIN, RANGE, INTERCEPTS, SYMMETRY, EVEN/ODD
Section 1.2 DOMAIN, RANGE, INTERCEPTS, SYMMETRY, EVEN/ODD zeros roots line symmetry point symmetry even function odd function Estimate Function Values A. ADVERTISING The function f (x) = 5x 2 + 50x approximates
More informationFinal Jeopardy! Appendix Ch. 1 Ch. 2 Ch. 3 Ch. 4 Ch. 5
Final Jeopardy! Appendix Ch. 1 Ch. Ch. 3 Ch. 4 Ch. 5 00 00 00 00 00 00 400 400 400 400 400 400 600 600 600 600 600 600 800 800 800 800 800 800 1000 1000 1000 1000 1000 1000 APPENDIX 00 Is the triangle
More informationCalculus and Structures
6 CHAPTER 1 LINES 7 Copyright Chapter 1 LINES 1.1 LINES A line is the easiest mathematical structure to describe. You need to know only two things about a line to describe it. For example: i) the y-intercept,
More informationMTH 103 Group Activity Problems (W1B) Name: Types of Functions and Their Rates of Change Section 1.4 part 1 (due April 6)
MTH 103 Group Activity Problems (W1B) Name: Types of Functions and Their Rates of Change Section 1.4 part 1 (due April 6) Learning Objectives Identify linear and nonlinear functions Interpret slope as
More informationSection 4.2: The Mean Value Theorem
Section 4.2: The Mean Value Theorem Before we continue with the problem of describing graphs using calculus we shall briefly pause to examine some interesting applications of the derivative. In previous
More informationMAT 122 Homework 7 Solutions
MAT 1 Homework 7 Solutions Section 3.3, Problem 4 For the function w = (t + 1) 100, we take the inside function to be z = t + 1 and the outside function to be z 100. The derivative of the inside function
More informationCalculus 221 worksheet
Calculus 221 worksheet Graphing A function has a global maximum at some a in its domain if f(x) f(a) for all other x in the domain of f. Global maxima are sometimes also called absolute maxima. A function
More informationWhich car/s is/are undergoing an acceleration?
Which car/s is/are undergoing an acceleration? Which car experiences the greatest acceleration? Match a Graph Consider the position-time graphs below. Each one of the 3 lines on the position-time graph
More informationChapter 2 Polynomial and Rational Functions
SECTION.1 Linear and Quadratic Functions Chapter Polynomial and Rational Functions Section.1: Linear and Quadratic Functions Linear Functions Quadratic Functions Linear Functions Definition of a Linear
More informationAP Calculus ---Notecards 1 20
AP Calculus ---Notecards 1 20 NC 1 For a it to exist, the left-handed it must equal the right sided it x c f(x) = f(x) = L + x c A function can have a it at x = c even if there is a hole in the graph at
More information5.1 Extreme Values of Functions. Borax Mine, Boron, CA Photo by Vickie Kelly, 2004 Greg Kelly, Hanford High School, Richland, Washington
5.1 Extreme Values of Functions Borax Mine, Boron, CA Photo by Vickie Kelly, 2004 Greg Kelly, Hanford High School, Richland, Washington 5.1 Extreme Values of Functions Borax Plant, Boron, CA Photo by Vickie
More informationINCREASING AND DECREASING FUNCTIONS
f (x) = x 3 x 2 2x + 3.5 5 4 3 2 1.5-0.5 0 0.5 1 1.5 2 increasing Walking uphill means the function is increasing f (x) = x 3 x 2 2x + 3.5 5 4 3 2 1.5-0.5 0 0.5 1 1.5 2 increasing decreasing Walking uphill
More informationInfinite Limits. Infinite Limits. Infinite Limits. Previously, we discussed the limits of rational functions with the indeterminate form 0/0.
Infinite Limits Return to Table of Contents Infinite Limits Infinite Limits Previously, we discussed the limits of rational functions with the indeterminate form 0/0. Now we will consider rational functions
More informationter. on Can we get a still better result? Yes, by making the rectangles still smaller. As we make the rectangles smaller and smaller, the
Area and Tangent Problem Calculus is motivated by two main problems. The first is the area problem. It is a well known result that the area of a rectangle with length l and width w is given by A = wl.
More informationLinear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)}
Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in
More informationUNIT 6 DESCRIBING DATA Lesson 2: Working with Two Variables. Instruction. Guided Practice Example 1
Guided Practice Eample 1 Andrew wants to estimate his gas mileage, or miles traveled per gallon of gas used. He records the number of gallons of gas he purchased and the total miles he traveled with that
More informationSection 3.2 Polynomial Functions and Their Graphs
Section 3.2 Polynomial Functions and Their Graphs EXAMPLES: P (x) = 3, Q(x) = 4x 7, R(x) = x 2 + x, S(x) = 2x 3 6x 2 10 QUESTION: Which of the following are polynomial functions? (a) f(x) = x 3 + 2x +
More informationf (x) = x 2 Chapter 2 Polynomial Functions Section 4 Polynomial and Rational Functions Shapes of Polynomials Graphs of Polynomials the form n
Chapter 2 Functions and Graphs Section 4 Polynomial and Rational Functions Polynomial Functions A polynomial function is a function that can be written in the form a n n 1 n x + an 1x + + a1x + a0 for
More informationMATH 114 Calculus Notes on Chapter 2 (Limits) (pages 60-? in Stewart)
Still under construction. MATH 114 Calculus Notes on Chapter 2 (Limits) (pages 60-? in Stewart) As seen in A Preview of Calculus, the concept of it underlies the various branches of calculus. Hence we
More informationEXAM 1 Review. 1. Find the distance between the points (2, 6) and ( 5, 2). Give the exact solution and an approximation to the nearest hundredth.
EXAM 1 Review 1. Find the distance between the points (2, 6) and ( 5, 2). Give the exact solution and an approximation to the nearest hundredth. 2. Find the midpoint of the line segment with end points
More informationBasic Equations and Inequalities
Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 1 - of 45 Topic 0: Definition: Ex. 1 Basic Equations and Inequalities An equation is a statement that the values of two expressions
More informationBob Brown Math 251 Calculus 1 Chapter 4, Section 1 Completed 1 CCBC Dundalk
Bob Brown Math 251 Calculus 1 Chapter 4, Section 1 Completed 1 Absolute (or Global) Minima and Maxima Def.: Let x = c be a number in the domain of a function f. f has an absolute (or, global ) minimum
More informationAlgebra I EOC Review (Part 2)
1. Let x = total miles the car can travel Answer: x 22 = 18 or x 18 = 22 2. A = 1 2 ah 1 2 bh A = 1 h(a b) 2 2A = h(a b) 2A = h a b Note that when solving for a variable that appears more than once, consider
More information1) Solve the formula for the indicated variable. P = 2L + 2W for W. 2) Solve the formula for the variable y. 5 = 7x - 8y
Math120 Cumulative Review This is to help prepare you for the 40 question final exam. It is not all inclusive of the material covered in your course. Therefore items not on this review may appear on the
More informationTransportation Costs (or Trucker s Dilemma) Math 1050 College Algebra Project
Name Carlos Vega Date 03/11/16 Transportation Costs (or Trucker s Dilemma) Math 1050 College Algebra Project A truck driving 260 miles over a flat interstate at a constant rate of 50 miles per hour gets
More information1. The graph of a quadratic function is shown. Each square is one unit.
1. The graph of a quadratic function is shown. Each square is one unit. a. What is the vertex of the function? b. If the lead coefficient (the value of a) is 1, write the formula for the function in vertex
More informationBonus Homework and Exam Review - Math 141, Frank Thorne Due Friday, December 9 at the start of the final exam.
Bonus Homework and Exam Review - Math 141, Frank Thorne (thornef@mailbox.sc.edu) Due Friday, December 9 at the start of the final exam. It is strongly recommended that you do as many of these problems
More informationPolynomial and Rational Functions. Copyright Cengage Learning. All rights reserved.
2 Polynomial and Rational Functions Copyright Cengage Learning. All rights reserved. 2.2 Polynomial Functions of Higher Degree Copyright Cengage Learning. All rights reserved. What You Should Learn Use
More information6-1 Slope. Objectives 1. find the slope of a line 2. use rate of change to solve problems
6-1 Slope Objectives 1. find the slope of a line 2. use rate of change to solve problems What is the meaning of this sign? 1. Icy Road Ahead 2. Steep Road Ahead 3. Curvy Road Ahead 4. Trucks Entering Highway
More information1 Lecture 25: Extreme values
1 Lecture 25: Extreme values 1.1 Outline Absolute maximum and minimum. Existence on closed, bounded intervals. Local extrema, critical points, Fermat s theorem Extreme values on a closed interval Rolle
More informationCH 2: Limits and Derivatives
2 The tangent and velocity problems CH 2: Limits and Derivatives the tangent line to a curve at a point P, is the line that has the same slope as the curve at that point P, ie the slope of the tangent
More informationRewriting Absolute Value Functions as Piece-wise Defined Functions
Rewriting Absolute Value Functions as Piece-wise Defined Functions Consider the absolute value function f ( x) = 2x+ 4-3. Sketch the graph of f(x) using the strategies learned in Algebra II finding the
More informationConcepts of graphs of functions:
Concepts of graphs of functions: 1) Domain where the function has allowable inputs (this is looking to find math no-no s): Division by 0 (causes an asymptote) ex: f(x) = 1 x There is a vertical asymptote
More informationSection 1.1: THE DISTANCE AND MIDPOINT FORMULAS; GRAPHING UTILITIES; INTRODUCTION TO GRAPHING EQUATIONS
PRECALCULUS I: COLLEGE ALGEBRA GUIDED NOTEBOOK FOR USE WITH SULLIVAN AND SULLIVAN PRECALCULUS ENHANCED WITH GRAPHING UTILITIES, BY SHANNON MYERS (FORMERLY GRACEY) Section 1.1: THE DISTANCE AND MIDPOINT
More informationSuppose that f is continuous on [a, b] and differentiable on (a, b). Then
Lectures 1/18 Derivatives and Graphs When we have a picture of the graph of a function f(x), we can make a picture of the derivative f (x) using the slopes of the tangents to the graph of f. In this section
More informationINTERMEDIATE VALUE THEOREM
THE BIG 7 S INTERMEDIATE VALUE If f is a continuous function on a closed interval [a, b], and if k is any number between f(a) and f(b), where f(a) f(b), then there exists a number c in (a, b) such that
More informationMATH 2053 Calculus I Review for the Final Exam
MATH 05 Calculus I Review for the Final Exam (x+ x) 9 x 9 1. Find the limit: lim x 0. x. Find the limit: lim x + x x (x ).. Find lim x (x 5) = L, find such that f(x) L < 0.01 whenever 0 < x
More informationChapter 3 The Integral Business Calculus 197
Chapter The Integral Business Calculus 97 Chapter Exercises. Let A(x) represent the area bounded by the graph and the horizontal axis and vertical lines at t=0 and t=x for the graph in Fig.. Evaluate A(x)
More informationAlgebra 1 S1 Lesson Summaries. Lesson Goal: Mastery 70% or higher
Algebra 1 S1 Lesson Summaries For every lesson, you need to: Read through the LESSON REVIEW which is located below or on the last page of the lesson and 3-hole punch into your MATH BINDER. Read and work
More informationNorth Carolina State University
North Carolina State University MA 141 Course Text Calculus I by Brenda Burns-Williams and Elizabeth Dempster August 7, 2014 Section1 Functions Introduction In this section, we will define the mathematical
More informationSection 2.1. Increasing, Decreasing, and Piecewise Functions; Applications
Section 2.1 Increasing, Decreasing, and Piecewise Functions; Applications Features of Graphs Intervals of Increase and Decrease A function is increasing when the graph goes up as you travel along it from
More informationPlease print the following information in case your scan sheet is misplaced:
MATH 1100 Common Final Exam FALL 010 December 10, 010 Please print the following information in case your scan sheet is misplaced: Name: Instructor: Student ID: Section/Time: The exam consists of 40 multiple
More informationQuarter 1 Calculus Test. The attached problems do not comprise a comprehensive test review. Test topics
Quarter 1 Calculus Test The attached problems do not comprise a comprehensive test review. As review, use: this packet your 3 quizzes and first test your 3 quiz review packets and first test review packet
More informationMATH 121: EXTRA PRACTICE FOR TEST 2. Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam.
MATH 121: EXTRA PRACTICE FOR TEST 2 Disclaimer: Any material covered in class and/or assigned for homework is a fair game for the exam. 1 Linear Functions 1. Consider the functions f(x) = 3x + 5 and g(x)
More information1) The line has a slope of ) The line passes through (2, 11) and. 6) r(x) = x + 4. From memory match each equation with its graph.
Review Test 2 Math 1314 Name Write an equation of the line satisfying the given conditions. Write the answer in standard form. 1) The line has a slope of - 2 7 and contains the point (3, 1). Use the point-slope
More informationAP Calculus Summer Homework MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
AP Calculus Summer Homework 2015-2016 Part 2 Name Score MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the distance d(p1, P2) between the points
More informationMath 131. Increasing/Decreasing Functions and First Derivative Test Larson Section 3.3
Math 131. Increasing/Decreasing Functions and First Derivative Test Larson Section 3.3 Increasing and Decreasing Functions. A function f is increasing on an interval if for any two numbers x 1 and x 2
More informationPosition, Speed and Velocity Position is a variable that gives your location relative to an origin. The origin is the place where position equals 0.
Position, Speed and Velocity Position is a variable that gives your location relative to an origin. The origin is the place where position equals 0. The position of this car at 50 cm describes where the
More informationChapter 2: Limits & Continuity
Name: Date: Period: AP Calc AB Mr. Mellina Chapter 2: Limits & Continuity Sections: v 2.1 Rates of Change of Limits v 2.2 Limits Involving Infinity v 2.3 Continuity v 2.4 Rates of Change and Tangent Lines
More informationMath 1241, Spring 2014 Section 3.3. Rates of Change Average vs. Instantaneous Rates
Math 1241, Spring 2014 Section 3.3 Rates of Change Average vs. Instantaneous Rates Average Speed The concept of speed (distance traveled divided by time traveled) is a familiar instance of a rate of change.
More informationChapter 1 Skills Points and Linear Equations
Example 1. Solve We have Chapter 1 Skills Points and Linear Equations t = 3 t t = 3 t for t. = ( t)( t) = ( t) = 3( t) = 4 4t = 6 3t = = t t = 3 ( t)( t) t Example. Solve We have = A + Bt A Bt for t. =
More informationMath 3 Variable Manipulation Part 7 Absolute Value & Inequalities
Math 3 Variable Manipulation Part 7 Absolute Value & Inequalities 1 MATH 1 REVIEW SOLVING AN ABSOLUTE VALUE EQUATION Absolute value is a measure of distance; how far a number is from zero. In practice,
More information2. To receive credit on any problem, you must show work that explains how you obtained your answer or you must explain how you obtained your answer.
Math 50, Fall 2011 Test 3 PRINT your name on the back of the test. Directions 1. Time limit: 1 hour 50 minutes. 2. To receive credit on any problem, you must show work that explains how you obtained your
More informationWednesday August 24, 2016
1.1 Functions Wednesday August 24, 2016 EQs: 1. How to write a relation using set-builder & interval notations? 2. How to identify a function from a relation? 1. Subsets of Real Numbers 2. Set-builder
More information(10) What is the domain of log 123 (x)+ x
EXAM 1 MASTER STUDY GUIDE MATH 131 I don't expect you to complete this. This is a very large list. I wanted to give you as much information about things that could be asked of you as possible. On Tuesday,
More informationGiven a polynomial and one of its factors, find the remaining factors of the polynomial. 4. x 3 6x x 6; x 1 SOLUTION: Divide by x 1.
Use synthetic substitution to find f (4) and f ( 2) for each function. 2. f (x) = x 4 + 8x 3 + x 2 4x 10 Divide the function by x 4. The remainder is 758. Therefore, f (4) = 758. Divide the function by
More information4.1 & 4.2 Student Notes Using the First and Second Derivatives. for all x in D, where D is the domain of f. The number f()
4.1 & 4. Student Notes Using the First and Second Derivatives Definition A function f has an absolute maximum (or global maximum) at c if f ( c) f ( x) for all x in D, where D is the domain of f. The number
More informationMath 112 Group Activity: The Vertical Speed of a Shell
Name: Section: Math 112 Group Activity: The Vertical Speed of a Shell A shell is fired straight up by a mortar. The graph below shows its altitude as a function of time. 400 300 altitude (in feet) 200
More informationMA Lesson 12 Notes Section 3.4 of Calculus part of textbook
MA 15910 Lesson 1 Notes Section 3.4 of Calculus part of textbook Tangent Line to a curve: To understand the tangent line, we must first discuss a secant line. A secant line will intersect a curve at more
More informationMTH 241: Business and Social Sciences Calculus
MTH 241: Business and Social Sciences Calculus F. Patricia Medina Department of Mathematics. Oregon State University January 28, 2015 Section 2.1 Increasing and decreasing Definition 1 A function is increasing
More informationOne-Variable Calculus
POLI 270 - Mathematical and Statistical Foundations Department of Political Science University California, San Diego September 30, 2010 1 s,, 2 al Relationships Political Science, economics, sociology,
More informationChapter 3: Inequalities, Lines and Circles, Introduction to Functions
QUIZ AND TEST INFORMATION: The material in this chapter is on Quiz 3 and Exam 2. You should complete at least one attempt of Quiz 3 before taking Exam 2. This material is also on the final exam and used
More information2.1 The Tangent and Velocity Problems
2.1 The Tangent and Velocity Problems Tangents What is a tangent? Tangent lines and Secant lines Estimating slopes from discrete data: Example: 1. A tank holds 1000 gallons of water, which drains from
More informationAnnouncements. Topics: Homework: - sections , 6.1 (extreme values) * Read these sections and study solved examples in your textbook!
Announcements Topics: - sections 5.2 5.7, 6.1 (extreme values) * Read these sections and study solved examples in your textbook! Homework: - review lecture notes thoroughly - work on practice problems
More informationUnit 3 Functions HW #1 Mrs. Dailey
HW#1 Name Algebra II Unit Functions HW #1 Mrs. Dailey 1) In each of the following, the variable pair given are proportional to one another. Find the missing value. (a) b = 8 when a = 16 b =? when a = 18
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Math 1325 Ch.12 Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the location and value of each relative extremum for the function. 1)
More informationIntroduction to Calculus
Introduction to Calculus Contents 1 Introduction to Calculus 3 11 Introduction 3 111 Origin of Calculus 3 112 The Two Branches of Calculus 4 12 Secant and Tangent Lines 5 13 Limits 10 14 The Derivative
More informationSection 3.7. Rolle s Theorem and the Mean Value Theorem
Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate of change and the average rate of change of
More informationIncreasing or Decreasing Nature of a Function
Öğr. Gör. Volkan ÖĞER FBA 1021 Calculus 1/ 46 Increasing or Decreasing Nature of a Function Examining the graphical behavior of functions is a basic part of mathematics and has applications to many areas
More informationFind all points where the function is discontinuous. 1) Find all vertical asymptotes of the given function. x(x - 1) 2) f(x) =
Math 90 Final Review Find all points where the function is discontinuous. ) Find all vertical asymptotes of the given function. x(x - ) 2) f(x) = x3 + 4x Provide an appropriate response. 3) If x 3 f(x)
More information