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The document discusses slope and average rate of change in multiple contexts. It provides examples of finding the slope of a line, the equation of a line parallel or perpendicular to another, and the average rate of change of functions. It discusses how average rate of change is the slope of a secant line between two points, and how this can be used to analyze real-world situations like changes in drug concentration over time.
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1.
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categorical.
4.
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1.
Stream quality is based on the levels of many variables, including the following. Which of these variables is quantitative?
The amount of dissolved oxygen
The number of distinct species present
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All of the above
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Weight of a fish
Length of a fish
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Geckos are lizards with specialized toe pads that enable them to easily climb all sorts of surfaces. A research team examined the adhesive properties of 7 Tokay geckos. Below are their toe-pad areas (in square centimeters, cm2).
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This document summarizes a presentation on recent advances in math interventions. It discusses key concepts like math proficiency, foundational math skills, and common math myths. It also outlines a multi-tiered system of supports including strategies for Tier 1 core instruction, Tier 2 small group interventions, and Tier 3 individualized supports. Specific evidence-based practices are recommended, such as explicit instruction, strategic use of math vocabulary, and targeting foundational skills like whole number knowledge before rational numbers.
Presentation1
This document discusses completing the square to factor quadratic trinomials. It provides examples of factoring expressions of the form x^2 + bx + c by finding the constant to add inside the square. These quadratic trinomials can be written as perfect squares minus/plus a constant. The document demonstrates completing the square on sample expressions and emphasizes that this method allows solving quadratic equations using square roots rather than the quadratic formula.
Sulpcegu5e ppt 14_5
This document discusses two objectives related to calculating areas and integrals. The first objective covers finding the area under a curve using integrals, while the second objective likely expands on this topic or covers related concepts.
Sulpcegu5e ppt 14_4
The document discusses finding the equation of the tangent line to the graph of f(x)=2x^2 at the point (1,2). It also discusses the motion of a ball thrown straight up in the air, with the equation h(t)=-16t^2 + 64t, where t is time. It asks for the time when the ball strikes the ground, passes the rooftop on the way down, and its instantaneous speed at various times.
Sulpcegu5e ppt 14_3
This section discusses one-sided limits and continuous functions. It has two objectives: 1) To define and evaluate one-sided limits and use them to determine if a function is continuous at a point. 2) To use limits to analyze the graph of a function near any discontinuities and determine if they are removable, jump, or essential discontinuities.
Sulpcegu5e ppt 14_2
This document outlines 5 objectives for learning algebra techniques for finding limits. The objectives cover skills for determining limits of functions algebraically by factoring, simplifying rational expressions, applying properties of exponents, and evaluating one-sided limits.
Sulpcegu5e ppt 14_1
This document discusses finding limits using tables and graphs. The first objective relates to using tables and graphs to find limits, while the second objective likely continues the discussion on finding limits through tables and graphs. In summary, the document covers finding limits through tables and graphs across two objectives.
Sulpcegu5e ppt 13_3
The document discusses probability and contains 4 objectives: 1) The probability of a head or tail in a fair coin toss. 2) The probability of having 3 boys and 1 girl in a 4-child family is 1 in 4. 3) The probability of drawing a heart from a standard deck is 13/52 and drawing an ace is 4/52. 4) The probability of rain is 30% so not raining is 70%, and the probability that at least 2 people in a group of 15 have the same birthday is over 50%.
Sulpcegu5e ppt 13_2
This document discusses permutations and combinations. It contains examples of calculating the number of possible codes, lineups, and committees given different conditions like allowing or not allowing repetition. It also provides examples of listing combinations and calculating the number of arrangements of different objects like letters, flags, and people on committees.
Sulpcegu5e ppt 13_1
The document discusses three counting objectives: 1) Listing all subsets of the set {1,2,3}, 2) Determining the number of students in a survey of 200 that had been to Florida or New York or neither, and 3) Calculating the number of possible three-symbol code words where the first symbol is an uppercase letter, second is a digit, and third is a lowercase letter.
Sulpcegu5e ppt 12_5
This document discusses the binomial theorem and provides two objectives: 1) Four Useful Formulas and 2) an unnamed second objective. The binomial theorem is a formula for expanding expressions of the form (a + b)^n where a and b are numbers or variables and n is a non-negative integer.
Sulpcegu5e ppt 12_4
Mathematical induction is a method of proof typically used to establish that a property holds for all natural numbers. The principle of mathematical induction states that to show a statement is true for all natural numbers n, it must be shown that the statement holds true for the base case n=1, and also that if the statement holds true for any natural number k, then it must also hold true for k+1. This allows proving statements about an infinite set by considering only a finite number of cases.
Sulpcegu5e ppt 12_3
This document discusses geometric sequences and series. It lists 4 objectives: 1) Identifying geometric sequences by listing the first term and common ratio, 2) and 3) Contain no information but indicate additional objectives, 4) Contains no information. The example geometric sequence given is 2, 8, 32, 128, ... with a first term of 2 and common ratio of 4.
Sulpcegu5e ppt 12_2
This document contains objectives and problems regarding arithmetic sequences. Objective 1 introduces arithmetic sequences and asks to identify the first term and common difference of sequences. Objective 2 asks to find specific terms in arithmetic sequences and identify the first term and common difference given additional information. Objective 3 asks to find sums of terms in arithmetic sequences and evaluate numeric expressions that represent arithmetic sequences. The final problem asks to calculate the total number of seats in a stadium section where each row has two more seats than the row in front of it.
Sulpcegu5e ppt 12_1
This document discusses sequences and provides 5 objectives: 1) introduce sequences, 2) define a recursively defined sequence and write out its first 5 terms, 3) another objective discussing sequences, 4) another objective discussing sequences, 5) another objective discussing sequences.
Sulpcegu5e ppt 11_8
A factory manufactures two types of ice skates - racing skates and figure skates. Racing skates require 8 hours of fabrication and 2 hours of finishing while figure skates require 4 hours of fabrication and 2 hours of finishing. The factory's fabrication department has 224 hours available per day and the finishing department has 72 hours available. The problem is to determine the number of each type of skate to manufacture daily to maximize profit, where racing skates yield $15 profit each and figure skates yield $12 profit each.
Sulpcegu5e ppt 11_8
A factory manufactures two types of ice skates - racing skates and figure skates. Racing skates require 8 hours of fabrication and 2 hours of finishing while figure skates require 4 hours of fabrication and 2 hours of finishing. The factory's fabrication department has 224 hours available per day and the finishing department has 72 hours available. The problem is to determine the number of each type of skate to manufacture daily to maximize profit, where racing skates yield $15 profit each and figure skates yield $12 profit each.
Sulpcegu5e ppt 11_7
This document discusses systems of inequalities and contains 3 objectives: the first objective covers half-planes, the second objective is not specified, and the third objective is also not specified.
Sulpcegu5e ppt 11_6
This document discusses solving systems of nonlinear equations. It recommends graphing the equations if there are two variables and the equations are easy to graph, as this allows you to see how many solutions there are and their approximate locations. It also notes that extraneous solutions can occur when solving nonlinear systems, so all apparent solutions must be verified.
Sulpcegu5e ppt 11_5
This document discusses partial fraction decomposition and outlines 4 objectives: 1) decompose rational functions into partial fractions, 2) find the coefficients of the partial fractions, 3) use partial fraction decomposition to evaluate integrals, and 4) use partial fraction decomposition to solve application problems.
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2 4 Bzca5e
- 1. Section 2.4 More on Slope
- 2. Parallel and Perpendicular Lines
- 3. y=2x+7
- 5. Example Write the equation in slope intercept form for a line that is parallel to 3x-4y=12 and passing through (5,2).
- 8. Example Write the equation in slope intercept form for a line perpendicular to 3x-4y=12 and passing through (5,2).
- 9. Slope as Rate of Change
- 10. Definition- Slope is defined as the ratio of a change in y to a corresponding change in x. or
- 11. Interpreting a real life situation The line graphs the percent of US adults who smoke cigarettes x years after 1997. a. Find the slope of the line segment from 1997 to 2007 Percent Adults X years after 1997 ( 0,24.7 ) ( 10,19.5 ) b. What does this slope represent? The percent of US adult cigarette smokers is decreasing by .52 percent each year. The change is consistent each year.
- 12. The Average Rate of Change of a Function
- 15. The slope of this line between the points (1,3.83) and (5,7.83) is (1,3.83) (5,7.83 )
- 16. The slope of this line between the points (1,3.83) and (4,7.34) is (1,3.83) (4,7.34 ) Continuation of same problem
- 17. The slope of this line between the points (1,3.83) and (3,6.5) is (1,3.83 ) (3,6.5) Continuation of same problem
- 19. Find the average rate of change of f(x)= x 3
- 20. Example Find the average rate of change of f(x)=3x-1 from x 1 =0 to x 2 =1 x 1 =1 to x 2 =2 x 1 =2 to x 2 =3
- 22. Average rate of change and the difference quotient Do you recognize the difference quotient that we studied in section 2.2? You will study more about the difference quotient in future math classes. Suppose x 1 =x and x 2 =x+h, then
- 24. Average Rate of Change Application 0.00 0.00 0.50 6.14 1.00 8.10 1.50 8.54 2.00 8.43 2.50 8.13 3.00 7.77 3.50 7.41 4.00 7.05 4.50 6.71 5.00 6.38 Sometimes it takes a while for a drug to diffuse sufficiently to affect the desired organ. The curve below is a close approximation of the concentration in that organ compared to the time after the drug was taken. a. What is the average rate of change from the time the drug was taken until the first hour? b. What was the average rate of change from the second hour until the fourth.? c. What is the interpretation of each answer? hours cncntr hours concentration Example
- 25. (a) (b) (c) (d)
- 26. (a) (b) (c) (d)
- 27. (a) (b) (c) (d)