37 Questions
What are some concepts covered in the chapter 'Preparation for Calculus'?
Sketching the graphs of equations and functions, graphing equations of lines, evaluating and graphing functions, fitting mathematical models to data
Which French mathematician revolutionized the study of mathematics by joining algebra and geometry with the coordinate plane?
René Descartes
What is the main advantage of representing points in the plane by pairs of real numbers?
It allows for analytic formulation of geometric concepts and graphical representation of algebraic concepts.
Graphical representation of equations shows the exact and complete graph of the equation.
False
The point (2, 1) is a solution point of the equation 3x - y = 7 because when 2 is substituted for x and 1 is substituted for y, the equation is ________.
satisfied
What is the mathematical model of the form y = at^2 - bt + c for the CPI data provided?
y = 0.5327t^2 - 1944.9t + 1813385
What is the predicted CPI for the year 2010 based on the model?
201.6
What is the mathematical model of the form y = at^2 - bt + c for the cellular phone subscriber data provided?
y = 53.9015t^2 - 263944.8t + 248631981
What is the predicted number of cellular phone subscribers for the year 2010 based on the model?
284
What is the term used for the points at which the graph intersects the x-axis or y-axis?
intercepts
Which type of symmetry involves the graph being a mirror image of the portion to the left of the y-axis to the right of the y-axis?
Symmetry with respect to the y-axis
An equation has an x-intercept when the value of y is zero.
True
To find the x-intercepts of a graph, let y be ______ and solve the equation for x.
zero
What is the condition for a quadratic function to have two x-intercepts?
b^2 - 4ac > 0 and a ≠ 0
What is the equation for the break-even point?
R = C
If b^2 - 4ac ≤ 0 and a ≠ 0, the graph of a quadratic function will have only one x-intercept.
True
The resistance of 1000 feet of solid copper wire at 77°F can be approximated by the model y = ____.
0.37x^2 + 5x + 100
What is the formula to calculate the slope of a line passing through two points?
m = (y2 - y1) / (x2 - x1)
What is a point of intersection of the graphs of two equations?
A point that satisfies both equations
How can you find the points of intersection of two graphs?
By solving their equations simultaneously
What are the two points of intersection for x^2 - y = 3 and x - y = 1?
(-1, -2) and (2, 1)
How can you check the points of intersection in Example 5?
By substituting into both of the original equations or by using the intersect feature of a graphing utility
What are the two conflicting goals when developing a mathematical model for real-life applications?
Simplicity and Accuracy
What was the prediction for the carbon dioxide concentration in Earth's atmosphere in the year 2035 according to the quadratic model?
470 parts per million
Based on the linear model for 1980–2007, was the 1990 prediction for the year 2035 accurate?
No
What is the interpretation of the slope of a line when the x- and y-axes have the same unit of measure?
ratio
What is stated about two distinct nonvertical lines being parallel?
Their slopes are equal
What is the interpretation of the slope of a line when the x- and y-axes have different units of measure?
rate
What is stated about two nonvertical lines being perpendicular?
Their slopes are negative reciprocals of each other
Lines parallel to the line 2x - 3y = 5 have a slope of ___.
3
The slope of a line is the ratio of its height (the rise) to the length of its base (the _____).
run
How can you determine if two nonvertical lines are parallel?
Having the same slope
What is the slope of a line perpendicular to the line 2x - 3y = 5?
-2
Find the equation of the vertical line with an x-intercept at 3.
x = 3
Show that the line with intercepts (a, 0) and (0, b) has the equation x/a + y/b = 1 (a ≠ 0, b ≠ 0).
x/a + y/b = 1 (a ≠ 0, b ≠ 0)
When is the line represented by ax - by = 4 parallel to the x-axis?
When b = 0
When is the line represented by ax - by = 4 parallel to the y-axis?
When a = 0
Study Notes
Preparation for Calculus
- Reviewing concepts helpful for calculus study
- Sketching graphs of equations and functions
- Fitting mathematical models to data
- Key concepts to learn:
- Identifying characteristics of equations and sketching their graphs
- Finding and graphing equations of lines, including parallel and perpendicular lines
- Evaluating and graphing functions and their transformations
- Fitting mathematical models to real-life data sets
Graphs and Models
- The graph of an equation is the set of all solution points
- Point plotting: a method to sketch a graph by plotting points and connecting them with a smooth curve
- Analytic approach: solving an equation for y to find the graph
- Graphical approach: using a graphing utility to graph an equation
- Importance of choosing a suitable viewing window when using a graphing utility
Intercepts of a Graph
- Intercepts: points at which the graph intersects the x- or y-axis
- x-intercepts: points of the form (a, 0)
- y-intercepts: points of the form (0, b)
- Methods to find intercepts:
- Analytic approach: letting y = 0 to find x-intercepts and x = 0 to find y-intercepts
- Graphical approach: using a graphing utility to approximate intercepts
Symmetry of a Graph
- Three types of symmetry:
- Symmetry with respect to the y-axis
- Symmetry with respect to the x-axis
- Symmetry with respect to the origin
- Tests for symmetry:
- Replacing x by -x yields an equivalent equation for y-axis symmetry
- Replacing y by -y yields an equivalent equation for x-axis symmetry
- Replacing x by -x and y by -y yields an equivalent equation for origin symmetry
- Importance of knowing symmetry when sketching graphs### Symmetry of Polynomial Graphs
- A polynomial graph has symmetry with respect to the y-axis if each term has an even exponent (or is a constant).
- A polynomial graph has symmetry with respect to the origin if each term has an odd exponent.
Testing for Symmetry
- To test for symmetry with respect to the y-axis, replace x with -x and see if the equation remains equivalent.
- To test for symmetry with respect to the origin, replace x with -x and y with -y and see if the equation remains equivalent.
Examples of Symmetry
- The graph of y = 2x^4 + x^2 + 2 has symmetry with respect to the y-axis.
- The graph of y = 2x^3 - x has symmetry with respect to the origin.
Points of Intersection
- A point of intersection of two graphs is a point that satisfies both equations.
- To find the point(s) of intersection, solve the equations simultaneously.
Example of Finding Points of Intersection
- Find the points of intersection of the graphs of x^2 - y = 3 and x - y = 1.
- Solve the first equation for y: y = x^2 - 3.
- Solve the second equation for y: y = x - 1.
- Equate the two expressions for y and solve for x.
- Substitute the values of x into either of the original equations to find the corresponding values of y.
Mathematical Models
- Mathematical models are used to represent real-life data.
- A good model should strive for two conflicting goals: accuracy and simplicity.
- Examples of mathematical models include quadratic and linear models.
Comparing Two Mathematical Models
- A quadratic model and a linear model can be used to predict the carbon dioxide level in the atmosphere.
- The quadratic model was used to predict the level in 2035, but the linear model suggests that the prediction was too high.
Exercises
- Exercises 1-4 involve matching equations with their graphs.
- Exercises 19-28 involve finding intercepts of various equations.
- Exercises 29-40 involve testing for symmetry with respect to each axis and the origin.
- Exercises 41-58 involve sketching the graph of an equation and identifying intercepts and symmetry.Here are the study notes on the text:
Slope of a Line
- The slope of a nonvertical line is a measure of the number of units the line rises (or falls) vertically for each unit of horizontal change from left to right.
- The slope of a nonvertical line is calculated using the formula: m = (y2 - y1) / (x2 - x1)
Properties of Slope
- If the slope is positive, the line rises from left to right.
- If the slope is zero, the line is horizontal.
- If the slope is negative, the line falls from left to right.
- If the slope is undefined, the line is vertical.
Equations of Lines
- Any two points on a nonvertical line can be used to calculate its slope.
- The point-slope equation of a line is given by: y - y1 = m(x - x1)
- This equation can be rewritten in the form y = y1 + m(x - x1)
Interpreting Slope
- The slope of a line can be interpreted as either a ratio or a rate.
- Slope represents the change in y (vertical change) over the change in x (horizontal change).
Example: Finding an Equation of a Line
- Find an equation of the line that has a slope of 3 and passes through the point (1, -2).
- Use the point-slope form: y - (-2) = 3(x - 1)
- Simplify: y = 3x - 5
Review key concepts before studying calculus, including graphing equations and functions, and fitting mathematical models to data.
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