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CHAPTER
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Review Questions and Answers
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Part One
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This is not a test! It s a review of important general concepts you learned in the previous nine chapters Read it through slowly and let it sink in If you re confused about anything here, or about anything in the section you ve just finished, go back and study that material some more
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1
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Question 1-1
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What s the difference between an open interval, a half-open interval, and a closed interval
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Answer 1-1
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All three types of intervals are continuous spans of values that a variable can attain between a specific minimum and a specific maximum, which are called the extremes But there are subtle differences between the three types as listed below: In an open interval, neither extreme is included In a half-open interval, one extreme is included, but not the other In a closed interval, both extremes are included
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Question 1-2
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Imagine two real numbers a and b, such that a < b These numbers can be the extremes of four different intervals: one open, two half-open, and one closed How can we denote these four intervals for a variable x
Answer 1-2
If we include neither a nor b, we have an open interval where a < x < b We can write
x (a,b) 172
Part One
which means x is an element of the open interval (a,b) If we include a but not b, we have a half-open interval where a x < b We can write x [a,b) which translates to x is an element of the half-open interval [a,b) If we include b but not a, we have an open interval where a < x b We write x (a,b] which means x is an element of the half-open interval (a,b] If we include both a and b, we have a closed interval where a x b We can write x [a,b] which means x is an element of the closed interval [a,b]
Question 1-3
What point of confusion must we avoid when working with interval notation
Answer 1-3
We must never confuse an open interval with an ordered pair, which uses the same notation If we pay close attention to the context in which the expression appears, we shouldn t have trouble
Question 1-4
Relations and functions are operations that map specific values of a variable into specific values of another variable There s an important distinction between a relation and a function What is it
Answer 1-4
In a relation, we can have more than one value of the dependent (or output) variable for a single value of the independent (or input) variable In a function, we re allowed no more than one output for any given input All functions are relations, but not all relations are functions
Question 1-5
The Cartesian plane can be used for graphing relations and functions between an independent variable and a dependent variable The plane is divided into four sections, called quadrants How do we identify them
Answer 1-5
In the first quadrant (usually the upper right), both variables are positive In the second quadrant (usually the upper left), the independent variable is negative and the dependent variable is positive In the third quadrant (usually the lower left), both variables are negative In the fourth quadrant (usually the lower right), the independent variable is positive and the dependent variable is negative
Review Questions and Answers Question 1-6
Suppose we have a point S in Cartesian two-space that is represented by the ordered pair (xs,ys) We can write this as S = (xs,ys) What s the straight-line distance ds between S and the coordinate origin What s the minimum possible distance between S and the origin Can the distance be negative What s the maximum possible distance
Answer 1-6
We can find the distance using the formula that we derived from the Pythagorean theorem in geometry In this situation, the formula is ds = (xs2 + ys2)1/2 The minimum possible distance between S and the origin is zero, which occurs if and only if xs = 0 and ys = 0, so that S = (xs,ys) = (0,0) We can never have a negative distance There is no maximum possible distance between S and the origin We can make it as large as we want by making xs or ys (or both) huge positively or huge negatively
Question 1-7
Imagine two points in Cartesian two-space, called S and T, such that S = (xs,ys) and T = (xt,yt) What s the straight-line distance dst going from S to T What s the straight-line distance dts going from T to S Does it make any difference which way we go
Answer 1-7
If we go from S to T, the distance between the points is dst = [(xt xs)2 + ( yt ys)2]1/2 If we go from T to S, the distance is dts = [(xs xt)2 + ( ys yt)2]1/2
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