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8 Using a calculator, plug in n = 2, n = 6, n = 10, and n = 20 to informally illustrate that Lim (1 + 2 + 3 + + n)/n 2 = 1/2 = 05
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and therefore that Lim
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Work out the partial sums to obtain decimal quantities Round off your results to five decimal places when you encounter repeating or lengthy decimals 9 Using a calculator, plug in n = 2, n = 6, n = 10, and n = 20 to informally illustrate that Lim (12 + 22 + 32 + + n 2)/n 3 = 1/3 = 033333
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i 2/n 3 = 1/3 = 033333
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Work out the partial sums to obtain decimal quantities Round off your results to five decimal places when you encounter repeating or lengthy decimals 10 Using a calculator, plug in n = 2, n = 6, n = 10, and n = 20 to informally illustrate that Lim (13 + 23 + 33 + + n 3 )/n 4 = 1/4 = 025
and therefore that Lim
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CHAPTER
Review Questions and Answers
Part Two
This is not a test! It s a review of important general concepts you learned in the previous nine chapters Read it though 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
11
Question 11-1
What s a mathematical relation
Answer 11-1
A relation is a clearly defined way of assigning, or mapping, some or all of the elements of a source set to some or all of the elements of a destination set Suppose that X is the source set for a relation, and Y is the destination set for the same relation In that case, the relation can be expressed as a collection of ordered pairs of the form (x,y), where x is an element of set X and y is an element of set Y
Question 11-2
What s an injection, also known as an injective relation
Answer 11-2
Imagine two sets X and Y Suppose that a relation assigns each element of X to exactly one element of Y Also suppose that, according to the same relation, an element of Y never has more than one mate in X (Some elements of Y might have no mates in X) In a situation like this, the relation is an injection
Question 11-3
What s a surjection, also called an onto relation
Review Questions and Answers Answer 11-3
Again, imagine two sets X and Y Suppose that according to a certain relation, every element of Y has at least one (and maybe more than one) mate in X, so that no element of Y is left out A relation of this type is a surjection from X onto Y
Question 11-4
What s a bijection, also called a one-to-one correspondence
Answer 11-4
A bijection is a relation that s both an injection and a surjection Given two sets X and Y, a bijection assigns every element of X to exactly one element of Y, and vice versa This is why a bijection is sometimes called a one-to-one correspondence
Question 11-5
What s a two-space function Is every two-space function a relation Is every two-space relation a function
Answer 11-5
A two-space function is a relation between two sets that never maps any element of the source set to more than one element of the destination set All two-space functions are relations However, not all two-space relations are functions
Question 11-6
What s the vertical-line test for the graph of a two-space function
Answer 11-6
The vertical-line test is a quick way to determine, based on the graph of a two-space relation, whether or not the relation is a function Imagine an infinitely long, movable line that s always parallel to the dependent-variable axis (usually the vertical axis) Suppose that we re free to move the line to the left or right, so it intersects the independent-variable axis (usually the horizontal axis) wherever we want The graph is a function of the independent variable if and only if the movable vertical line never intersects the graph at more than one point
Question 11-7
Based on the vertical-line test, which of the curves in Fig 20-1 are functions of x within the span of values for which 6 < x < 6
Answer 11-7
Only f is a function of x If we construct a movable vertical line (always parallel to the y axis), it never intersects the curve for f at more than one point over the span of values for which 6 < x < 6 However, the movable vertical line intersects the curve for g at more than one point for some values of x where 6 < x < 6 The same is true of the curve for h
Question 11-8
Suppose we re working in the polar coordinate plane, and we encounter the graph of a relation where the independent variable is represented by q (the direction angle) and the dependent
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