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Draw a two-dimensional graph of this relation as it appears when we look broadside at the plane containing it 6 Consider a relation in Cartesian xyz space described by the system of parametric equations x=t y = 7 z = t2/2 5 Draw a two-dimensional graph of this relation as it appears when we look broadside at the plane containing it 7 Consider a relation in Cartesian xyz space described by the system of parametric equations x = 4 cos t y = 4 sin t z=1 Draw a two-dimensional graph of this relation as it appears when we look broadside at the plane containing it 8 Consider a relation in Cartesian xyz space described by the system of parametric equations x = 5 cos t y=0 z = 5 sin t Draw a two-dimensional graph of this relation as it appears when we look broadside at the plane containing it 9 Consider a relation in Cartesian xyz space described by the system of parametric equations x = 5 cos t y = 3 sin t z=p Draw a two-dimensional graph of this relation as it appears when we look broadside at the plane containing it
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10 Consider a relation in Cartesian xyz space described by the system of parametric equations x = 2 cos t y = t /(2p ) z = 2 sin t Draw a perspective view of this relation s three-dimensional graph Here s a hint: You can probably tell that the graph is a circular helix, but as you draw it, pay attention to the orientation, the pitch, and the sense of rotation
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CHAPTER
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Have you ever tried to find the missing number in a list Have you ever figured out how much money an interest-bearing bank account will hold after 10 years Have you ever calculated the value that a function approaches but never reaches If you can answer Yes to any of these questions, you ve worked with sequences (also called progressions), series, or limits
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A sequence is a list of numbers Some sequences are finite; others are infinite The simplest sequences have values that repeatedly increase or decrease by a fixed amount Here are some examples: A = 1, 2, 3, 4, 5, 6 B = 0, 1, 2, 3, 4, 5 C = 2, 4, 6, 8 D = 5, 10, 15, 20 E = 4, 8, 12, 16, 20, 24, 28, F = 2, 0, 2, 4, 6, 8, 10, The first four sequences are finite The last two are infinite, as indicated by an ellipsis (three dots) at the end
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Arithmetic sequence In each of the sequences shown above, the values either increase steadily (in A, C, and E ) or decrease steadily (in B, D, and F ) In all six sequences, the spacing between numbers is constant throughout Here s how each sequence changes as we move along from term to term:
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The values in A always increase by 1 The values in B always decrease by 1 The values in C always increase by 2
Sequences, Series, and Limits
The values in D always decrease by 5 The values in E always increase by 4 The values in F always decrease by 2 Each sequence has an initial value After that, we can easily predict subsequent values by repeatedly adding a constant If the constant is positive, the sequence increases If the added constant is negative, the sequence decreases Suppose that s0 is the first number in a sequence S Let c be a real-number constant If S can be written in the form S = s0, (s0 + c), (s0 + 2c), (s0 + 3c), then it s an arithmetic sequence or an arithmetic progression In this context, the word arithmetic is pronounced err-ith-MET-ick The numbers s 0 and c can be integers, but that s not a requirement They can be fractions such as 2/3 or 7/5 They can be irrational numbers such as the square root of 2 As long as the separation between any two adjacent terms is the same wherever we look, we have an arithmetic sequence, even in the trivial case S0 = 0, 0, 0, 0, 0, 0, 0,
Arithmetic series A series is the sum of all the terms in a sequence For an arithmetic sequence, the corresponding arithmetic series can be defined only if the sequence has a finite number of terms For the above sequences A through F, let the corresponding series be called A+ through F+ The total sums are as follows
A+ = 1 + 2 + 3 + 4 + 5 + 6 = 21 B+ = 0 + ( 1) + ( 2) + ( 3) + ( 4) + ( 5) = 15 C+ = 2 + 4 + 6 + 8 = 20 D+ = ( 5) + ( 10) + ( 15) + ( 20) = 50 E+ is not defined F+ is not defined Now consider the infinite series S0+ = 0 + 0 + 0 + 0 + 0 + 0 + 0 + We might think of S0+ as infinity times 0, because it s the sum of 0 added to itself infinitely many times It s tempting to suppose that S0+ = 0, but we can t prove it When we add up any finite number of nothings , we get nothing , of course However, when we try to find the sum of infinitely many nothings, we encounter a mystery The best we can do is say that S0+ is undefined