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Graph y=(1/3)cos(2x-n/3) 2 ~ - ~ / 3
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Solution: The h c t i o n shown in Fig 1-31 has another little twist to it, which has to do with the minus sign Set up the chart and make 2 x - x l 3 = 0 for the first point This point is
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cos(2x - n/3) 1 0
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x = n16 or 2x1 12 The next point is for
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2x- x / 3 = x l 2 This (second) point is
Set up the x-y coordinate system and place the first quarter of the cosine function between 2x112 and 5x112 With this section of the cosine function complete,
MATHEMAlTCAl BACKGROUND
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1-37
Graph y = tan(x - n/4)
Solution: If you are a all unfamiliar with the t tangent function go back and review it in the trigonometry section The important features as far as graphing is concerned are that tan8 is zero a8 when 8 is zero and t n is 1 when 8 is d 4 The tangent curve goes infinite when 0 goes to d 2 , but a point at infinity is not an easy one to deal with
For the function shown in Fig 1-32, set up a chart and find the values of x that make x - d 4 equal zero and d 4 These two points allow construction of the fkction
Ix-d4 I
Solve x - n / 4 = 0
x I tan(x-d4) In/4I 0
Fig 1-32
for x = z / 4 for x = x/2 = 2n/4
Solve x - lr/4 = n/4
Be carefbl graphing the tangent function, especially this one This tangent function is zero when x = 4 4 , and 1 when x = 2x/4 The standard mistake is to take the function to infinity at x = 2n/4
LIMITS AND CONTINUITY
The concept of the limit in calculus is very important It describes what happens to a function as a particular value is approached The derivative, one of the major themes of calculus, is defined in limit terms This short chapter will help you to think in terms of limits The first thing to understand about limits is that a limit of a function is not the value of the hction The change in thinking (fiom value to limit) is important because most functions are understood as a series of mathematical operations that can be evaluated at certain points simply by substitution The (polynomial) h c t i o n y = x2 + 2x + 3 can be evaluated for any real number: replace x with the number and perform the indicated operations Askmg the limit of this h c t i o n as x approaches 2, for example, is an uninteresting question The fhction can be evaluated at 2 or any point arbitrarily close to 2 by substituting and performing the operations Other functions, such as polynomial fractions, cannot be evaluated at certain points and these functions are best understood by thinking in terms of limits The h c t i o n y = (x2 - 4)/(x + 2) can be evaluated for any real number except -2 Replacing x by -2 produces the meaningless statement 0 / 0 Remember that any number times 0 is 0, but any number divided by 0 is "meaningless" (including 010) Looking at the limit of the hction, as x approaches -2, tells us about the h c t i o n in the vicinity of -2 - The limit of the firnction is a convenient phrase for the question, "What happens to the function as a certain value is approached " Writing this in mathematical notation we get the following: lim
X-b-2
____
x2 - 4
= lim
x+-2
(x + 2)(x - 2)
= lim(x-2)=-4
x+--2
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