Part Two 419

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length p /2 The positive peak amplitude of f is equal to 1 The negative peak amplitude of f is also equal to 1 The domain of f is the set of all real numbers except the integer multiples of p /2 The range is the set containing the single element 1

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Consider a function f of a real-number variable q such that f (q) = tan2 q + cot2 q What are the period, the positive peak amplitude and the negative peak amplitude of f What are the domain and range of f

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The period of f is p /2, which is half that of the tangent squared, and is also half that of the cotangent squared The positive peak amplitude of f is undefined, because the function blows up positively at all integer multiples of p /2 The negative peak amplitude of f is equal to 2, which occurs whenever q is an odd-integer multiple of p /4 The domain of f is the set of all real numbers except the integer multiples of p /2 The range of f is the set of real numbers f (q) such that f (q) 2

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Consider a function f of a real-number variable q such that f (q) = (tan2 q)/(cot2 q) What are the period, the positive peak amplitude and the negative peak amplitude of f What are the domain and range of f

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The period of f is p, which is the same as the period of the tangent squared, and is also the same as the period of the cotangent squared The positive peak amplitude of f is undefined, because the function blows up positively at all odd-integer multiples of p /2 The negative peak amplitude of f is undefined as well, although f (q) approaches 0 whenever q approaches any integer multiple of p from either side (We can t say that the negative peak amplitude is 0, because the function never actually attains that value) The domain of f is the set of all real numbers except the integer multiples of p /2 The range of f is the set of all positive real numbers

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16

Question 16-1

What s a parameter What s a set of parametric equations

Answer 16-1

A parameter is an independent variable on which other variables depend A set of parametric equations is a collection of equations, at least one of which has one or more variables that

Review Questions and Answers

depend on the parameter The parameter, which is often symbolized t, plays the role of master controller for the other variables in the system

Question 16-2

Consider the pair of parametric equations x = 3t and y = t where t is the parameter on which both x and y depend How can we sketch a Cartesian graph of this system How can we find an equivalent equation in terms of the variables x and y only, based on the graph

Answer 16-2

We can input various values of t to both equations, and plot the ordered pairs (x,y) that come out of those equations Following are some examples: When t = 2, we have x = 3 ( 2) = 6 and y = 1 ( 2) = 2 When t = 1, we have x = 3 ( 1) = 3 and y = 1 ( 1) = 1 When t = 0, we have x = 3 0 = 0 and y = 1 0 = 0 When t = 1, we have x = 3 1 = 3 and y = 1 1 = 1 When t = 2, we have x = 3 2 = 6 and y = 1 2 = 2

When we plot the (x,y) ordered pairs based on this list as points on a Cartesian plane and then connect the dots, we get a line through the origin with a slope of 1/3, as shown in Fig 20-9 In slope-intercept form, the line can be represented as y = ( 1/3)x

Question 16-3

Consider again the pair of parametric equations x = 3t and y = t How can we use algebra alone (without the help of a graph) to determine the equivalent equation in terms of x and y only

Answer 16-3