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How can we verbally describe the graph of y = ln (1/x) in the Cartesian xy plane
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Answer 14-6
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The graph is a smooth, continually decreasing curve that crosses the y axis at (1,0) The domain is the set of positive real numbers, and the range is the set of all real numbers The curve is entirely contained within the first and fourth quadrants As we move to the left from the point (1,0), the curve blows up positively, approaching the y axis but never reaching it As we move to the right from (1,0), the graph falls at an ever-decreasing rate In fact, the curve representing y = ln (1/x) has exactly the same shape as the curve for y = ln x but is reversed top-to-bottom with respect to the x axis, so the two graphs are vertical mirror images of each other
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Question 14-7
How can we informally describe the graphs of the common-log functions y = log10 x and y = log10 (1/x) in the Cartesian xy plane
Answer 14-7
The graphs of these functions closely resemble the graphs of the functions y = ln x and y = ln (1/x) respectively Both common-log graphs cross the x axis at (1,0), just as the natural-log graphs do However, the contours differ The natural-log curves are somewhat steeper than the commonlog curves
Part Two 415 Question 14-8
How can we visually and qualitatively compare the graphs of the four functions described in Questions 14-5 through 14-7
Answer 14-8
We can graph them all together on a generic rectangular-coordinate grid such as the one shown in Fig 20-8
Question 14-9
How can we plot the graph of the sum of two functions or the difference between two functions
Answer 14-9
There are two ways in which this can be done First, we can graph the original functions separately, and then add or subtract their values visually to infer the sum or difference graph Second, we can calculate several outputs for each function using inputs that we ve selected to get a good sampling Then we can add or subtract these outputs arithmetically Based on that data, we can graph the sum or difference function
Question 14-10
Texts don t always agree in the denotation of logarithmic functions How can we avoid confusion when we write our own papers
+y y = ln x y = log 10 x (1, 0)
y = log 10 (1/x) y = ln (1/x) y
Figure 20-8
Illustration for Question and Answer 14-8
Review Questions and Answers Answer 14-10
We should always clarify the logarithmic base when we write log followed by anything For example, we should write log10 or loge instead of log (unless we can t portray the subscript within the constraints of a text-editing or Web site building program) We don t need to write a subscript when we write ln to denote the natural logarithm, because ln means natural log or base-e log all the time
15
Special note
If you want to see graphical illustrations of the answers to the following 10 questions, feel free to look back at Chap 15 Try to envision or draw the graphs yourself before you look back!
Question 15-1
Consider a function f of a real-number variable q such that f (q) = sin q + cos q What are the period, the positive peak amplitude and the negative peak amplitude of f What are the domain and range of f
Answer 15-1
The period of f is 2p That s the same as the period of the sine It s also the same as the period of the cosine The positive peak amplitude of f is 21/2 The negative peak amplitude of f is 21/2 The domain of f is the set of all real numbers The range of f is the set of all real numbers f (q) such that 21/2 f (q) 21/2
Question 15-2
Consider a function f of a real-number variable q such that f (q) = sin q cos q What are the period, the positive peak amplitude and the negative peak amplitude of f What are the domain and range of f
Answer 15-2
The period of f is p, which is equal to half the period of the sine, and is also half the period of the cosine The positive peak amplitude of f is 1/2 The negative peak amplitude of f is 1/2 The domain of f is the set of all real numbers The range of f is the set of all real numbers f (q) such that 1/2 f (q) 1/2
Part Two 417 Question 15-3
Consider a function f of a real-number variable q such that f (q) = sin2 q + cos2 q What are the period, the positive peak amplitude and the negative peak amplitude of f What are the domain and range of f
Answer 15-3
In this case, the function f has a constant value of 1 The period is not defined, because the function s output value never changes, and is defined for all inputs 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 The range of f is the set containing the single number 1
Question 15-4
Consider a function f of a real-number variable q such that f (q) = sec q + csc q What are the period, the positive peak amplitude and the negative peak amplitude of f What are the domain and range of f
Answer 15-4
The period of f is 2p, which is the same as the period of the secant, and the same as the period of the cosecant The positive and negative peak amplitudes of f are not defined, because f blows up in both the positive and negative directions whenever q is an integer multiple of p /2 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 real numbers
Question 15-5
Consider a function f of a real-number variable q such that f (q) = sec q csc q What are the period, the positive peak amplitude and the negative peak amplitude of f What are the domain and range of f
Answer 15-5
The period of f is p, which is half the period of the secant, and half the period of the cosecant The positive and negative peak amplitudes of f are undefined, because f blows up both positively and negatively at all integer multiples of p /2 The domain of f is the set of all real numbers except the integer multiples of p /2 The range is the set of all real numbers f (q) such that f (q) 2 or f (q) 2
Review Questions and Answers Question 15-6
Consider a function f of a real-number variable q such that f (q) = sec2 q + csc2 q What are the period, the positive peak amplitude and the negative peak amplitude of f What are the domain and range of f
Answer 15-6
The period of f is p /2, which is half the period of the secant squared, and half the period of the cosecant squared The positive peak amplitude of f is undefined, because f blows up positively at all integer multiples of p /2 The negative peak amplitude of f is equal to 4, 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 is of f the set of all real numbers f (q) such that f (q) 4
Question 15-7
Consider a function f of a real-number variable q such that f (q) = tan q + cot q What are the period, the positive peak amplitude and the negative peak amplitude of f What are the domain and range of f
Answer 15-7
The period of f is p, which is the same as that of the tangent, and the same as that of the cotangent The positive and negative peak amplitudes of f are both undefined, because f blows up positively and negatively at all integer multiples of p /2 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 real numbers f (q) such that f (q) 2 or f (q) 2
Question 15-8
Consider a function f of a real-number variable q such that f (q) = tan q cot q What are the period, the positive peak amplitude and the negative peak amplitude of f What are the domain and range of f
Answer 15-8
This particular function presents a strange situation The graph of f is a horizontal, straight line with single-point gaps wherever q is an integer multiple of p /2 The period of f is p /2, because the graph consists of infinitely many open line segments placed end to end, each of
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