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8 Plot a rectangular-coordinate graph of the product of the natural log function and the common log function The curve is represented by the following equation: y = (ln x) (log10 x) What is the domain of this function What is its range Include all values of the domain from 0 to 10, and all values of the range from 0 to 5 9 Draw the graphs of the three functions from Fig B-18 (the illustration for the solution to Problem 7) on a y-linear semilog coordinate grid Portray values of x over the two orders of magnitude from 01 to 10 Portray values of y from 5 to 5 10 Draw the graphs of the three functions from Fig B-19 (the illustration for the solution to Problem 8) on a y-linear semilog coordinate grid Portray values of x over the single order of magnitude from 1 to 10 Portray values of y from 0 to 25
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If you ve taken basic algebra and geometry, you re familiar with the trigonometric functions You also got some experience with them in Chap 2 of this book Now we ll graph some algebraic combinations of these functions
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Let s find out what happens when we add, multiply, square, and divide the sine and cosine functions
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Sine and cosine: example 1 Figure 15-1 shows superimposed graphs of the sine and cosine functions along with a graph of their sum You can follow along by inputting numerous values of q into your calculator, determining the output values, plotting the points corresponding to the input/output ordered pairs, and then filling in the curve by connecting the dots In Fig 15-1, the dashed gray curves are the individual sine and cosine waves The solid black curve is the graph of the sum function:
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f (q) = sin q + cos q The sum-function wave has the same period (distance between the corresponding points on any two adjacent waves) as the sine and cosine waves In this situation, that period is 2p The new wave also has the same frequency as the originals The frequency of any regular, repeating wave is always equal to the reciprocal of its period The peaks (recurring maxima and minima) of the sine and cosine waves attain values of 1 The peaks of the new wave attain values of 21/2, which occur at values of q where the graphs of the sine and cosine cross each other By definition, the peak amplitude of the new function is 21/2 times the peak amplitude of either original function The new wave appears sine-like, but we can t be sure that it s a true sinusoid on the basis of its appearance in this graph The domain of our function f includes all real numbers The range of f is the set of all reals in the closed interval [ 21/2,21/2]
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Figure 15-1
Graphs of the sine and cosine functions (dashed gray curves) and the graph of their sum (solid black curve) Each division on the horizontal axis represents p /2 units Each division on the vertical axis represents 1/2 unit
Sine and cosine: example 2 In Fig 15-2, we see graphs of the sine and cosine functions along with a graph of their product The dashed gray curves are the superimposed sine and cosine waves The solid black curve is the graph of the product function:
f (q) = sin q cos q The new function s graph has a period of p, which is half the period of the sine wave, and half the period of the cosine wave The peaks of the new wave are 1/2, which occur at values of q where the graphs of the sine and cosine intersect As in the previous example, the new wave looks like a sinusoid, but we can t be sure about that by merely looking at it The domain of the product function spans the entire set of reals The range is the set of all reals in the closed interval [ 1/2,1/2]
Sine and cosine: example 3 Figure 15-3 shows the graphs of the sine function (at A) and the cosine function (at B) along with their squares The dashed gray curve at A is the sine wave; the dashed gray curve at B is the cosine wave In illustration A, the solid black curve is the graph of