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Use a scientific calculator and input various values of x Plot several points for each function and then draw curves through them, interpolating as you go Be sure that your calculator is set for the natural logarithmic and exponential functions (that is, base e), not common logarithm or common exponential functions (base 10) You should get graphs that look like those shown in Fig 14-16 In two cases, only a single point of the function shows up in the coordinate spans portrayed here You d have to expand the linear axis (the x axis) at A beyond 1 to see any of the graph for y = ln x You d have to expand the linear axis (the y axis) at B beyond 1 to see any of the graph for y = ex
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Figure 14-16 Simple functions in x-linear semilog
coordinates (at A), y-linear semilog coordinates (at B), and log-log coordinates (at C)
Practice Exercises
Practice Exercises
This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App B The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! When plotting graphs here, feel free to use a calculator, locate numerous points, and connect the dots 1 When we discussed the range of the natural exponential function, we claimed that no real-number power of e is equal to 0 Prove it Here s a hint: Use the technique of reductio ad absurdum, in which we assume the truth of a statement and then derive something obviously false or contradictory from that assumption 2 Set up a rectangular coordinate system like the one in Fig 14-1A, where the values of x are portrayed from 25 to 25, and the values of y are portrayed from 0 to 10 Sketch the graphs of the following functions on this coordinate grid: y = ex y = e x y = ex e x 3 Set up a rectangular coordinate system like the one in Fig 14-1B, where the values of x are portrayed from 1 to 1, and the values of y are portrayed from 0 to 10 Sketch the graphs of the following functions on this coordinate grid: y = 10x y = 10 x y = 10x/10 x 4 Draw the graphs of the three functions from Problem 3 on an x-linear semilog coordinate grid, where the values of x are portrayed from 1 to 1, and the values of y are portrayed over the three orders of magnitude from 01 to 100 5 Plot a rectangular-coordinate graph of the function we get when we raise 10 to the power of 1/x The curve is represented by the following equation: y = 10(1/x) What is the domain of this function What is its range Include all values of the domain from 10 to 10 6 We ve claimed that the natural log of 0 isn t a real number Prove it Here s a hint: Use reductio ad absurdum, and use the solution to Problem 1 as a lemma (a theorem that helps in the proof of another theorem) 7 Plot a rectangular-coordinate graph of the sum 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 Include all values of the range from 5 to 5
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