121616949-math.88

# 121616949-math.88 - 6 Diﬀerentiate f x = x ln x 7...

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74 Chapter 4 Transcendental Functions whenever x = p/q ; if we were to graph this we’d see something like this: . But this is a poor picture, because you can’t see that the “curve” is really a whole lot of individual points, above the rational numbers on the x -axis. There are really a lot of “holes” in the curve, above x = π , for example. But (this is the hard part) it is possible to prove that the holes can be “filled in”, and that the resulting function, called 2 x , really does have the properties we want, namely that 2 x 2 y = 2 xy and (2 x ) y = 2 xy . Exercises 1. Expand ln(( x + 45) 7 ( x - 2)). 2. Expand ln x 3 3 x - 5 + (7 /x ) . 3. Sketch the graph of y = ln( x - 7) 3 + 14. 4. Sketch the graph of y = ln | x | for x = 0. 5. Write ln 3 x + 17 ln( x - 2) - 2 ln( x 2 + 4 x + 1) as a single logarithm. 6. Differentiate f ( x ) = x ln x . 7. Differentiate f
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Unformatted text preview: 6. Diﬀerentiate f ( x ) = x ln x . 7. Diﬀerentiate f ( x ) = ln(ln(3 x )). 8. Sketch the graph of ln( x 2-2 x ). 9. Solve ln(1 + √ x ) = 6 for x . 10. Solve e x 2 = 8 for x . 11. Solve ln(ln( x )) = 1 for x . 12. Sketch the graph of f ( x ) = e 4 x-5 + 6. 13. Sketch the graph of f ( x ) = 3 e x +6-4. 14. Sketch the graph of y = 3 6 x-1 + 5. 15. Sketch the graph of y =-(1 / 2)-3 x . 16. Sketch the graph of y = 4 log 2 (12 x + 6)-2. 17. Sketch the graph of y = a x in the three cases a > 1, a = 1, and 0 < a < 1. What happens to the graph as a → + ? What happens to the graph as a → ∞ ? 18. Sketch the graph of y = log a x in the two cases a > 1 and 0 < a < 1. What happens to the graph as a → + ? What happens to the graph as a → ∞ ?...
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