121616949-math.94 - lim Δ x → f x Δ x-f x Δ x both the...

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80 Chapter 4 Transcendental Functions Exercises In 1–19, find the derivatives of the functions. 1. 3 x 2 2. sin x e x 3. ( e x ) 2 4. sin( e x ) 5. e sin x 6. x sin x 7. x 3 e x 8. x + 2 x 9. (1 / 3) x 2 10. e 4 x /x 11. ln( x 3 + 3 x ) 12. ln(cos( x )) 13. p ln( x 2 ) /x 14. ln(sec( x ) + tan( x )) 15. x sin( x ) 16. x ln x 17. ln(ln(3 x )) 18. 1 + ln(3 x 2 ) 1 + ln(4 x ) 19. x 8 ( x - 23) 1 / 2 27 x 6 (4 x - 6) 8 20. Find the value of a so that the tangent line to y = ln( x ) at x = a is a line through the origin. Sketch the resulting situation. 21. If f ( x ) = ln( x 3 + 2) compute f ( e 1 / 3 ). We have defined and used the concept of limit, primarily in our development of the deriva- tive. Recall that lim x a f ( x ) = L is true if, in a precise sense, f ( x ) gets closer and closer to L as x gets closer and closer to a . While some limits are easy to see, others take some ingenuity; in particular, the limits that define derivatives are always difficult on their face, since in lim
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Unformatted text preview: lim Δ x → f ( x + Δ x )-f ( x ) Δ x both the numerator and denominator approach zero. Typically this difficulty can be re-solved when f is a “nice” function and we are trying to compute a derivative. Occasionally such limits are interesting for other reasons, and the limit of a fraction in which both nu-merator and denominator approach zero can be difficult to analyse. Now that we have the derivative available, there is another technique that can sometimes be helpful in such circumstances. Before we introduce the technique, we will also expand our concept of limit. We will occasionally want to know what happens to some quantity when a variable gets very large or “goes to infinity”....
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