By Smith G., Mclelland G.

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**Additional resources for On the Shoulders of Giants: A Course in Single Variable Calculus**

**Example text**

The sequence of function values or output sequence is . The terms of this output sequence are → ✑ ✼ ✓ ✒ ✰ ❈ ❞ ✖ ✲ ✪ ✑ ✼ ✓ ✒ ✰ ❈ ⑨ ❞ ✲ ✪ ✑ ✼ ✓ ✒ ✰ ❈ ⑨ ❞ ✲ ✓ ✑ ✓ ✓ ✞ ✑ The terms in brackets converge to 2, so the output sequence converges to , which is the function value at 2. We have shown that there is a sequence with for which . Deﬁnitely not! The deﬁnition of continuity requires Can we conclude that is continuous at that the sequence of function values converges to the actual function value for any input sequence, and we have only tried one sequence.

17. 18. ④ ④ ✥ ✥ 19. Let be a differentiable function and let to the quotient ❨ ❨ ✰ ✥ be a small positive number. Give a physical meaning ✜ ❞ ✜ ✲ ❇ ❨ ✰ ✥ ✲ ✜ in the following cases. (a) ✥ ✲ ❨ ✰ ✥ ✲ ❨ ✰ ✥ ✲ (b) ✰ (c) is the height of a missile ❨ seconds after liftoff. ✥ is the population of a city at time . ✥ is the cost in dollars of producing units of a product. 2 RULES FOR DIFFERENTIATION We have identiﬁed the derivative of a differentiable function as one of its important properties. 1 on page 44, which is often a cumbersome process.

In Exercises 7–12, differentiate the given function by using the deﬁnition of a derivative. In each case, plot the graph of the derivative. 7. 9. 11. ✍ ✰ ✥ ✲ ✡ ✛ ❨ ✰ ✥ ✲ ✡ ✛ 8. ❨ ✰ ✥ ✲ ✡ ❈ ✥ ❇ ✥ ❈ ❞ 12. ✖ ✑ 13. Show that if ❨ ✰ ✥ ✲ ✡ ❨ ✰ ✥ ✲ ✡ 10. ❨ ✰ ✥ ✲ ✡ ⑤ ✥ ⑤ ✥ ✖ ✰ ✥ ✲ ✡ ✥ ✈ ❇ ð ✰ ✥ ✲ ✡ ✖ Plot the graph of ❨ ð . At what points is ❨ ð ✖ ⑦ ✖ ✰ ✥ ❞ ❈ ✲ ✞ ✜ , then ❨ ❞ if if ✥ ❑ ✙ ✥ ▼ ✙ differentiable? ✓ ❞ ✼ ✥ ❞ ✖ 48 DIFFERENTIATION 14. Decide which, if either, of the two functions ; justify your answers.