Precalc0110to0111-page11

Precalc0110to0111-page11 - (Section 1.10: Difference...

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(Section 1.10: Difference Quotients) 1.10.11 FOOTNOTES 1. Assumptions made about a function. When defining the average rate of change of a function f on an interval a , b ± ² ³ ´ , where a < b , sources typically do not state the assumptions made about f . The formula fb () ± fa b ± a seems only to require the existence of and , but we typically assume more than just that. • Although the slope of the secant line on a , b ± ² ³ ´ can still be defined, we need more for the existence of derivatives (i.e., the differentiability of f ) and the existence of non-vertical tangent lines, as we will see in Section 1.11. • We ordinarily assume that f is continuous on a , b ± ² ³ ´ . Then, there are no holes, jumps, or vertical asymptotes on a , b ± ² ³ ´ when f is graphed. (See Section 1.5.) • We may also assume that f is differentiable on a , b ± ² ³ ´ . Then, the graph of f makes no sharp turns and does not exhibit “infinite steepness” (corresponding to vertical tangent lines). However, this assumption may lead to circular reasoning, because the ideas of secant lines and average rate of change are used to develop the ideas of derivatives,
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This document was uploaded on 12/29/2011.

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