(Section 1.10: Difference Quotients) 1.10.11FOOTNOTES1.Assumptions made about a function. When defining the average rate of change of a functionfon an interval a,b±²³´, where a<b, sources typically do not state the assumptions made aboutf. The formula fb()±fab±aseems 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±²³´can still be defined, we need more for the existence of derivatives(i.e., the differentiabilityoff) and the existence of non-vertical tangent lines, as we will see in Section 1.11. • We ordinarily assume thatfis continuouson a,b±²³´. Then, there are no holes, jumps, or vertical asymptotes on a,b±²³´whenfis graphed. (See Section 1.5.) • We may also assume thatfis differentiableon a,b±²³´. Then, the graph offmakes 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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