5Unit 5: Analytical Applications of Differentiation

If a function is smooth over an interval, then at some interior point its instantaneous rate (tangent slope) exactly equals its average rate (secant slope) over the interval.

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A continuous function on a closed interval must reach a highest and lowest value; those extremes happen at critical points or at the endpoints.

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Where the slope is positive the function climbs; where it is negative the function falls.

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At a critical point, watch how the slope changes sign: from up to down is a peak; from down to up is a valley.

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To find the overall highest and lowest values on a closed interval, just test all the candidates: critical points and both endpoints.

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The second derivative tells you how the curve bends: $f''>0$ bends upward (like a cup), $f''<0$ bends downward.

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At a point where the slope is zero, the bending tells you the type: concave up means a valley, concave down means a peak.

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The three graphs are linked: where $f'=0$, $f$ has a horizontal tangent; where $f'>0$, $f$ rises; where $f''>0$, $f$ is concave up.

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Given any one of the three graphs, you can deduce features of the others — position, slope, and bending are all connected.

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To optimize a real quantity, first write it as a function of a single variable using the constraint that ties the variables together.

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Once the objective is a single-variable function, use calculus: find critical points, confirm which is the max or min, and report the quantity that was asked.

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For a curve given implicitly, the tangent slope comes from implicit differentiation.

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