Unit 5: Analytical Applications of Differentiation

5.1 The Mean Value Theorem

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.

5.2 Extreme Value Theorem & critical points

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

5.3 Increasing and decreasing intervals

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

5.4 The first derivative test

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

5.5 The candidates test (absolute extrema)

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

5.6 Concavity and inflection points

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

5.7 The second derivative test

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

5.8 Sketching f and its derivatives

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.

5.9 Connecting f, f′, and f″

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

5.10 Setting up optimization problems

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

5.11 Solving optimization problems

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.

5.12 Behaviours of implicit relations

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