Applications of Derivatives Unit 4 Applications of Derivatives

Types of Maximums and Minimums 4.1 Critical Number Test Types of Maximums and Minimums Absolute/Global Extreme Values Maximum – Minimum – Local/Relative Extreme Values

4.1 Critical Number Test Critical Number/Value –

4.1 Critical Number Test For the following picture, find the absolute extrema, local extrema and critical numbers.

4.1 Critical Number Test Extreme Value Theorem – If f is continuous on a closed interval [a, b], then f has on that interval.

Critical Numbers Test for Finding Extrema 4.1 Critical Number Test Critical Numbers Test for Finding Extrema If f is continuous on a closed interval [a, b], then it’s absolute extreme values are paired with critical numbers or endpoints. Steps for finding the absolute extrema. 1. 2. 3. 4.

4.1 Critical Number Test Find the critical numbers of f. Then use the critical number test to determine the absolute extreme values of f. 1)

4.1 Critical Number Test Find the critical numbers of f. Then use the critical number test to determine the absolute extreme values of f. 2)

4.2 Mean Value Theorem

4.2 Mean Value Theorem Determine if f satisfies the conditions of the mean value theorem. If so, find all possible values of c. 1)

4.3 First Derivative Test

4.3 First Derivative Test

4.3 First Derivative Test

4.3 First Derivative Test

4.3 First Derivative Test

4.4 Second Derivative Test If the second derivative is positive, a curve looks like If the second derivative is negative, a curve looks like

4.4 Second Derivative Test f is concave up on (a, b) if and only if the slopes of the tangents to f are increasing which means f is concave down on (a, b) if and only if the slopes of the tangents to f are decreasing

4.4 Second Derivative Test (c, f(c)) is an inflection point if and only if 1. 2.

4.4 Second Derivative Test

4.4 Second Derivative Test

4.4 Second Derivative Test Relationship of f’ and f’’ to f f

4.4 Second Derivative Test