C.D.3 - Apply theorems and rules of differentiation to solve mathematical and real-world problems. a. Explain geometrically and verbally the mathematical and real-world meanings of the Extreme Value Theorem and the Mean Value Theorem. b. Write an equation of a line tangent to the graph of a function at a point. c. Explain the relationship between the increasing/decreasing behavior of 𝑓 and the signs of 𝑓′. Use the relationship to generate a graph of 𝑓 given the graph of 𝑓′, and vice versa, and to identify relative and absolute extrema of 𝑓. d. Explain the relationships among the concavity of the graph of 𝑓, the increasing/decreasing behavior of 𝑓′ and the signs of 𝑓′′. Use those relationships to generate graphs of 𝑓, 𝑓′, and 𝑓′′ given any one of them and identify the points of inflection of 𝑓. e. Solve a variety of real-world problems involving related rates, optimization, linear approximation, and rates of change.

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