STAGE 7: EXPLORING THE Landscape

In Stages 5 and 6 we learned how to compute derivatives of Field Guide functions. The derivative can be interpreted as an instantaneous rate of change or as a slope. In Stage 7 we apply derivatives and second derivatives to determine where functions are increasing and decreasing, where their graphs are "concave up" and "concave down", and where they have local and absolute maxima and minima or "inflection points". We explore the art of curve sketching -- making a sketch of a function which shows the important qualitative attributes, and we meet and prove one of the basic theorems in Calculus, the Mean Value Theorem.

Applications of finding maxima and minima are the subject of Stage 8.

Objectives

Determine the monotonicity (i.e., increasing/decreasing) and concavity properties of a function. Apply the First Derivative Test and draw conclusions about the first and second derivatives from these properties.

Find local extreme points and points of inflection. Apply the Second Derivative Test.

Find the critical points of a function. Find absolute maxima and minima.

LESSON
What To Do:
Go to the Onward and Upward area for a description of the graded activities that you must complete in this Stage. There are no graded activities in the Lesson or Practice areas. You will need to complete these graded activities before moving on to the next Stage:

Stage 7 Quiz
Stage 7 Communication Activity
We recommend going to the Onward and Upward area to look at the Communications Activity right away. The Communications Activity depends on your knowledge of the Mean Value Theorem, and you might consider reading through the section on the MVT first. The Stage 7 quiz concentrates on the remainder of the Lesson.
Side Trails: There are very few enrichment pages in Stage 7.

PRACTICE

ONWARD
and UPWARD

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