Curve Sketching

Question 1

How to tell if a function is increasing during an interval?

Answer 1

If f(x2) > f(x1) when x2>x1.
Test: If f1(x)>0 for all x during that interval, it is increasing.

Question 2

How to tell if a function is decreasing during an interval?

Answer 2

If f(x2) < f(x1) when x2>x1.
Test: If f1(x)<0 for all x during that interval, it is decreasing.

Question 3

Absolute Max/Min

Answer 3

The highest/lowest value for f(x) within some domain.

** Can occur at endpoints **

Question 4

How to find intervals of concavity?

Answer 4

If f2(x)>0 it is concave up.
If f2(x)<0 it is concave down.
If f2(x)=0 (and cc changes) it is Point of Inflection

Question 5

Classify Critical Points

Answer 5

If f1(c)=0 and f2(c)>0, it is local max
If f1(c)=0 and f2(c)<0, it is local min
If f1(c)=0 and f2(c)=0, it is a special critical PI.**

Question 6

What does f1(x) do?

Answer 6

Find when a function is inc/dec and locate CPs (critical points)

Question 7

What does f2(x) do?

Answer 7

Find and describe concavity, locate PIs (point of inflection) and classify critical points.

Question 8

Interval Charts

Answer 8

Show Intervals, Test (sometimes), y1/y2, and description.
For int of inc/dec, y1, show all CPs AND discontinuities.
For int of cc, y2, show all PIs AND discontinuities.

Question 9

Optimization Steps

Answer 9

1. Identify what is being optimized.
2. Identify constraints.
3. Write a combined function.
4. Graph (if necessary).
5. Identify optimal (max/min) value using calc.
6. State results.