Curve Sketching

Some Important Reminders Regarding Curve Sketching

(I will post solutions to both HW problems tomorrow night)

Increasing and Decreasing Functions

The derivative of a function can tell us where the function is increasing and where it is decreasing. If

a) f'(x) > 0 on an interval I, the function is increasing on I.

b) f'(x) < 0 on an interval I, the function is decreasing on I.

The intervals of increase and decrease will occur between points where f'(x) = 0 or f'(x) is undefined. However, these points are not necessarily critical numbers because we include x even if it is not in the domain of f. We simply want to find the intervals of increase and decrease around x, even if the function is not defined at that point.

The graph to the right illustrates this theorem. From A to B, the slope of the tangent lines are all negative, so the derivative, f'(x) is negative from A to B. The theorem above states that the function is decreasing from A to B. The graph shows that the values of the function are decreasing between A and B. Similarly, the function is also decreasing between C and D. From B to C however, the slopes of the tangent lines are positive. Therefore, the derivative is positive from B to C. The graph shows that the values of the function are increasing between B and C.

Concavity and Points of Inflection

A graph is called concave upward (CU) on an interval I, if the graph of the function lies above all of the tangent lines on I. A graph is called concave downward (CD) on an interval I, if the graph of the function lies below all of the tangent lines on I.

The second derivative of a function can tell us whether a function is concave upward or concave downward. If

a) f''(x) > 0 for all x in an interval I, the graph is concave upward on I.

b) f''(x) < 0 for all x in an interval I, the graph is concave downward on I.

The intervals of concavity will occur between points where f''(x) = 0 or f''(x) is undefined. We test the concavity around these points even if they are not included in the domain of f.

The graphs below illustrate the different forms of concavity. Remember that there are two ways in which a graph can be concave upward or concave downward. Graphs A and C illustrate the types of concavity when the function is increasing on the interval, while graphs B and D illustrate concavity when the function is decreasing on the interval. These 4 graphs cover every different form of concavity.

A point P on a curve is called a point of inflection if the function is continuous at that point and either

a) the function changes from CU to CD at P
b) the function changes from CD to CU at P

Points of inflection may occur at points where f''(x) = 0 or f''(x) is undefined, where x is in the domain of f. We must test the concavity around these points to determine whether they are points of inflection.

The graph to the right illustrates a curve with a point of inflection.