A Related Rates Problem

Gravel is being dumped from a conveyor belt at a rate of 30 cubic ft/min and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal.

How fast is the height of the pile increasing when the pile is 10 ft. high?

4.2Finding intervals of increase and decrease; Concavity

Greg Kelly, Hanford High School, Richland, WashingtonModified: Mike Efram HHS 2004

Photo by Vickie Kelly, 1995

Old Faithful Geyser, Yellowstone National Park

In the past, one of the important uses of derivatives was as an aid in curve sketching. Although we can usually use a calculator or computer to draw complicated graphs, it is still important to understand the relationships between derivatives and graphs.Technology is not always perfect…for example, if the important part of the graph is at x = 100, we often do not see any details (because our focus is around the origin!!)Today we will begin to learn how the derivative will help usSketch curves without the use of technology.

First derivative:

y is positive Curve is increasing (rising).

y is negative Curve is decreasing (falling).

y is zero Possible local maximum or minimum.

Second derivative:

y is positive Curve is concave up.

y is negative Curve is concave down.

y is zero Possible inflection point(where concavity changes).

y is undefined Possible local maximum or minimum.

“holds water”

“spills water”

Example:Given: 2( ) 5 6f x x x

Find: a) Intervals where f is increasing b) Intervals where f is decreasing c) Intervals where f is concave up d) Intervals where f is concave down e) Any inflections?

Example:Given:

23( )f x x

Find: a) Intervals where f is increasing b) Intervals where f is decreasing c) Intervals where f is concave up d) Intervals where f is concave down e) Any inflections?

Example:In each part, sketch a continuous curve, f(x),with the stated properties:

Given: (2) 4, (2) 0, ( ) 0 for all f f f x x

Example:In each part, sketch a continuous curve, f(x),with the stated properties:

Given: (2) 4, (2) 0,

( ) 0 for 2, ( ) 0 for 2

f f

f x x f x x

Example:In each part, sketch a continuous curve, f(x),with the stated properties:

Given:

2 2

(2) 4, ( ) 0 for all 2,

lim ( ) , lim ( )x x

f f x x

f x f x