Graphing the

Derivative, Application

sSection 3.1b

Remember, that in graphical terms, the derivative of afunction at a given point can be thought of as the slopeof the curve at that point…

Therefore, we can get a good idea of what the graph oflooks like by estimating the slopes at various pointsf

along the graph of …f

Graph the derivative of the function f whose graph is shown below.Discuss the behavior of f in terms of the signs and values of f .

A

B C

D

E F

Point Estimated SlopeA 4B 1C 0D –1E –1F 0

A

BC

D E

F

Note: We do not have a formula foreither the function or its derivative, butthe graphs are still very revealing…

Graph the derivative of the function f whose graph is shown below.Discuss the behavior of f in terms of the signs and values of f .

A

B C

D

E F

A

BC

D E

F

We notice that f is decreasing where f is negative and increasingwhere f is positive.Where f is zero, the graph of f has a horizontal tangent, changingfrom inc. to dec. (point C) or from dec. to inc. (point F).

Suppose that the function below represents the depth y (in inches)of water in a ditch alongside a road as a function of time x (in days).

A

B CD

E F

A

BC

D E

F

1. What does the graph on the right represent? What units would you use along the y -axis?

The graph represents the rate of change of the depth of the waterwith respect to time. The derivative dy/dx would be measured ininches per day.

Suppose that the function below represents the depth y (in inches)of water in a ditch alongside a road as a function of time x (in days).

A

B CD

E F

A

BC

D E

F

2. Describe what happened to the water in the ditch over the course of the 7-day period.Water is 1 in deep at the start of day 1 and rising rapidly, cont. torise until end of day 2, where it’s max. depth is 5 in. Then thewater level goes down until it reaches a depth of 1 in at the endof day 6. During day 7 the water rises to about 2 in.

Suppose that the function below represents the depth y (in inches)of water in a ditch alongside a road as a function of time x (in days).

A

B CD

E F

A

BC

D E

F

3. Can you describe the weather during the 7 days? When was it the wettest? When was it the driest?

The weather appears to have been wettest at the beginning ofday 1, and driest at the end of day 4.

Suppose that the function below represents the depth y (in inches)of water in a ditch alongside a road as a function of time x (in days).

A

B CD

E F

A

BC

D E

F

4. How does the graph of the derivative help in finding when the weather was wettest or driest?The highest point on the graph of the derivative shows where thewater is rising the fastest, while the lowest point on this graphshows where the water is declining the fastest.

Sketch the graph of a function f that has the following properties:

0 0;f i.f ii. the graph of is shown below;

fiii. is continuous for all x.

y f x y f x

Find the lines that are (a) tangent and (b) normal to the givencurve at the given point. 27 2 3f x x x 2x

2

0

7 2 2 3 2 152 lim

h

h hf

h

2

0

7 4 2 12 12 3 15limh

h h hh

0lim 10 3h

h

10Tangent: m = –10, point (–2,15)

10 5y x 15 10 2y x Normal: m = 1/10, point (–2,15)

1 7610 5

y x 115 210

y x

Try #14, 15, and 16 on p.102…14. The function that is its own derivative: xy e

How did we arrive at this answer graphically???

(0,1)

xy y e

Try #14, 15, and 16 on p.102…15. (a) The amount of daylight is increasing the fastest when the slope is greatest sometime around April 1.

Rate at this time4 hours24 days

16

= hours/day

(b) The rate of change of daylight appears to be zero whenthe tangent to the graph is horizontal January 1 and July 1.

(c) Positive: January 1 through July 1 Negative: July 1 through December 31

Try #14, 15, and 16 on p.102…16. The slope of the graph is zero at about x = 1 and x = –2…

f x

The derivative graph includes the points (–2,0) and (1,0).

The slopes at x = –3 and x = 2 are about 5 and the slope atx = –0.5 is about –2.5…

The derivative graph includes the points (–3,5), (2,5), (–0.5,–2.5).