# Applications of Graphing and Derivatives : Critical Points, Concave, Increasing and Decreasing

Using a calculator or computer, graph the functions. Describe briefly in words the interesting features of the graph including the location of the critical points and where the function in increasing/decreasing. Then use the derivative and algebra to explain the shape of the graph.

9.
Try graphing the x values from -5 to 5 with the major unit of 1, and the y values from -40 to 40 with the major unit of 10.

First Derivative:
Critical Points:

The critical points divide our graph into how many regions? three
-2.49 -1.41 1.41 2.4

You will now substitute a point from each of these regions into the derivative to determine whether the function is increasing or decreasing on that interval.

f is ??? f is ??? f is ???

Second Derivative:

You can now substitute your critical points into the derivative to determine if the graph is concave up or concave down at that point.

Concave ??? Concave ???
Local Min/Max Delete the incorrect one! Local Min/Max
Point: (??, ?? ) (??, ??)

13.
Try graphing the x values from 0.1, 1, 2, 3, 4, 5 with the major unit of 0.5, and the y values from -1 to 8 with the major unit of 1.

First Derivative:
Critical Points:

The critical point divides our graph into (how many?) regions:

?? ??

You will now substitute a point from each of these regions into the derivative to determine whether the function is increasing or decreasing on that interval.

f is ??? f is ???

You can now substitute your critical points into the derivative to determine if the graph is concave up or concave down at that point.

Second Derivative:

Concave ???
Local Min/Max (delete the incorrect one!)
Point: (??, ??)

Section 4.2 p. 175: 9, 17

Use the first derivative to find all critical points and use the second derivative to find all inflection points. Use a graph to identify each critical point as a local maximum, a local minimum, or neither.

9.

First Derivative: Critical Points:

Second Derivative:

You can now substitute your critical points into the derivative to determine if the graph is concave up or concave down at that point.

Concave ??? Concave ??? Concave ???
Local Min/Max Local Min/Max Local Min/Max

To find inflection points, you must set the second derivative equal to zero and solve it!

Inflection Points:

Try graphing the x values from -4 to 4 with the major unit of 1, and the y values from -20 to 60 with the major unit of 20.

17. For f(x) = x3 – 18×2 – 10x + 6, find the inflection point algebraically. Graph the function with a calculator or computer and confirm your answer.

First Derivative:
Second Derivative:
Inflection Points:

Try graphing the x values from -4 to 20 with the major unit of 2, and the y values from -1000 to 500 with the major unit of 500.

Section 4.3 p. 180: 13, 22

13. For f(x) = x – ln(x), and , find the value(s) of x for which:

(a) f(x) has a local maximum or minimum. Indicate which ones are maxima and which ones are minima.

First Derivative:
Critical Points:

Second Derivative:

Concave ???
Local Min/Max (delete the incorrect one!)

There is much debate over labeling an endpoint of an interval as a “local” minimum or maximum. Therefore, we will not classify x = 0.1 or x = 2 as such.

(b) f(x) has a global maximum or global minimum.

Critical Point and End Point Values:
??? is the global max
??? is global min

22. The distance, s, traveled by a cyclist, who starts at 1 pm, is given in Figure 4.34. Time, t, is in hours since noon.

(a) Explain why the quantity, s/t, is represented by the slope of a line from the origin to the point (t, s) on the graph.

(b) Estimate the time at which the quantity s/t is a maximum.

At the point (?, ?), the value of s/t is ?? km/hr – ?:?? pm.

(c) What is the relationship between the quantity s/t and the instantaneous speed of the cyclist at the time you found in part (b).

The quantity s/t is the ?????.

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