Mathematics 161                                  Laboratory 8                             October 22, 1992
	  
Name: _____________________________   Lab Partner: ___________________________
	  
Consulted with:  ____________________________________________________________

© Lafayette College, 1994

Concavity & Derivatives

In this lab we will define concavity, and describe its relationship to the first and second derivatives.

Part 0: What is Concavity?

Preliminaries

As in preceding labs, we will use some special commands contained in the "DerivativeGraphics" package; recall, we input the package as follows:

<<i:\course\root\math161\derivative.m

What Concavity Looks Like

Consider the trigonometric function t(x) defined below:

t[x_] := Tan[Pi x/2]

Use the command below to plot the graph of the function and its tangent line at (1/2, 1) over the interval (0, 3/4):

TangentLine[t,1/2,{x,0,3/4}]

Notice that the graph "sits on top" of its tangent line.  That is, over an interval containing the point of intersection (tangency), the values of t are larger than the values of the function defined by the tangent line.  At any point on a graph where such a line can be drawn, the function is called concave up.  Notice that as we move from left to right along the graph near x = 1/2, the slopes of the tangent lines get steeper.

Now plot the function and its tangent line at (–1/2, –1), using a similar command.

TangentLine[t,-1/2,{x,-3/4,0}]

This time, in an interval containing the point of intersection, the graph remains below the tangent line, and the values of t are less than those of the function defined by the tangent line.  At any point on a graph where such a line can be drawn, the function is called concave down.  Notice that as we move from left to right along the graph near x = –1/2, the slopes of the tangent lines get less steep.

A straight line lies neither always above nor always below any other intersecting line, and so it has no concavity or zero concavity.

Part 1:  Concavity and the Second Derivative

Plot the second derivative of t(x) over the interval (–3/4, 3/4), and make a hypothesis about the relationship between the concavity of a function and the sign of its second derivative.  Make as many tangent line plots (like the two above) as you need to formulate your hypothesis.

Write your hypothesis here.

Hypothesis 1:

Plot the first derivative of t over the same interval.  Make a hypothesis about the relationship between the concavity of a function and the behavior of its first derivative.  (Hint:  The second derivative is the derivative of the first derivative.  What does the sign of a derivative have to do with the behavior of its function?  This is lab 5 again!)

Hypothesis 1a:

Part 2:  Testing Hypothesis 1

Define in Mathematica the following function:

Use the command below to plot the graphs of a and its second derivative over the interval (0, 2).  This may take a moment.  If you have defined a correctly, Mathematica will complain extensively about this plot, concluding Plot::plnr: CompiledFunction ... is not a machine-size real number at x = 1.Which of the two functions is not defined at x = 1?  Explain how you know.

Plot[{a[x], a''[x]}, {x, 0, 2}, PlotStyle->{Red,Blue},
     PlotRange->All]

What can you say about the concavity of a based on the  plot above and Hypothesis 1?

Plot a(x) and the line y = 1/2 together.  Based on this graph, what can you say about the concavity of a at the point (1, 1/2)?  (Look at the definitions of "concave up" and "concave down" if you need inspiration.)

Based on your investigations, is Hypothesis 1 true?  Explain.

Plot the derivative of a(x). Describe its behavior (increasing, decreasing, or constant), including at x = 1.

Based on your investigations, is Hypothesis 1a true?  Explain.

If necessary, revise your hypotheses to reflect what you have learned studying a(x).

Hypothesis 2:

Hypothesis 2a:

Part 3:  Inflection Points

A point of inflection for a continuous function f(x) is a point on the graph where the function changes from concave down to concave up or vice versa).

Based on your new hypotheses and the graphs of t '(x) and t ''(x) generated in part 0 of this lab, list the points of inflection for t(x) with a brief justification.

Draw a "concavity chart" for the function t on the number line below.  That is, indicate on the line the intervals and points where it is concave up, those where it is concave down, and those where it has no concavity.  Label the x values corresponding to inflection points.

    <____________________________________________>

Is (1, 1/2) a point of inflection for a(x) in part 1?  Explain.

Draw a concavity chart the function a below.

    <____________________________________________>

Part 4:  An Interesting Function

Use the command on the next page to define the function

Cbrt is a special function defined in the DerivativeGraphics package you loaded at the start of the lab (is it loaded now?) that forces Mathematica to give the real cube root of a real input (Mathematica's default is cube roots in the right half-plane, for those of you familiar with complex numbers).

q[x_] := Cbrt[(x^5/20 - 2x^3/3 + 3x)^2]

Draw a "concavity chart" for this function.  Support your chart with plots of the graphs of the function q and its derivatives as needed.  Write a short essay describing how you determined the concavity for each interval and point.