1. The second derivative.

The first derivative of a function is the function . The second derivative of the function is the first derivative of the function . So the notation for the second derivative is of course . This is a function, and so can be evaluated. For example, to find the first and second derivatives of with Maple, we proceed as follows.

> f:=x->x^4-3*x^3+16*x;

> D(f)(x); D(D(f))(x);

We can also evaluate the second derivative at a specified value, for example 1,

> D(D(f))(1);

Warning! Beginners often use the bogus syntax and , so try not to make this mistake.

Let .

(a) Graph f and f'' on separate axis, but with the same domain.

(b) On what intervals is f'' > 0 ? On those intervals how is the graph of f related to its tangent lines?

(c) On what intervals is f'' < 0 ? On those intervals how is the graph of f related to its tangent lines?

Submission worksheet:

In this exercise, we show how to compute higher derivatives in Maple, and to apply these higher derivatives to study the rate of change of a function. Let us suppose that a particle is moving along a path whose position is given by the following function.

> s:=t->4-7*t+6*t^2-t^3;

To see what the path of the particle looks like, let us graph the function. By experimenting with domains, the domain -1..5 seems to capture the interesting part of the graph.

> plot(s,-1..5);

Next let us define the velocity and acceleration functions by taking derivatives.

> v:=D(s);

> a:=(D@@2)(s);

Look up the D command and see some other ways there are to correctly enter the command for second derivatives. Give an example of such a definition.

In the text, there is a terminology for the third derivative of the position function, or the first derivative of the acceleration, which is the same thing, called the jerk of the function. Let us define this as well. We could define the jerk by , or equivalently by , or .

> j:=D(D(D(s)));

This says that for this problem, the jerk is a constant function. Now let us graph the position function, its velocity and its acceleration all on one graph.

> plot([s,v,a],-1..5,color=[red, blue, green]);

There are various connections that we can see between these graphs. First, note that when the red graph has a horizontal tangent line, the blue graph is crossing the x -axis. Also, when the slope of the red graph is positive (negative), the value of the function in the blue graph is positive (negative). Similarly, when the blue graph is increasing (decreasing), the green graph is above (below) the x -axis, and the green graph crosses the x-axis precisely at the point where the slope of the tangent line to the blue graph is zero. It is thus pretty easy to identify which function represents the position, the velocity and the acceleration just from their relationships that you can see in the picture.

Submission:

The position function of an object traveling in a straight line over the time interval [ ] is . Graph f, its velocity function and its acceleration function on the same set of axis. Discuss the objects behavior in relation to the signs and values of and . Include in your discussion the following items:

(a) When is the object momentarily at rest.

(b) When does it move up and when does it move down (assume down is a negative direction and up is a positive direction).

(c) When does it change direction?

(d) When does it speed up and slow down?

(e) When is it moving fastest? When is it moving slowest?

(f) When is it farthest from the origin (the point ( )).

Submission worksheet:

3. More on the second derivative and a review of "Non-differentiability."

Recall that the Maple command for absolute value is abs, thus f(x)=abs(x), is computed by

> f:=x->abs(x);

Submission:

If , find f' and f'' . Plot the graph of all three functions, be sure to use the option discont=true. What are the domains of each function? Explain why the domains are what they are, use words like "corner", "jump discontinuity", etc. .

Submission worksheet: