1. How can a function fail to be Differentiable?

In this activity you will make several plots of a function together with its derivative. By looking at the graphs, you will begin to see the important interplay between the graph of a function and of its derivative.  Consider the following six functions.

• 

• 

• 

• 

•  f5(x) = 0.2 if x is less than or equal to , if is between and , and if is greater than or equal to 4.

• if does not = and 0 if .

Begin by executing these commands. We have defined the functions for you and set up the plots of the function/derivative pairs.

> f1:=x->abs(x); plot([f1(x),D(f1)(x)],x=-3..3,discont=true,scaling=constrained);

> f2:=x->piecewise(x<0,-(-x)^(1/3),x^(1/3)); plot([f2(x),D(f2)(x)],x=-3..3,y=-3..3,axes=framed,discont=true,scaling=constrained);

> f3:=x->abs(x)^(2/3); plot([f3(x),D(f3)(x)],x=-3..3,y=-3..3,axes=framed,discont=true,scaling=constrained);

> f4:=x->abs(x-6); plot([f4(x),D(f4)(x)],x=-3..9,y=-3..3,axes=framed,discont=true,scaling=constrained);

> f5:=x->piecewise(x<0,0.2,x<4,5-x,1/(5-x));f(x); plot([f5(x),D(f5)(x)],x=-3..7,y=-5..5,axes=framed,discont=true,scaling=constrained);

> f6:=x->x*sin(1/x);f6(0):=0; plot([f6(x),D(f6)(x)],x=-1..1,y=-3.3..3.3,axes=framed);

Submission :

These "bad" features should be chosen from this list: a jump discontinuity, a corner, a vertical asymptote, a vertical tangent line, or an oscillation discontinuity.

Submission worksheet:

2. More interplay between the graph of a function and its Derivative.

Consider the function . We can calculate its derivative by taking the limit of the difference quotient.

> f:=x->sqrt(x-1);

> limit((f(x+h)-f(x))/h,h=0);

Thus we have = .  We may check to see that this result is reasonable by looking at correspondences between the graph of the function and of its derivative. This is the beginning of a very important thread in this course and its sequel-namely understanding graphically correspondences between a function and its derivative.

> D(f):=x->1/(2*sqrt(x-1));

> plot(f,labels=[x,`f`]);

> plot(D(f),labels=[x,`D(f)`]);

We see that when the slope of f is very steep the value of D(f) is very high, and when the slope of f decreases so does the value of D(f) . Also notice that the slope of f is always positive which is reflected in a positive valued function for D(f).

Submission:

If , find , then check to see that your answer is reasonable by comparing the graphs of and . Strive to write a good response. Spend time on making this short piece of technical writing coherent, concise, and complete. You will find the syntax for plotting a function/derivative pair in the previous activity.

Submission worksheet:

Define the function f in Maple by

> f := x->3*x^15-5*x^3+3;

In order to differentiate this function in Maple, we use the D command.

> D(f);

> D(f)(x);

Notice that D(f) is a function, so you can evaluate it at . However, you cannot apply the D operator to the expression . You can only apply it to functions. Thus D(f(x)) will not be interpreted correctly by Maple. (We can use the diff command to differentiate expressions, but don't use it in this exercise.)  Let us plot the function and its derivative on the same graph, and see what features convince us that Maple has given us the correct derivative.

> plot([f(x),D(f)(x)],x=-1.25..1.25,y=-10..10,color=[red,blue]);

Starting from the left, note that f is increasing at first, and the derivative is positive. Then the function reaches a maximum value, and starts decreasing. Its derivative vanishes at the maximum value, and then is negative. At x=0, the function levels off, and its derivative again vanishes, but then it heads down again, and its derivative is negative. At the place where the function reaches a minimum value and then starts increasing, the derivative vanishes and then becomes positive.

Submission:

Use the ideas above to do study the relationship between the graph of the function and its derivative for the following functions.

(a)

(b)

Submission worksheet: