Submission:

Use Maple to compute the derivative of with respect to x , ( you should be able to calculate this derivative with paper and pencil using the product rule), then compute the equation of the tangent line at ( ) .  Recalling the syntax for plotting two curves together: plot([f(x),g(x)],x=a..b) ; plot the graph of the curve and the tangent line on the same set of axis.

Submission worksheet:

2. Derivatives of trigonometric functions .

Consider the function given by the rule

> f:=x->x*tan(x);

By using the product rule, we obtain that . Maple can also compute this derivative by using the D operator as follows.

> D(f);

Notice that the answer is given in function notation, with the arrow. To simply find the expression , we type in

> D(f)(x);

Note that Maple gives a different expression than our hand calculation, but using the trigonometric identity , we can see that the answer from Maple is consistent. It is possible to get Maple to simplify the result to the same form as we got by hand by reminding it of the trigonometric identity that we used above, as follows.

> simplify(D(f)(x),{1+tan(x)^2=sec(x)^2});

If you think carefully about it, we did not gain too much, because we could not simplify the Maple result without knowing in advance how it could be simplified to our result. Nevertheless, there are circumstances where the technique above can be used in more complicated situations to help us decide when two expressions represent the same quantity.  Let us graph the function and its derivative and see how the graphs are related.

> plot([f(x),D(f)(x)],x=-4*Pi..4*Pi,y=-20..20,color=[red,blue],discont=true);

Notice that only at (0,0) does the function reach a minimum value and then start increasing again, and this is exactly where the derivative vanishes. When x>0, note that the function is increasing on each interval of definition, with the slope reaching a minimum value somewhere in the middle of the interval, but then increasing again. A similar pattern occurs when x<0, except that this time the function is decreasing on each interval of definition, and its slope is therefore always negative. We can even ask Maple to solve for the points at which the derivative vanishes.

> solve(D(f)(x)=0,x);

Of course, in this case, the answer that Maple gives us is not too helpful. Note that if you plug into the formula for the derivative, then you can see immediately that the derivative is zero. Do this by hand, and then have Maple calculate it by asking it to evaluate for you. Note that the picture only suggests that the derivative is zero only at . The picture alone can not tell you for sure, because we don't know what is happening outside our window. Nevertheless, the picture is very suggestive.

Submission:

(a) Let .

1. Calculate the derivative of f by hand, and show how you did this in your submission report.

2. Calculate the derivative of f using Maple.

3. Calculate the equation of the tangent line to f at ( )

4. Plot the function, f, and the tangent line on the same set of axis.

(b) Let .

1. Calculate the derivative of g by hand, and show your computations in the report.

2. Calculate the derivative of g using Maple.

3. Plot the function g and g' on the same set of axis over the interval ( ).

4. Write an explanation of why the graph of the derivative seems reasonable for the function, by pointing out key features of the graph of the function that are reflected in the graph of its derivative.

Submission worksheet:

3. More on derivatives of trig functions.

Submission:

(a) Find the derivative of .

(b) Graph both f and its derivative on the same set of axis over the interval ( ) .

(c) Explain why the derivative expression is reasonable in view of the two graphs.

Because of vertical asymptotes, you will need to prevent Maple from "auto-scaling" the y -axis. So you will need to use the syntax plot([f(x),g(x)],x=a..b,y=c..d);

Submission worksheet:

Consider the function defined below.

> f:=x->x^2-cos(x);

Let us plot the function, and see if we can judge from the graph where the derivative is equal to zero, that is where the tangent line is horizontal.

> plot(f);

From the picture, it looks like this occurs when . Of course, there could be other places outside of the domain shown in the plot. To determine where the derivative is equal to zero, let us compute the derivative, and set it equal to zero.

> D(f)(x);

> solve(D(f)(x)=0,x);

As a warning, it should be pointed out that Maple will not always find all the solutions to the problem. But let us think about the equation carefully. In that case, we have , and since the right hand side is never larger in absolute value than , this equality could only have a solution when . So we can rule out the possibility that solutions exist outside of the domain plotted in the picture. However, the graph looks pretty flat near , so in order to be confident about the behavior of the function near zero, it is probably a good idea to plot it over a smaller domain.

Submission:

(a) Determine if the following functions have any horizontal tangents in the interval . If they do where are they. If they do not, why not?

(b) Plot the graph of the functions.

Submission worksheet:

5. Applications of derivatives of trigonometric functions.

Consider an object of weight being dragged along a horizontal surface by a rope acting with a force that depends on the angle which the rope makes with the plane of the surface. This force can be computed by the formula  where is a constant called the coefficient of friction .

(a) Write a Maple procedure defining the function F as a function of the angle , for these specific values of and .

(b) Find the rate of change of F with respect to .

(c) When is the rate of change equal to 0?

(d) Let us suppose that lb and . Plot F ( ) and use it to locate the value of theta for which F'( )=0 . Is this value consistent with your answer to part (c)? Be sure to explain why it is consistent.

Submission worksheet: