Math 114, Calculus I, Laboratory 5
The Derivative as a Function

1.  Read pages 115-118. Here is the basic process for finding the slope of the tangent line to the graph of at the point :

form the slope of the secant line between the points and and then take the limit of the slope of this secant line as . Let’s illustrate example 6 using MapleV:


What is this strange looking D(f)(x); ? Look at the top of page 118 for a discussion of D. The basic idea is that D stands for “derivative”, and represents the process of taking a function f and turning it into another function called D(f), or sometimes . See what happens when you enter D(f); instead of D(f)(x); Also see what happens when you enter D(f)(2); and D(f)(1); and D(f)(0); Go through the same steps shown in the MapleV worksheet above with EXAMPLE 7 on page 117 and problem 19 on page 121. Notice how in these cases we have two graphs: the graph of the function and the graph of its derivative. Think about this: Is there any relationship between the domain of the function and the domain of its derivative? For example, does one domain always contain the other?

SUBMIT

two printouts showing you imitated the process illustrated above for EXAMPLE 7 page 117 and problem 19 on page 121, a paragraph that explains the difference between D(f)(x) and D(f). Write D(f)(2), D(f)(1) and D(f)(0) using the f' notation. Also include your answer to the question about the domains of f and D(f).

2.  Maple knows lots of differentiation rules. Look at the table of rules on page 131, and enter the following commands into Maple:
>assume(f, procedure ); assume(g, procedure); assume(c, constant);

Now go to Options, Assumed Variables, and select No Annotation. Next enter
D(f+g); D(f+g)(x);

Notice you get one of the differentiation rules on page 131. Try to get Maple to recite the other differentiation rules on that page. At times you may need to use simplify. You need to memorize these rules. After completing this activity, enter restart; to clear out the assumptions from MapleV's brain.

SUBMIT for each of the seven rules on page 131, a verbal description of the rule along with MapleV's recitation of the rule.

3.  Look at problem #61 on page 133. Let's take the domains of f and g to be [0,7]. Do you notice that it is actually easier to read off the values of the derivatives of f and g from the graph than it is to read off the values of the functions themselves? Fill in the table of values on the attached page.
Here's one way to enter f into MapleV:

f:=proc(x)

if 0<=x and x<=2 then 2*x;
elif x>=2 then –(x-2)/3+4;

fi;

end;

Write MapleV procedures for g, fprime and gprime. Now plot the function, derivative pairs over the domain 0<x<7 on the same coordinate axis so you can easily complete the table. The syntax is plot([f, fprime], 0..7); If we take the domains of f and g to be , then what are the domains of u and v?

SUBMIT the filled in table. How did the rules for differentiation help you to fill in the table? Also give us a printout showing your MapleV code for the functions f, fprime and g, gprime, the graphs, and your answer to the question about the domains of u and v.

4.  Read problem #26 on page 143. Use MapleV to get good graphs of the functions R and S (pay attention to tickmarks, labeling of axes, scale, title, etc.). What is R(0)? Can you see it on your graph? In view of the fact that R is decreasing, what can you say about S? Why? What is and ? What is and ?

SUBMIT your graphs of R and S, your answers to the posed questions, and your comment about the values of R and S at low levels of brightness.

x

/ f '(x) / g '(x) / f (x) / g (x) / u(x) / v(x) / u '(x) / v '(x)
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