given by the equation
, consider the Maple procedure below, which we shall call
deriv
, which should take an input
in the domain of the function
and
output the slope of the tangent line at the point (
).
1. The Derivative as a function.
For the function
given by the equation
, consider the Maple procedure below, which we shall call
deriv
, which should take an input
in the domain of the function
and
output the slope of the tangent line at the point (
).
> deriv:=proc(x);limit((f(x+h)-f(x))/h,h = 0);end;
In order for you to use your procedure, it is necessary that you pre-define f as a function before executing the procedure .
Submission:
A plot which shows the graphs of
and deriv(
) on the same axes on the domain -2..2. Then write a paragraph explaining this plot, addressing the following question:
How can you use only the graph of
to explain the graph of
=deriv(
)?
You will be assessed on the clarity of your exposition as well as the correctness of your assertions.
Submission worksheet:
Maple is really good at differentiating functions. Read about Maple's diff command from the help screen. Also read about Maple's subs command from the help screen.
Submission:
Use the diff command to differentiate:
(a)
(b)
= (
)(
)
(c)
(d) Use Maple's
subs
command to find the slope of the tangent line of
at (
) .
Submission worksheet:
There is another way to view a derivative. It is a function itself - which enjoys functions as inputs and returns another function as an output. For example D(sqrt) gives an output which might at first sight look confusing, but notice that the output is a proc , i.e., a function. So, you know how to evaluate functions - wrapping the input up in parentheses to feed to the function: D(sqrt)(x), D(sqrt)(4) . Notice how you don't have to use the substitution command - since we are actually evaluating a function instead of substituting into an expression .
Submission:
Use the D command to find the slope of the tangent line of the given function at the given point. Write the equation of the tangent line and plot the tangent line and the function on the same set of axis (be sure to choose the range on your graphs to demonstrate that your line is indeed tangent at the point).
(a)
at (0,5).
(b)
at (-2,0) .
Submission worksheet: