1. Define the function . Use the diff command to find
the derivative of P.
2. Find all critical points of P and make a chart indicating where P is
increasing and where it is decreasing. Determine which critical
points give local maxima, which give local minima, and which
give neither.
3. Use the diff command to find the second derivative of P.
4. Find all x-values where the second derivative of P is 0, and
make a chart indicating where the graph of P is concave up and
where it is concave down. Find all inflection points of P.
5. Use the information from steps 2 and 4 to sketch the graph of P.
(This step is done by hand.)
6. Use the plot command to draw the graph of P for 3 < x < 6.
Compare this graph to your sketch in step 5.
Part II. Newton’s Method
7. Use MAPLE to define the two functions f (x) = e
2x and
g(x) = x + 2. Make a plot on the same set of coordinate axes of f
and g, being sure to choose x values so that you see both places
where the graphs of the two functions cross. Note that the
MAPLE command for e
2x
is exp(2*x).
8. Use Newton’s method (applied to the function f(x) - g(x)) to find
all the x values where e
2x
(x + 2) = 0. Your choice of starting
values should be motivated by the graph in part (1). Continue
the iterations until there is no change in the values at least to the
seventh decimal place. Show on your output all the iterations
you calculate.
Part III. Implicit Functions
9. Define the equation cos(x) sin(y) = .25, by using
>f:= cos(x-y)*sin(y)=.25
Right click on the expression you get after defining f, and use the
“plots →plots builder” command to plot the curve. First plot the
curve with domain and range [-10,10] and use the grid [30, 30].
Then do the same, but using the grid [100, 100]. State what you
thinkthe “grid” does.
10. Find all values of y that correspond to the value x = 1. This
can bedone by defining
> c:=1;
>subs(x=c,f);
>evalf(solve(%))
(where the % imports the information from the preceding line).
Make sure you get all the y-values for this x-value in the given
window (check against your graph).
11. Find dy for the function above, using MAPLE’s implicitdiff
commandand the subs command. You will need to use
>subs({x=1,y=??},implicitdiff(f,y,x))
where you obtain the value for y from the preceding question.
Do this for every y-value in [ 10, 10] when x = 1. What pattern
would you expect for these derivatives by looking at the graph