MATH 1170
MATHEMATICS FOR LIFE SCIENTISTS
Computer Assignment VI
Due October 7, 2003
We will use the awesome graphical powers of Maple to study the relationships between
functions, their derivative, and their tangent lines.
PR OBLEMS
1. Consider the function H (y) = py. Our goal is to nd the tangent line at y = 0. It
will help you to write the denition of the derivative in this case to gure out where
the y goes.
(a) Find and graph the secant line with y = 1, along with the function H itself.
(b) Find and graph the slope of the secant as a function of y.
(c) What is the limit as y ! 0? What is the equation of the tangent line to the
curve?
2. Consider the function A(x) = j sin(x)j. This can be entered as
> A := x -> abs(sin(x));
(a) Graph this function for ?2 x 2 and identify one point where it is not
dierentiable. Call this point the base point.
(b) Find the slope of the secant as a function of the distance x from this base point.
Graph the slope function for ?2 x 2.
(c) What is the limit of the slope as x ! 0?
(d) Maple has a feature to nd left and right-hand limits. If you called the slope
function slope, then
> limit(slope(Dx),Dx=0,left);
gives the left hand limit. Find the left-hand and right-hand limits of your slope.
(e) Find the two tangent lines associated with these slopes. Add them to your graph
of A(x).
3. Consider yet again the wiggly function G(x) = cos(x2 ) ? sin(5x) ? cos(7x). If you input
this as a function, Maple can compute the derivative G (x) (here called Gp) with the
command
0
> Gp := D(G);
(a) Plot both G(x) and G (x) for 0 x 2.
(b) Mark all regions where G(x) is increasing. What do they correspond to on the
graph of G (x)?
(c) Mark where G(x) takes on its smallest value. What is G (x) at that value of x?
(d) Find where G (x) takes on its smallest value. What is happening to G(x) at that
value of x?
0
0
0
0