MATH 1170
MATHEMATICS FOR LIFE SCIENTISTS
Computer Assignment VII
Due October 13, 2003
PR OBLEMS
Exercise 1. Consider the function
( ) = 8 5 ? 18 4 ? 3 + 18 2 ? 7
Our goal is to nd all critical points and points of inection.
a. Graph the function. You may have to adjust your axes to make the graph look
nice { you should be able to see ve points where ( ) = 0.
b. Have Maple nd the derivative of ( ) with the command dP := D(P);. Graph
the derivative and indicate all regions where the function ( ) is increasing.
c. Locate the critical points on your graph. Use the fsolve command to make
Maple solve for the value at each critical point.
d. Have Maple nd the second derivative of ( ) (take the derivative of the derivative). Graph it, and indicate all regions where the function ( ) is concave up.
e. Locate the points of inection on your graph. Use Maple to solve for the value
at each point of inection.
Exercise 2. Consider the function
( ) = (( )) = 2 +1 +2 + 3
a. Make one graph of ( ) and ( ) for 0 1, and another of ( ). Could you
have guessed the shape of ( ) from looking at the graphs of ( ) and ( )?
b. What happens on the graph of and at the critical point of ?
c. Find the exact location of the critical point of and set it equal to r .
d. Compare (( r)) with (( r)) . Why are they equal?
P x
x
x
x
x
x:
P x
P x
P x
x
P x
P x
x
u x
r x
u x
x
v x
x
v x
x
:
x
r x
r x
u x
u
v
r
r
0
u
v
x
u xr
0
v x
x
x
v xr
Exercise 3. Consider the function
( ) = ( ) ( ) = (1 + )(2 + 2 + 3 )
using the same ( ) and ( ) as in Exercise 2.
a. Make one graph of ( ) and ( ) for ?2 0, and another of ( ). Could
you have guessed the shape of ( ) from looking at the graphs of ( ) and ( )?
b. What happens on the graph of and at the critical point of ?
c. Find the exact location of the critical point of and set it equal to p.
d. Compare (( p)) with (( p)) . Why are they equal but of opposite sign?
p x
u x
u x v x
x
x
x
v x
u x
v x
x
p x
p x
u
u x
v
p
p
0
u
x
u xp
v
0
x
v xp
x
v x