
Due Wednesday, June7th Estimating
functions and integrals.
Often information about a function is limited to the value of the function
at a few points or the values of the function and some of its derivatives
at a single point. The following problems ask you to construct alternative
functions matching the same given information and compute related
integrals.
1. Suppose f is a function with f (0) = 0, f (1) = 1, and f (2)
= 0 .
A Find a trigonometric function trig(x) so that trig(0) = f (0)
= 0, trig(1) = f (1) = 1, and trig(2) = f (2) = 0. Graph the function trig
and find .
B. Find a quadratic polynomial function q(x) so that q(0) = f (0) =
0, q(1) = f (1) = 1, and q(2) = f (2) = 0. Graph the function q and
find
.
2. Suppose f is a function with f (0) = 1, f '(0) = 1, and f
''(0) = 1 .
A. Find a trigonometric function rig(x) so that rig(0) = f(0) = 1,
trig '(0) = f '(0) = 1, and trig ''(0) = f ''(0) = 1. Graph the function
trig and find
.
B. Find a quadratic polynomial function q(x) so that q(0) = f(0)
= 1, q '(0) = f '(0) = 1, and q ''(0) = f ''(0) = 1. Graph the function
q and find
.
3. Suppose f is a function with f (0) = 1, f '(0) = 1, f ''(0)
= 1, f '''(0) = 1, and f ''''(0)=1.
A. Find a quadratic polynomial function q(x) so that q(0) = f(0)
= 1, q '(0) = f '(0) = 1, and q ''(0) = f ''(0) = 1. Graph the function
q and find
.
B.Find a cubic polynomial function c(x) so that c(0) = f(0) = 1, c
'(0) = f '(0) = 1,
c ''(0) = f ''(0) = 1, and c '''(0) = f '''(0) = 1. Graph the function
c and find
.
C. Find a quartic polynomial function r(x) so that r(0) = f(0) = 1,
r '(0) = f '(0) = 1,
r ''(0) = f ''(0) = 1, r '''(0) = f '''(0) = 1, and r ''''(0)
= f ''''(0) = 1.
Graph the function r and find .

Due Wednesday, June 14. Euler's method and estimating
trignometric functions.
The sine function satisfies the differential equation y'' = y. In
other words, if y = f(x) and y' = f '(x) =
g(x),
and g'(x) =  f(x) with y(0) = f (0)
= 0 and y'(0) = g(0) = 1 then a solution for y is y(x)
= f(x) = sin(x). We can therefore apply Euler's
method in estimating the sine as the solution to the differential equation
y''
= y with the initial conditions y(0) = 0 and y'(0) = 1.
The tangent function satisfies the differential equation y' =
1+y ^{2} with y(0)=0. This allows us to estimate
the tangent function with Euler's method as the solution to this differential
equation.

Use Euler's method with n = 10 applied to y'' = y with the initial
conditions y(0) = 0 and y'(0) = 1 to estimate sin(1). Compare
your result to the value for sin(1) on your calculator. [Understimate?
overestimate? why?]

Use Euler's method with n = 10 applied to y'' = y with the initial
conditions y(0) = 1 and y'(0) = 0 to estimate cos(1).Compare
your result to the value for cos(1) on your calculator. [Understimate?
overestimate? why?]

Use the previous estimates to estimate tan(1).

Use Euler's method with n = 10 applied to y'=1+y ^{2}
with y(0)=0 to estimate tan(1). Compare your result to the value
for tan(1) on your calculator. [Understimate? overestimate? why?]
SOLUTION (.xls
file)

Due Wednesday, June 21. Comparing
Functions and Polynomials.
One way to select a polynomial to use for approximating a nonpolynomial
function is to select coefficients for the polynomial to minimize some
measure of the difference between the two functions.



Find the value(s) for the coefficient m of a linear function f(x)
= mx to minimize the largest difference between f(x) and the function
p(x)
= x^{2} for all x in the interval [0,1]. Explain
your result. Draw a figure showing the point between 0 and 1 where the
difference is largest.

Find the value(s) for the coefficient m of a linear function f(x)
= mx to minimize the largest difference between f(x) and the function
p(x)
= x^{3} for all x in the interval [0,1]. Explain your
result. Draw a figure showing the point between 0 and 1 where the difference
is largest.

What if p(x)= x^{n} where n is
a positive integer?

What if p(x)= e^{x} ? [Correction For use
next time :) use p(x)= e^{x} 1.]

What can you say in general about this problem?



Find the value(s) for the coefficient m of a linear function f(x)
= mx to minimize the value of the definite integral,
. Explain your result.

Find the value(s) for the coefficient m of a linear function f(x)
= mx to minimize the value of the definite integral,
. Explain your result.

What if the integral is
where n is a positive integer?

What if the integral is ?

What can you say in general about this problem?

Due Thursday, July 6 A review trip.

Let .

Show that g takes on its smallest value when

Find the values of A that minimize g(A) in part a)
a) when f(x) = x and
b) when f(x) = exp(x).

FOR THIS PROBLEM ASSUME f '(x) = g(x) and
g'(x)=
f(x)
for all x, and that
f(0) = 0 and
g(0) = 1.

Using Euler's method with n = 4, estimate f(1) and g(1).

Show that [g(x)]^{ 2 } [f(x)]^{
2}=1 for all x. [Hint: Consider P(x)=[g(x)]^{
2 } [f(x)]^{ 2}.]

Let .
Show S''(x) = S(x), S(0) = 0 and S'(0) = 1.

Using your knowledge of calculus and the exponential function, graph the
following:
(a) S(x).
(b) S'(x).
In your graphs show and explain such features as extrema,
concavity, symmetry, etc.

Due Wednesday, July 12, 2000 Darts!

Darts: Imagine a circular board of radius 60 centimeters which is a powerful
magnet. See Figure 1. This magnet is very strong so that when I turn my
back to it and release a magnetic dart, the dart will be drawn to the board
and will land at random somewhere on the board. Let's suppose for simplicity
the following uniformity condition:
If we have any two regions on the board of equal area then the dart
would land in those regions about the same number of times with a large
number of throws.
In other words the two events of the dart falling in two regions of
equal area are equally probable or equilikely. Now I'll throw a magnetic
dart and mark where it lands on the board. This is our basic experiment.
For each of the following questions include a discussion of how you arrived
at your answer with your response.

If the board is divided into two regions by a concentric circle of radius
30 and I repeat the experiment a large number of times, what proportion
of markings from the experiment is likely to lie in the inner region?
See Figure 1.

If the board is divided into four regions by concentric circles of radius
15, 30, and 45 cm and I repeat the experiment a large number of times,
what proportion of the markings from the experiments is likely to lie in
each of the regions?
 If the board is divided into n regions by concentric circles of radius
(k/n) 60 cms with k = 1,2 ... n1, and I repeat the experiment a large
number of times, what proportion of the markings from the experiments
is likely to lie in each of the regions?

What proportion of the markings from repeating the experiment is likely
for the dart to lie within A cms of the center where 60>A>0? See Figure
2

Darts and Averages. Consider the same circular magnetic dart board with
radius 60 centimeters. Draw 5 concentric circles on the board with radii
10,20,30,40, and 50 creating 6 regions (one disc and five bands). Consider
the following game: Throw the dart at the board and score 5 points for
landing in the inner disc, 15 points for landing in the smallest band (between
the circles of radius 10 and 20), 25 points for landing in the next smallest
band (between the circles of radius 20 and 30), 35 points for landing in
the band between the circles of radius 30 and 40, 45 points for landing
in the band between the circles of radius 40 and 50, and 55 points for
landing in the outermost band (between the circle of radius 50 and and
the edge of the dart board). [In this game a low score is considered evidence
of greater skill.]
We allow a player to throw a dart 36 times and find the total score
for the player as well as the average score (the total divided by 36).
Notice the average will be a number between 5 and 55. Give a total score
and an average that are likely for a player with no special skill. [These
numbers are described as the expected score and the expected average for
the game.] Discuss your reasoning and show the work leading to your proposed
solution.
Suggestion: You might investigate the similar problem with 2 regions
and 4 throws and then 4 regions and 16 throws as a way to begin thinking
about the problem. How many darts do you think would fall in each region?
If we measure the distance each of the 36 darts fall from the center,
what do you think the average distance would be for a player with no special
skills. Explain your reasoning, any connections with the expected average
score in the game, and whether you have any belief about the accuracy of
your response. (Is it an underestimate or an overestimate?)

Due Wednesday 71900. Differential Equations,
MacLaurin Polynomials, and Euler's Method.
Suppose f is a
function with f (0) = 1, f '(0) = 1 and f ''(x)=
2f '(x) + 3f (x).

Find the MacLaurin polynomial of degree
6 for f.
[Hint: First find an equation for
f
'''(x).]

Estimate the value of f (1) using
this polynomial.

Use Euler's method with n = 4 to estimate
f
(1).

Due Thursday 72700. MacLaurin Polynomials
and Series

Let g(x) = (1+x)^{1/2} . Find the Maclaurin polynomial of
degree 4 for g. Use your result in part i to estimate g(1). Discuss the
error in this estimate.

Suppose g is a
function with g''(x) =  g(x) for all x and g(0) = 2 and g'(0) = 1.
Find the MacLaurin series for g

....

Find the Maclaurin polynomial of degree 8 for cos 2x.

Find the Maclaurin polynomial of degree 8 for sin^{ 2}x. (Hint:
sin^{ 2} x = (1  cos (2x) )/2.)

Estimate ò_{x=0}^{
x=1}(sin (x) /x )^{2} dx using the first three nonzero terms
of the Maclaurin polynomial in b). Explain how you handled the fact that
this is an improper integral in your estimate.

Estimate the integral in part c) using Simpson's rule. Explain how you
handled the fact that this is an improper integral in your estimate.

Discuss the error in your estimate in part c). [Use Taylor's theorem.]

Fitting curves.
Making Curves Fit together Smoothly: One way to make a curve that passes
through several points and looks smooth is to draw several curves that are defined
by a small number of points and make sure that when the curves are joined together
they have the same tangent lines, making the connections appear smooth.

Use two curves defined by quadratic functions to make a single smooth curve
that passes through the four points, (1,0), (0,0), (1,2) and (2,0) as
in the figure. Discuss briefly the strategy you used to find your solution.
Suggestion: Let f (X)= A X^{2} +BX+C and g(X)=
P X^{2} +QX + R. Have the pair of curves meet at (0,0).

Find a second pair of quadratic polynomials which can be used to make a
single smooth curve that passes through the same four points.

Find a single cubic polynomial that passes through the same four points.

Use some technology to sketch your solutions for 1,2, and 3. Discuss briefly
any noticeable differences (such as "vertices") between the three curves.

Find a cubic and a quadratic polynomial that can be used to make a single
smooth curve that passes through the five points (2,0), (1,0), (0,0),
(1,2) and (2,0).

Read
Derivatives
and Chemistry
(Taken from the UPenn problems related to the reading)
A certain overall reaction in aqueous solution, has the form: