Math 110 Problems of the Week
(sometimes with hints, etc.)
Last updated: 11/15  (Work in Progress, subject to change)
 1 Darts! Due Friday 8-31-2011 2. Euler and Trig.Due 9-13 3. A review trip. Due:9-28 4 Some Integrals Due: 10-12 5. MacLaurin Polynomials and DE's. Due:  10-29 6. Comparing functions.Due : 11-27 7.Probability and More integration. Due: 8.A Very Flat Function  Due: Due: Derivatives and Chemistry Due: Estimating integrals Due: Fitting Curves Due:

1.  Due Friday 8-31-2011  Darts

1. Due 9-13    Euler's method and estimating trigonometric functions.

2. 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.

1. 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. [Underestimate? overestimate? why?]

2.
3. 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. [Underestimate? overestimate? why?]

4.
5. Use the previous estimates to estimate tan(1) = sin(1)/cos(1).

6.
7. 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. [Underestimate? overestimate? why?]
3. Due 9-28    A review trip.

4. 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.
1. Using Euler's method with n = 10, estimate f(1) and g(1).
2. Using the information in the assumption, show that [g(x)] 2 - [f(x)] 2 =1 for all x. [Hint: Consider P(x)=[g(x)] 2 - [f(x)] 2. Use the assumption to show that P(x) is a constant.]
3. Let `S(x) = {e^x-e^{-x}}/2`. Show S''(x) = S(x), S(0) = 0 and S'(0) = 1.
4. 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.
5. Due 10-12  Some Integrals .

a)   Find
`int {tan^{-1}(x)}/( 1+x^2 ) dx`.
b)   Find  `int x^3 sin(x) dx`.
c)   Find `int (x^3+3x^2+1)/( x^3+x^2+x+1 ) dx`.
d)   Find  `int_1^e  [ln(x)]^3 dx`.
6. Due 10-29  .  Differential Equations, MacLaurin Polynomials, and Euler's Method.
1. Suppose is a "`C^{oo}` " function with f (0) = 1,  f '(0) = -1 and f ''(x)= 2f '(x) + 3f (x).
1. Find the MacLaurin polynomial of degree 6 for f.[Hint: First find an equation for f '''(x).]
2. Estimate the value of f (1) using this polynomial.
3. Use Euler's method with n = 4 to estimate f (1).
1. Let `g(x) =  (1-x)^{1/2}= sqrt{1-x)` . Find the Maclaurin polynomial of degree 4 for `g`. Use your result to estimate `g(1/2)`. Discuss the error in this estimate.
2. ....
1. Find the MacLaurin polynomial of degree 6 for `tan^{-1}(x)`.
2. Use the polynomial from part i) to estimate `tan^{-1}(1/2)` and `tan^{-1}(1/3)`. Discuss the error in these estimates.

7. Due:11-27 .   Comparing Functions by integrating differences squares and the beginning of  "Bad" function
1. Find the value(s) for the coefficient m of a linear function  f(x) = mx to minimize the value of the definite integral,  `int_0^1 (mx-x^2)^2 dx`  . Explain your result.
2. Find the value(s) for the coefficient m of a linear function  f(x) = mx to minimize the value of the definite integral, `int_0^1 (mx-x^3)^2 dx` . Explain your result.
3. Let `f(x) =  e^(-1/{x^2} )`  when x is not 0  and `f(0)=0`.
a) Explain why` f` is continuous at ` x = 0`.
b) Find` f '(x)` when ` x` is not 0.
c) Find `f '(0)`. (Hint: Use the def'n of the derivative and L'Hospital's Rule.)
d) Find ` f ''(x)` when x is not 0.
e) Find `f ''(0)`.(Hint: Use the def'n of the derivative and L'Hospital's Rule.)
f) Sketch the graph of  `f `.
g) Show that ` f '''(0) = 0` .

Due:
Probability and more integration.
a)Find exactly or estimate the value of a > 0 so that f (x) = e - 2x  is a probability density function of a random variable on the interval [-a, a].
Determine the mean for this random variable.
b) Keeping things small by parts:
Let  `g(A) =  int_{-pi}^pi (f(x) - A sin(x) )^2 dx` .
1. Show that g takes on its smallest value when  `A = 1/pi  int_{-pi}^pi f(x) sin(x)  dx` .
2. Find the values of A that minimize g(A) in part i)
a) when f(x) = x and                    b) when f(x) = ex

Due: 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 trig(x) so that trig(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 .

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 `int_0^1 ({sin (x)} /x )^2 dx`  using the first three non-zero terms of the Maclaurin polynomial in ii). Explain how you handled the fact that this is an improper integral in your estimate.
Discuss the error in your estimate in part iii). [Use Taylor's theorem.]
Estimate the integral in part iii) using Simpson's rule. Explain how you handled the fact that this is an improper integral in your estimate.

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

The initial rate of this reaction was measured at 25° C, as a function of initial concentrations (in M) of A, B, and C. The data are as follows:

Trial [A]_0 [B]_0 [C]_0 Initial Rate (in M/s)
#1 0.02 0.02 0.02 1.414
#2 0.06 0.02 0.02 12.726
#3 0.06 0.06 0.02 4.242
#4 0.03 0.06 0.03 1.299
1. Deduce the experimental rate law. Find the order with respect to A, with respect to B, with respect to C, and the overall order.
2. Calculate the experimental rate constant, kexp, in the appropriate units.
3. If equal volumes of 0.100 M aqueous solutions of each reactant are mixed at t = 0, find the time that it takes B to decrease to a value of 2.57 x 10¯ 3; M. What percent of B has been consumed at this point?