1 Darts! Due Thursday 2-2-2011 | 2. Euler and Trig.Due:2-9-2012 | 3. A review trip. Due:2-23-2012 |
4 Some Integrals Due: 3-8-2012 |
5. MacLaurin Polynomials and
DE's. Due: 3-29-2012 |
6. Comparing functions and polynomials.Due 4-30-2012 modified 4-24 |
7.Probability and More integration. Due: | 8.A Very Flat Function Due: | Due: | Derivatives and Chemistry Due: |
Estimating
integrals Due: |
Fitting Curves Due: |
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.
(a) S(x). (b) S'(x).
In your graphs show and explain such features as extrema, concavity, symmetry, etc.
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 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 .
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 |