Brandy Williams
MATH 2214 D1 25
Doyle, Duane
January 20, 2016
Problem Set
1. Find each of the following in exact form. Do not approximate.
a. tan1 (1) = b. sin1 (sin(
3
4
) =
c. cosh (0) = 1
2. Find the derivatives for the following.
a. = sin1 (1 )
()
b.
6
Honors Calculus I
Assignment #3
Due before 3 pm. on September 18
Limit Laws
0 1. If limm(x) exists and lim n(x) exists, then the limit laws state that both lim[m(x)+ n(x)]
.\'>a .\?L/
and lim[m(x)n(x)] exist. Suppose instead that limm(x) does not exist
1/
Honors Calculus |
Assignment #8
Due before 3 pm. on October 30
GA 7
Pharmacokinetics
Pharmacokinetics describes the process by which drugs are assimilated by the body. Ifwe
assume that an entire dose is immediately absorbed into the blood, the eliminat
Honors Calculus |
Assignment #6
Due beforeB pm. on October 1%
1. Unbroken Chain
Differentiate f(x) = (x + (x + (x + x3)4)5)6.
7/ C
Consider the equation x3 + y3 = 6xy.
[1/
2. Implicit Differentiation
y/ a. Use the ContourPlot command in Mathematica to pl
Honors Calculus l 7 t, (x lg;
Assignment #9 ' "
Due before 3 pm. on November 13
Applying Concepts
(You may include sketches to support your conclusions.)
1. The goal of investing in the stock market is to buy low and sell high. But, how can you tell
wheth
Honors Calculus | \V"
Assignment #2
Due before 3 pm. on September 5 7 f
Inverse Functions I
1. Show that the following function is its own inverse. What does this result tell you about the graph off?
4/ f(x)=3_x
1x
\ 2. Let f(x) = 2x3 + 5x + 3. Find x i
Honors Calculus |
Assignment #5 x" .,
Due before 3 pm. on October 2 _ it!
3 , , .
1. Help! .A":
I need the eighteenth derivative of the very simple looking function f(x) = x. I do not have the time (or
patience!) to repeat the power rule, eighteen times.
\ /B
Honors Calculus l k )ng
Assignment #4 [f .3
Due before 3 pm. on September 25 " "3; /
Velocity
1. The figure below shows the position versus time curves of four different particles moving
\ along a straight line. For each particle, determine whether
.K \
-4
r?
"7
Honors Calculus | .J
Assignment #7
Due before 3 pm. on October 23
1. The Right Procedure at the Right Time
Mathematics problems are like jigsaw puzzles because you must have the correct pieces
(definitions, theorems, procedures, etc.) _an_d
1 Taylor Polynomials
Dr. Jeongho Ahn
Department of Mathematics & Statistics
ASU
Dr. Jeongho Ahn
Jeongho.ahn@csm.astate.edu
Outline of Chapter 1
1
The Taylor Polynomial
2
The Error in Taylors Polynomials
3
Polynomial Evaluation
Dr. Jeongho Ahn
Jeongho.ahn@
3.2 Newtons Method
This method (called the Newton-Raphson Method) converges
much faster than the bisection method does.
In Newtons method, we assume that f C 1 and f (xn ) = 0
for any n 0.
Newtons method will be obtained, based on the linear
approximation
3.3 Secant Method
Secant Method replaces the tangent line with an
approximation using the secant line.
Recall the Newtons method formula:
f (xn )
xn+1 = xn
for n = 0, 1, 2, 3,
f (xn )
Then approximating f (xn ) = (f (xn ) f (xn1 ) / (xn xn1 )
provides t
3 Solving Non-linear Equations
Dr. Jeongho Ahn
Department of Mathematics & Statistics
ASU
Dr. Jeongho Ahn
Jeongho.ahn@mathstat.astate.edu
Outline of Chapter 3
1
The Bisection Method
2
Newtons Method
3
Secant Method
4
Fixed Point Iteration
5
Ill-Behaving R
2 Error and Computer Arithmetic
Dr. Jeongho Ahn
Department of Mathematics & Statistics
ASU
Dr. Jeongho Ahn
Jeongho.ahn@mathstat.astate.edu
Outline of Chapter 2
1
Floating-Point Numebers
2
Errors: Denitions, Sources, and Examples
3
4
Propagation of Error
S
1.2 The Error in Taylor's Polynomial
When we use the Taylor polynomial to approximate certain
functions, we need to consider its accuracy.
Theorem
(Taylor's remainder) Assume that f
interval
f
[ , ]
and a
[ , ].
(x ) pn (x ) =
is a smooth function on an
1.3 Polynomial Evaluation
Consider polynomials
p ( x ) = a0 + a1 x + a2 x 2 + + an x n
which you need to evaluate for many values of x .
How do we evaluate the polynomials eciently?
We use Nested Multiplication (or Horners method) to
save the computationa
3.4 FPI (Fixed Point Iteration)
Consider the nonlinear equation
x cos x = 0.
Then you may use the bisection method or Newtons
method(secant method) to nd a (approximate) solution of
the equation.
However, the xed point equation cos x = x which is
equivale
4 Interpolation and Approximation
Dr. Jeongho Ahn
Department of Mathematics & Statistics
ASU
Dr. Jeongho Ahn
Jeongho.ahn@mathstat.astate.edu
Outline of Chapter 4
1
Polynomial Interpolation
2
Error in Polynomial Interpolation
3
Interpolation Using Spline F
4.6 A Near-Minimax Approximation Method
Remember that a uniform spaced set of interpolation nodes on
the some interval provides a very poor approximation to the
Runge function f (x ) = 1/ x 2 + 1 . In order to improve the
situation, we will use the minima
4.7 Least Squares Approximation
In the previous section, we considered a near minimax
polynomial approximation based on suitable chosen nodes.
Another approach is to nd an approximation p (x ) with a
small average error over the interval [a, b]. A conveni
4.5 Chebyshev Polynomials
The choice of partitions may have a signicant eect on the
interpolation error.
The Chebyshev Interpolation provides a particular optimal way
to partition the interval.
The Chebyshev Interpolation can be used to understand a
near-
4.4 The best approximation problem
In this section we consider a best possible approximation. For
example, how do we improve Taylor approximations
(polynomials)?
Let f (x ) be a given function that is continuous on a given
interval [a, b]. Then we conside