1
DOE functions for R (version #4)
The file DOE_functions_v4.R contains a collection of R functions that simplify some of the
most common analyses in DOE, and especially in RSM:
Notation
The symbols <> are used to express the contents of an R object.For e
First comments on some things that appeared during the class.
1) Can you have legends on plot? Yes you can, and I added an example in Spectra.r that can
be found in the address given below. I also added an example of using mathematical
annotation in the l
The files that I created today (15.4.2016) can now be found in
http:/users.metropolia.fi/~velimt/Chemometrics/Classes/%234/
Your fourth assignment is to repeat the analyses we did today with the iris data using the wines
data. You may imitate the analyses
The files that I created today (8.4.2016) can now be found in
http:/users.metropolia.fi/~velimt/Chemometrics/Classes/%233/
Your third assignment is to repeat the modelling of Nt as a function of the NIR spectra using some
spectral pretreatment. Use the pr
MTH 1320
9.3 Rates of Change and Derivatives
1/9
For linear functions, we have seen that the slope, m, of the line measures the average rate of change of the function
y2 y1
and can be found from any to points on the line via m =
.
x2 x1
For a function tha
MTH 1320
10.1 Relative Maxima and Minima: Curve Sketching
1/7
Recall: The graph of a quadratic function, f (x) = ax2 + bx + c with a = 0, is a parabola. The graph has a turning
point located at its vertex, which is found via the vertex formula:
The vertex
MTH 1320
10.3 Optimization in Business and Economics
1/6
Denition: Absolute Extrema
The value f (a) is the absolute maximum of f if
for all x in the domain of f (or over
the interval of interest).
The value f (a) is the absolute minimum of f if
for all
MTH 1320
10.2 Concavity; Points of Inection
1/5
Denition: Concavity
A curve is said to be concave up on an interval (a, b) if at each point on the interval the curve is
its tangent at the point. If the curve is
all its tangents on a given interval, then i
MTH 1320
9.5 The Product and Quotient Rules
1/6
In the last section we discussed numerous dierentiation rules. We stated that unlike constant multiples, addition,
and subtraction, dierentiation does not play nice with multiplication and division. Again, c
MTH 1320
9.2 Continuous Functions; Limits at Innity
1/6
In the previous section, we found that if f (x) is a polynomial function, then lim f (x) = f (c) for all values of c. Any
xc
function for which this special property holds is called a continuous func
MTH 1320
12.2 The Power Rule
1/5
Using the only integration technique we have studied (the Powers of x Rule), can we evaluate the following integral?
(4x2 7)3 8x dx
Denition: Dierentials
If y = f (x) is a dierentiable function with dy/dx = f (x), then the
MTH 1320
9.1 Limits
1/9
In various applications we have seen the importance of the slope of a line as a rate of change. We were able to discuss
the slope of linear functions because their slope is constant; that is, the steepness of the line did not depen
MTH 1320
9.6 The Chain Rule and the Power Rule
1/7
Thus far we have only investigated how to take a derivative of fairly simple functions like those below.
x2 3
f (x) =
x+1
7
g(x) = 4x5 x3 3x2 + 1
2
y=
4x3 + 2x2 x 3
33x
But what if we wanted to take the
MTH 1320
11.1 Derivatives of Logarithmic Functions
Logarithms are concerned with powers (or exponents)!
Recall: Logarithm Basics
The logarithm denition: by = x y = logb x
The logarithmic function y = logb x has base b and domain (0, ).
The natural loga
MTH 1320
9.9 Applications: Marginals and Derivatives
1/5
Revenue and Marginal Revenue
If the demand function for a product is p = f (x) where p denotes the price per unit, then the total revenue
from the sale of x units is
.
If R = R(x) is the total rev
MTH 1320
12.1 Indenite Integrals
1/5
So far, we have studied procedures and applications of nding derivatives of a given function. We now turn our
attention to the reverse direction. That is, given a derivative f (x), how do we nd f (x)? This process is c
MTH 1320
11.2 Derivatives of Exponential Functions
Recall: Exponential Functions
The function y = bx is an exponential function with base b, where b > 0 and b = 1.
The natural exponential function y = ex has base e 2.71828.
Derivative of y = ex
If y = e
MTH 1320
9.4 Derivative Formulas
1/6
As one can imagine, nding the derivative of a function via the denition is going to become extremely tiresome.
Luckily, we do not always need to resort to such tedious endeavors! Consider the following derivatives:
If
MTH 1320
9.8 Higher-Order Derivatives
1/4
Because the derivative of a function is itself another function, one might ask how is the derivative changing? To
answer that question, one would just take a derivative of the derivative since derivatives tell us
MTH 1320
12.3 Integrals Involving Exponential and Logarithmic Functions
Exponential Formula
If u is a dierentiable function of x, then
In particular,
eu u dx =
ex dx =
Example 1: Evaluate each indenite integral.
(a)
2e4x1 dx
Your Turn 1: Evaluate each ind
12/6/16, 3:40 PM
Comp-XM Exam
Board Query 1
Points: 10
You have answered 1 out of 10
questions. Select your next
question or close the Quiz
when you are done.
2006 Capsim Management Simulations, Inc.
1
10
Investing $2,000,000 in TQM's Channel Support Syst
Ashley Gazewood Math 1600 MW 2:30
Cap #2
Problem Statement: How to find the area of various geometric figures, using the same formula Area = Base times Height. Describing how these formulas were developed and what do they mean? Activity Log: For thi