consider a finite rectangle in the plane. we will draw some number of lines that cut through the rectangle. In section 28.7.6 we define what we mean when we say that a map can be colored using two colors. Treat the rectangle that we just drew as a map, with regions defined by the lines that cut through it. Use induction to prove that, no matter how many lines we draw, the rectangle can be colored using two colors.
In: Advanced Math
Consider the following rational function:
f(x) = 18 (x-1)/(x^2 - 9)
Which of the following are true?
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A. The function touches but does not cross the horizontal axis at: x= 1. |
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B. When x is very large and positive, the value of the f(x) approaches to zero. |
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C. The function cuddles up to the vertical lines which pass through x = - 3 and x =3. |
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D. The function cuts the horizontal axis at x = 3 and x = -3. |
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E. The vertical intercept of the function is (0,2) and the horizontal intercept is (1, 0). |
In: Advanced Math
Euler’s Method and Introduction to MATLAB
>> 2+2
If you want to suppress the output of a command follow it with ‘;”. For example >>2+2;
Practice evaluating a few expressions in the command window. (In MATLAB multiplication is represented by * so 3*2=6).
Variables can be vectors. For example >>x=[1 2 3] creates a row vector with three components: x(1)=1, x(2)=2, and x(3)=3. Try it. What happens when you set x=[1;2;3]?
Now that you have created the function g(x,y) you can evaluate it at different x and y values. For example, to compute g(1,2), just type: >>g(1,2) and hit enter. Also compute g(2,2). Why is g(1,2)=g(2,2)?
When you open the file euler_method.m you will see that the lines of the function are numbered along the left hand side of the window. Look at the first line:
function [x,y] = euler_method(f,h,x0,y0,xn).
This line creates a function called euler_method in MATLAB.
The variables in square brackets are the outputs of the function. These variables are returned to the command window after the function is called. This function returns variables called x and y. The variables in round brackets are the inputs of the function. This function has five inputs. What are they?
The % symbol is used to comment out text. This means that whatever appears after a % sign is not executed as part of the code. Explanations are placed after a % symbol. Read the comments that describe how the function euler_method.m works.
Note: MATLAB updates the value of the index i with each iteration of the loop, so the command i=i+1 need not appear inside the loop.
In the command window type [x,y]=euler_method(g,h,0,1,1) to solve the initial value problem, dy/dx=g(x,y); y(0)=1, for h=.25, .125, .0625, and .03125. Each time you run euler_method, a plot showing the approximate solution as a function of x will be produced. Edit these plots to include axes labels and titles, and insert them here.
In: Advanced Math
1. prove that if{xn} is decreasing an bounded from below, then {xn} is convergent.
In: Advanced Math
Suppose that x is real number. Prove that x+1/x =2 if and only if x=1.
Prove that there does not exist a smallest positive real number. Is the result still true if we replace ”real number” with ”integer”?
Suppose that x is a real number. Use either proof by contrapositive or proof by contradiction to show that x3 + 5x = 0 implies that x = 0.
In: Advanced Math
Significant figures and unit conversion are important to scientist. Please elaborate/expand on why these skills are important. Examples?
In: Advanced Math
I need an in-depth explanation for quaternion geometry in game developing?
In: Advanced Math
Please find the question at the following link:
bit.ly/2s4jgE5
Thanks.
In: Advanced Math
In the Dihedral Group D4, determine all left cosets and right cosets of <pt>
(the t is tau and the p is rho)
In: Advanced Math
Determine the Pade approximation of degree 6 for f(x)=sinx, and compare the results at xi = 0.2i for i = 0,1,2,3,4,5 , with f(x) and with its sixth Maclaurin polynomial wit,h
n=2 and m=4
n=3 and m=3
n=4 and m=2
Please do this USING MATLAB. Thanks
...USING MATLAB?
In: Advanced Math
Prove that τ(n) < 2 n for any positive integer n. This is a question in Number theory
In: Advanced Math
Let τ ∈ Sn be the cycle (1, 2, . . . , k) ∈ Sn where k ≤ n.
(a) For σ ∈ Sn, prove that στσ-1 = (σ(1), σ(2), . . . , σ(k)).
(b) Let ρ be any cycle of length k in Sn. Prove that there exists an element σ ∈ Sn so that στσ-1 = ρ.
In: Advanced Math
In: Advanced Math
Find all of the inversions inversion taking (3, −5) to (3,5).
Give all of the possible centers and their
corresponding radii.
In: Advanced Math