Other Math Answers

Let f : A → B be a function. Show that the following two conditions are equivalent: 1. f is one-to-one.

2. For each a1, a2 ∈ A, whenever f(a1) = f(a2), then a1 = a2.

Let f, g : A → A be functions with the same domain and codomain A.

1. Give a brief justification of why g ◦f, f ◦g : A → A both have the same domain and codomain A.

2. Either give a proof that g◦f and f ◦g are equal or show that they are not equal by constructing a

Let f : A → B, g : B → C, and h : C → D be functions.

1. State what you need to show to conclude that h ◦ (g ◦ f) = (h ◦ g) ◦ f. 13

2. Consider now some a ∈ A. Calculate h((g ◦ f)(a)) and (h ◦ g)(f(a)). Are they equal?

3. Use your solutions to (1)–(2) to conclude that h ◦ (g ◦ f) = (h ◦ g) ◦ f.

Sketch the graph of the functions:

1. f(x) = 5

2. h(x) = 3x + 2

3. q(x) = x2 + 6x -7

4. k(x) =

5. h ₒ g = 2x3 + x2 – 10x - 5

6. g ₒ f = (2x + 1)(x – 3) = 2x2 – 5x -3

Take the first number as 0.00000000000003234 and assume any second number to demonstrate the concept of overflow or underflow for the given representation. (You may assume any second number to demonstrate overflow or underflow).

What is use of bias in binary floating point representation.

Explain the concept of bias with the help of an example for

binary floating point numbers

Numerically solve the problem of the propagation of a pulse in a non-homogeneous string. One half of the string has mass density "\\mu"₁ and the other half a mass density "\\mu"₂ . The pulse is incident on the interface from one side. Choose different relative values of the two densities.

Carry out the integrals

x(t)= "\\intop"₀^t (cos("\\pi"u²/2)du) ; y(t)= "\\intop"₀^t (sin("\\pi"u²/2)du

for a large number of values of t in the interval [0, N], where N is a large number (20, say, which you can change) and plot the values on a x-y plane. Get the beautiful Cornu spiral, which you can get on the internet. Use both Simpson rule and Gaussian quadrature.

Use Runge-Kutta(4th order) method to solve the differential equation for a forced, damped SHO given by

dx²/dt² + "\\gamma"dx/dt + "\\omega"₀²x = Acos"\\omega"t

Choose appropriate values for the constants. By changing the value of "\\omega" in comparison with "\\omega"₀ , demonstrate resonance. The initial conditions could be x(0)=0, "\\dot{x}"(0)=0. Plot the displacement as a function of time.