The force of interest varies as a linear function from t = 0 until t = 5 and as a quadratic function From t = 5 until t =10 (t in years) and is continuous at all time points.

Suppose the force of interest is 2% p.a. at t = 0, 4.5% p.a. at t =5, 8% p.a. at t = 8 and 10% p.a. at t = 10.

a) Determine the functional form of the force of interest over the 10 year period

b) A company receives a cash-flow from t=1 until t=5 at the rate of (100+25t) per unit time.

The company pays out a cash flow from t = 6 until t = 8 at the rate of (-t^2 +48t - 80 )per unit time.  Calculate the net present value of the company’s cash flow position at t = 0

Investigate the following theorems

(h) For sets A, B and C we have

i. A\(B ∪ C) = (A\B) ∩ (A\C),

ii. A\(B ∩ C) = (A\B) ∪ (A\C),

iii. A ̸= B if and only if (A\B) ∪ (B\A) ̸= ∅,

iv. A ∪ B ⊆ C if and only if A ⊆ C and B ⊆ C.

What happens in the extreme case(s) where some (or all) sets are empty?

a) Prove that n 3 − 91n 2 − 7n − 14 = Ω(n 3 ). Your answer must clearly specify the constants c and n0.

b) Let g(n) = 27n 2 + 18n and let f(n) = 0.5n 2 − 100. Find positive constants n0, c1 and c2 such that c1f(n) ≤ g(n) ≤ c2f(n) for all n ≥ n0. Be sure to explain how you arrived at the constants.

An object of mass 2 kg is released from rest from a platform 30 m above the water and allowed to fall under the influence of gravity. After the object strikes the water, it begins to sink with gravity pulling down and a buoyancy force pushing up. Assume that the force due to gravity (g = 9.81 m/s2) is constant, no change in momentum occurs on impact with the water, the buoyancy force is 1/2 the weight (weight=mg), and the force due to air resistance is proportional to the velocity, with proportionality constant b1 = 10 N-sec/m in the air and b2 = 100 N-sec/m in the water. Find the equation of motion of the object. What is the velocity of the object 1 min after it is released?

Please describe how you have prepared for your intended major, including your readiness to succeed in your upper-division courses once you enroll at the university.

I'm major at Math

Let A be an infinite set and let B ⊆ A be a subset. Prove:

(a) Assume A has a denumerable subset, show that A is equivalent to a proper subset of A.

(b) Show that if A is denumerable and B is infinite then B is equivalent to A.

Residents were surveyed in order to determine which flowers to plant in the new Public Garden. A total of N people participated in the survey. Exactly 9/14 of those surveyed said that the colour of the flower was important. Exactly 7/12 of those surveyed said that the smell of the flower was important. In total, 753 people said that both the colour and smell were important. How many possible values are there for N?

Please explain clearly.

How to find the unit vectors for the following equation: r(t) = <e^t,2e^-t,2t>

A) Compute the unit Tangent Vector, unit Normal Vector, and unit Binomial Vector.

B) Find a formula for k, the curvature.

C) Find the normal and osculating planes at t=0

Use the preliminary test to decide whether the following series are divergent or require further testing. Sum (3+2i)^n/n!

2. For each 3*3 matrix and each eigenvalue below construct a basis for the eigenspace Eλ.

A= (9 42 -30 -4 -25 20 -4 -28 23), λ = 1,3

A= (2 -27 18 0 -7 6 0 -9 8), λ = −1,2

3. Construct a 2×2 matrix with eigenvectors(4 3) and (−3 −2) with eigen-values 2 and −3, respectively.

1. Let f (t) be the amount of yeast (in grams) in a culture of yeast at time t (in minutes), and let g (t) represent the rate at which the culture of yeast is growing (in grams/min) at time t (in minutes).

   1. (a) Write an expression in terms of f (t) for the amount the culture grew (in grams) from 25 to 30 minutes.

   2. (b) Write an expression in terms of g (t) for the amount the culture grew (in grams) from 25 to 30 minutes.

   3. (c) Write an expression in terms of f (t) for the average amount of yeast in the culture (in grams) from 25 to 30 minutes.

   4. (d) Write an expression in terms of f (t) for the average rate at which the culture of yeast grew (in grams/minute) from 25 to 30 minutes.

   5. (e) Write an expression in terms of g (t) for the average rate at which the culture of yeast grew (in grams/minute) from 25 to 30 minutes.

   6. (f) Suppose at 20 minutes the amount of yeast in the culture was 100 grams. Write an expression using g (t) representing the amount of yeast in the culture at 30 minutes.

((sqrtx)+1)dy/dx=(ysqrtx)/(x) + sqrt(y/x), y(2)=1

give the compound nucleus resulting from protons bombarding an aluminum target