Finance (Quantitative Tech)

Q1. Derive the equation of a straight line that passes through (-2, 3) and (1, -5).

Answer:

(x₁, y₁) = (−2, 3) and (x₂, y₂) = (1, -5), so

a₁ = y₂ - y₁x₂ - x₁

= -5-31-(-2)

= - 83

a₀ = y₁ − a₁ x₁

= 3 – ( - 83 ) (-2)

= 3 – 163

= - 73

y = - 73 - 83 x

3y = -7 – 8x or 3y = -8x – 7

Q2. Where are the axis of symmetry, x-intercept and y-intercept for y = 4x² + 2x – 1.

Answer:

y = 4x² + 2x – 1

a₀ = -1

a₁ = 2

a₂ = 4, so for the axis of symmetry:

x = -a₁2a₂

= -22(4)

= - 28

d = a₁² - 4 a₂ a₀

= (2)² - 4 (4) (-1)

= 20, there are 2 intercepts.

The x-intercepts are the solution to 0 = 4x² + 2x – 1, which can be solved by the quadratic equation:

x = -a₁ ± a₁² - 4 a₂ a₀2a₂

= -2 ± 2² - 4 (4) (-1)2(4)

= -2 ± 208

= - 0.809 or 0.309

The y-intercept are the solution to y = a₂ x² + a₁ x + a₀,

x = 0

y = a₀

y = 2(0)² - 3(0) + 1

= 1

y-intercept is 1.

Q3. Find the first and second derivatives for

a. y = 3x + 1.

b. y = log 3.

c. y = log x² - x² + 3 exp (4x²).

Answer:

a. y = 3x + 1

yʹ = 3

yʺ = 0

b. y = log 3

yʹ = 0

yʺ = 0

c. y = log x² - x² + 3 exp (4x²)

yʹ = 2xx² - 2x + 24x exp 4x²

= 2x - 2x + 24x exp 4x²

yʺ = - 2x-2 - 2 + 192x² exp 4x²

Q4. Find the equation for the tangent line at x = 1 for y = 1/x.

Answer:

In this case, f(x) = 1/x, x₀ = 1 and y₀ =1/ x₀ = 1. Moreover, f ʹ(x) = - 1x² which equal to -1 when x = 1.

Moreover, the tangent must pass through the point of interest which is (1, 1). So, the tangent line is

y - y₀ = f ʹ(x₀)(x - x₀)

y – 1 = - 1 (x – 1)

y = - x + 1 + 1

y = - x + 2

Q5. Find all the first and second order partial derivatives for

y = Ax1α x2β, A > 0, 0 < α + β ≤ 1.

Demonstrate that the Young’s theorem holds in this case. Using total differential, show how much would y change if x₁ changes from 1 to 1.01 and x₂ changes from 1 to 0.99 simultaneously?...