Math Answers

The present value of a perpetuity, with the first cash flow paid in 4 years time, is equivalent to receiving $100,000 in 15 years time. The perpetuity and the lump sum are of equivalent risk and have a required rate of return of 10%. What is the annual cash flow associated with the perpetuity?

Given triangle ABC has vertices at (-2,4),(-2,-4) and (0,-2), respectively, find the circumcenter of the triangle.

The number of letters in the language of a weird country is 5 and no one in that
country uses more than 3 letters to make a word. What is the highest number of
words one can make in that language?

Solve the following ODE by using the Laplace transform: y'(x) - 4y(x) = 0, y(0) = 1

how do i write a slope intercerpt form of a equaton?

1. Find the indicated derivatives:
a) u =
x+y
y+z
, x = p + r + t, y = p − r + t, z = p + r − t;
∂u
∂r .
b) y
5 + x
2
y
3 = 1 + yex
2
;
dy
dx .
c) ln(x + yz) = 1 + xy2
z
3
; ∂z/∂y
2. Let f(x, y) = x
2
y + x
3
y
2 and suppose you dont know what φ(t) = (x(t), y(t)) is, but you
know φ(2) = (1, 1), dx
dt (2) = 3, and dy
dt (2) = 1. Find the derivative of f(φ(t)) when t = 2.
3. Show that the following functions are functionally dependent and find a relation connect￾ing them:
f(x, y, z) = x + y + z, g(x, y, z) = x
2 + y
2 + z
2
, h(x, y, z) = xy + yz + xz
4. Find the local maxima, minima, and saddles of the functions h(x, y) = (2x−x
2
)(2y −y
2
).
5. Find the largest volume of a box with an open top, and surface area 100m2
.
6. Find the absolute minimum of f(x, y) = x
2 + y
2 + 2y − 1 on D = {(x, y)|x
2 +
y
2
4 ≤ 1}

Q. Compute the torsion of the following curves
(i) γ(t)=4/5cos t, 1-sin⁡t,(-3)/5cos t
(ii) γ(t)=(t, cosht)
(iii) γ(t)=4/5 (〖cos〗^3t,〖sin〗^3t)

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Q. Compute the torsion of the following curves
(i) γ(t)=4/5cos t, 1-sin⁡t,(-3)/5cos t
(ii) γ(t)=(t, cosht)
(iii) γ(t)=4/5 (〖cos〗^3t,〖sin〗^3t)
For the astroid in (iii), show that the curvature tends to ∞ as we approach one of the points (±1,0), (0,±1)
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Q.Torsion is defined only when K(S)≠0 (why?)

Q. Show that if curvature K(t) of a regular curve γ(t) is >0 every where, then k(t) is a smooth function of t. Give an example to show that this may not be the case without the assumption that k>0.