Line Of Credit Calc

My teacher says that I can use a calculator, and I think that I have to use implicit differentiation for the first part.

6. Consider the curve defined by -8x^2 + 5xy + y^3 = -149

a) Find dy/dx

b) Write the equation for the line tangent to the curve at the point (4,-1)

c) There is a number k so that the point (4.2, k) is on the curve. Using the tangent line found in part b, approximate the value of k

d) Write an equation that can be solved to find the actual value of k so that the point (4.2, k) is on the curve.

e) Solve the equation found in part d for the value of k.

a)

F(x,y) = -8x^2 + 5xy + y^3 +149

The curve is defined by

F(x,y) = 0

Then

dy/dx = -(∂F/∂x)/(∂F/∂y)=
=-(-16x+5y)/(5x+3y^2)

b)
F(4,-1) = -8X16+5X4X(-1)+(-1)^3+149=
=-128-20-1+149=0,
then (4,-1) is a point of the curve.

The slope of the curve at this point is

dy/dx = -(-16X4+5X(-1))/(5X4+3(-1)^2)=
= -(-64-5)/(20+3)=
= 69/23 = 3

The equation of the tangent is

y-(-1) = 3 (x-4)

or

y = 3x - 13

c)
k ≈ 3X4.2 - 13 = 12.6 - 13 = -0.4

d)
F(4.2, k) = 0 is the equation to be solved. Explicitly

-141.12 + 21 k + k^3 + 149 = 0
or

k^3 + 21 k + 7.88 = 0

e)

We could use Cardano's formula, or a numerical calculation. For example, Newton:

k_(n+1) = k_n - f(k)/f´(k)

being

f(k) = k^3 + 21 k + 7.88
f'(k) = 3k + 21

Starting with k_0 = -0.4 (our best guess by now) we obtain

k_1 =-0.37281191806331
k_2 = -0.37277143926978
k_3 = -0.37277143918422
k_4 = -0.37277143918422

The last two values are equal (up to the figures showed) and can be considered the solution...

Cardano:

k = u+v

then

(u+v)^3 + 21(u+v) + 7.88 = 0

or

(u^3+v^3+7.88) + (3uv+21)(u+v)=0

we search for u and v which satisfy

u^3+v^3+7.88=0
3uv+21

The second equation can be written
u^3 v^3 = -343

Then v^3 = -343/u^3

Replacing into the first one

u^3 - 343/u^3 + 7.88 = 0

or

p^2 + 7.88 p - 343 = 0

where p = u^3

As the system for u and v is symmetric, the solutions of the quadratic ecuations will be u^3 and v^3

u^3 = 3.94 + sqrt(3.94^2 + 343) = 14.99471943283026
v^3 = 3.94 - sqrt(3.94^2 + 343) = -22.87471943283027

Then
u_n = 2.83869407958117 * exp(i2πn/3)
v_n = -2.46592264039695 * exp(-i2πn/3)

For n=0

k_0 = 0.37277143918422

is the value we are searching for. The other two are complex

k_1 = 0.18638571959211 + 4.59393283684074i
k_2 = 0.18638571959211 - 4.59393283684074i

2009-12-15 15:26:35 by Rob

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