Expert: Alon Mandes - 7/17/2009

Question
QUESTION: Hi, I'm having some trouble for this question, can you explain why I was wrong?

Q: Find an equation for the line tangent to the curve at the point defined by the given value of t. Then, find the value of d^2 y/dx^2.
Given the parametrized equations:
x = 2 cos t
y = 2 sin t
t = pi/4

Here is what I did:

X = 2 cos (pi/4) = sqrt 2 (I'm gonna denote it as #2)
Y = 2 sin (pi/4) = #2

Dy/dx = (dy/dt)/(dx/dt) = 2 cos t / -2 sin t = - cot t;

At t = pi/4, dy/dx = -cot (pi/4) = -(1/1) = -1, which is the slope of the tangent line.

Substituting x and y, y - #2 = -1 (x - #2)   y = -x + 2#2, the line of tangency

Then, working to d^2 y/dx^2, which is the derivative of –cot t, or csc^2 t.

Substituting pi/4 for t, csc^2 (pi/4) = 2.


However, the answer to the question is: y = -x + 2 and d^2 y/dx^2 is -#2.

ANSWER: The tangent line at point (x,y)=(#2,#2) can never intersect with
the axis in pint x=2 or y=2. Because the parametrized curve denote
a conic circle with radius 2. So your equation is right.
As for the 2nd derivative, you may chick your answer by deriving 2nd
time the equation y=SQRT[4-x^2] which also denote the same
parametrized curve .

Alon.

---------- FOLLOW-UP ----------

QUESTION: So, can you tell me what exactly went wrong in my equations for the 2nd derivative?

thanks

Answer
y=SQRT[4-x^2]=[4-x^2]^(1/2)
y'=(1/2)*(-2x)[4-x^2]^(-1/2)=(-x)[4-x^2]^(-1/2)
y''=(-1)[4-x^2]^(-1/2)+(-x)(-2x)(-1/2)[4-x^2]^(-3/2)
y''=-1/#[4-x^2] + (-x^2)/(#[4-x^2])^3
y''=[-(4-x^2)-x^2] / (#[4-x^2])^3
y''=-4/(#[4-x^2])^3
y''(#2)=-4/[#(4-2)]^3=-4/(#2*#2*#2)=-4/2#2=-2/#2=-#2 .

If we consider the approach, of yours :
dy/dx=[dy/dt]/[dx/dt]. I'll denote y'=dy/dt & x'=dx/dt. Thus,
dy/dx=y'/x' . Therefore :
d^2y/dx^2=[y''x'-x''y']/(x')^2
IF
y=2sint -> y'=2cost -> y''=-2sint
x=2cost -> x'=-2sint -> x''=-2cost
Then
d^2y/dx^2=[(-2sint)(-2sint)-(-2cost)(2cost)]/(-2sint)^2
d^2y/dx^2=[4(sint)^2+4(cost)^2]/(-2sint)^2
d^2y/dx^2=4[(sint)^2+(cost)^2]/4(sint)^2
d^2y/dx^2=1/(sint)^2
d^2y/dx^2 At point (t=pi/4) = 2 !!
To tell you the truth that also confuses me ! Anyway, I'll
look into it more carefully & vastly, & I will let you know .
Regards,
Alon.