Calculus question

Question

I missed my class today, so I'm not sure on how to do these 2 questions that are on the homework.

Find the equation of the tangent to the curve at x = 3 for the parametric equations below:

 , with t > 0.

For this one, I got y=6x-11.

 

Find  for the curve given by x = t² and y = t³.

For this one, I'm not sure at all, but I somehow ended with 4t/3.

I don't know if either one is right. Could someone please help?

Answer 1

dy/dx = (dy/dt)/(dx/dt) = (2t - 2/t³)/(1 - 1/t²) = 2(t⁴ - 1)/(t² - 1)t = 2(t² + 1)/t= 2(t + 1/t). Slope m = 2x = 6;

t + 1/t = 3. If we square this we get : t² + 2 + 1/t² = 9; so y = t² + 1/t² = 9 - 2 = 7.

Equation of tangent line: y - 7 = 6(x - 3); y = 6x - 11;

dy/dx = 2x; d²y/dx² = d/dx(dy/dx) = d/dx(2x) = 2

Answer 2

It's been a while since I looked at parametric equations, but here we go. For the first problem, if we differentiate x with respect to t, dx/dt = 1-(1/t^2). similarly dy/dt = 2t-2/t^3 by the power rule. To find dy/dx we divide dy/dt by dy/dx but it would be simpler to come up with a value for M (the slope of the tangent line) if we first knew what t was. Given in the problem x = 3, we can solve the first equation to determine that t (for x=3) must be 2.61795 (use quadratic formula for this part). So at that t, we can determine y to be 2.999928 by plugging t into the y equation. We now have a point (3, 2.999928).

so using this t = 2.61795, we can determine dy/dt = 2*(2.61795) - 2/(2.61795)^3 or dy/dt at t=2.61795 = 5.12443. Similarly, we determine dx/dt at the same t or dx/dt = 1-1/(2.61795)^2 = 0.854093. Now divide the two numbers and you get M = 6. Now use the point slope formula (you have a point and slope) to determine equation of the line.

For the second problem, let's solve for t in both equations. T is just t - think of it as time. It tells you were the point is at any given time. if x = t² then t = sqrt(x). Similarly t = y^(1/3). So since t is t,

x^(1/2) = y^(1/3). Cube both sides of equation

y = x^(3/2)

Therefore by the power rule

dy/dx = 3/2 * x^(1/2) then differentiate again to solve

= 3/(4*sqrt(x))