For what value of t is the slope of the curve undefined for the graph defined by x = 10 – t2, y = t3 – 12t?
Type your answer in the space below to the nearest whole number. (ex: 4)
When I write -t2 and t3, I mean -t^2 and t^3 by the way.
did this one already
dy/dx is undefined for x=0
I am sorry but I don't get it
Please can you help me with this one because I don't understand at all
To find the value of t for which the slope of the curve is undefined, we need to determine when the derivative of y with respect to x, or dy/dx, becomes undefined.
First, let's find dy/dx by differentiating both equations with respect to t and then expressing dy/dx in terms of t:
x = 10 – t^2,
Differentiating with respect to t: dx/dt = -2t.
Solving for dt/dx: dt/dx = -1 / (2t).
y = t^3 – 12t,
Differentiating with respect to t: dy/dt = 3t^2 – 12.
Solving for dt/dy: dt/dy = 1 / (3t^2 – 12).
Now, we'll set dt/dx equal to dt/dy and solve for t:
-1 / (2t) = 1 / (3t^2 – 12).
Cross-multiplying to get rid of fractions:
-(2t) = (3t^2 – 12).
Rearranging the equation:
3t^2 – 2t – 12 = 0.
To solve this quadratic equation, we can factor or use the quadratic formula. Factoring may not work in this case because the factors of 12 (2 and 6) don't combine to give us a middle term of 2. So, let's use the quadratic formula:
t = (-(-2) ± √((-2)^2 - 4(3)(-12))) / (2(3)).
Simplifying:
t = (2 ± √(4 + 144)) / 6,
t = (2 ± √148) / 6,
t = (2 ± 2√37) / 6.
We have two possible values for t: (2 + 2√37) / 6 and (2 - 2√37) / 6.
Now, let's evaluate those values to the nearest whole number:
(2 + 2√37) / 6 ≈ 1.63,
(2 - 2√37) / 6 ≈ -1.96.
Since we are looking for the nearest whole number, the value of t for which the slope of the curve is undefined is approximately -2.