The last question please
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)
Since dy/dx = (dy/dt)/(dx/dt)
dy/dx will be undefined if dx/dt = 0 and dy/dt is not zero.
dy/dx = (3t^2-12)/(-2t)
we must exclude t=0
So you mean the answer is "0"?
To find the value of t for which the slope of the curve is undefined, we need to find the value of t when the derivative of y with respect to x is undefined.
Given x = 10 - t^2 and y = t^3 - 12t, we can find the derivative of y with respect to x by using the chain rule:
dy/dx = (dy/dt)/(dx/dt)
To find dx/dt, we can differentiate x = 10 - t^2 with respect to t:
dx/dt = -2t
To find dy/dt, we can differentiate y = t^3 - 12t with respect to t:
dy/dt = 3t^2 - 12
Now, we can find dy/dx by dividing dy/dt by dx/dt:
dy/dx = (3t^2 - 12)/(-2t)
The slope of the curve is undefined when the denominator of this expression, -2t, equals zero. So, we need to solve for t when -2t = 0:
-2t = 0
Dividing both sides by -2 gives us:
t = 0
Therefore, the value of t for which the slope of the curve is undefined is 0 (to the nearest whole number).
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 (dy/dx) is undefined.
First, let's express y in terms of x by substituting the first equation into the second equation:
x = 10 – t^2
=> t^2 = 10 - x
=> t = ±√(10 - x)
Now that we have t in terms of x, we can differentiate y with respect to x.
dy/dx = d(t^3 - 12t)/dx
Using the chain rule, we can express the derivative as:
dy/dx = d(t^3 - 12t)/dt * dt/dx
To find dt/dx, we can differentiate x = 10 – t^2 with respect to t:
dx/dt = -2t
=> dt/dx = 1/(dx/dt)
=> dt/dx = 1/(-2t)
=> dt/dx = -1/(2t)
Substituting dt/dx into the expression for dy/dx, we have:
dy/dx = (d(t^3 - 12t)/dt) * (-1/(2t))
=> dy/dx = (3t^2 - 12) * (-1/(2t))
=> dy/dx = (12 - 3t^2) / (2t)
For the slope to be undefined, the denominator must be equal to zero. Hence, we set 2t = 0 and solve for t:
2t = 0
=> t = 0
Therefore, the slope of the curve is undefined for t = 0.