For what value of t is the slope of the curve undefined for the graph defined by x = 10 – t^2, y = t^3 – 12t?
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How many times we gonna do this?
x = 10 – t^2, y = t^3 – 12t
dy/dx = (dy/dt)/(dx/dt) = (3t^2-12)/(-2t)
As with all rational functions, the value is undefined when the denominator is zero. Division by zero is not defined.
So, dy/dx is undefined when t=0
https://www.wolframalpha.com/input/?i=plot+x+%3D+10+%E2%80%93+t%5E2,+y+%3D+t%5E3+%E2%80%93+12t+for+0+%3C%3D+t+%3C%3D+1
You can see from the plot that when t=0 the tangent at (10,0) is vertical.
No problem at all! I'm here to help. To find the value of t for which the slope of the curve is undefined, we need to determine the value(s) of t at which the derivative of y with respect to x does not exist.
To find the derivative, we need to differentiate the equation for y with respect to x. Start by differentiating x = 10 – t^2 with respect to t. The derivative of x with respect to t will give us dx/dt, which represents the rate of change of x with respect to t.
Differentiating x = 10 – t^2 with respect to t:
dx/dt = d/dt(10 – t^2)
= -2t
Now, differentiate y = t^3 – 12t with respect to t. The derivative of y with respect to t will give us dy/dt, which represents the rate of change of y with respect to t.
Differentiating y = t^3 – 12t with respect to t:
dy/dt = d/dt(t^3 – 12t)
= 3t^2 – 12
To determine when the slope of the curve is undefined, we need to find the values of t for which dx/dt is zero. In this case, dx/dt = -2t.
Setting -2t = 0:
-2t = 0
t = 0
Therefore, the slope of the curve is undefined when t = 0.