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olganol [36]
3 years ago
8

Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to chec

k your work. (Enter your answers as a comma-separated list of ordered pairs.)x = t^3 - 3t, y = t^2 - 4

Mathematics
1 answer:
klasskru [66]3 years ago
5 0

Answer:

There is a horizontal tangent at (0,-4)

The tangent is vertical at (-2,-3) and (2,-3).

Step-by-step explanation:

The given function is defined parametrically by the equations:

x=t^3-3t

and

y=t^2-4

The tangent function is given by:

\frac{dy}{dx}=\frac{\frac{dy}{dt} }{\frac{dx}{dt} }

\implies \frac{dy}{dx}=\frac{2t}{3t^2-3}

The tangent is vertical at when \frac{dx}{dt}=0

\implies \frac{3t^2-3}{2t}=0

\implies 3t^2-3=0

\implies 3t^2=3

\implies t^2=1

\implies t=\pm1

When t=1,

x=1^3-3(1)=-2 and y=1^2-4=-3

When t=-1,

x=(-1)^3-3(-1)=2 and y=(-1)^2-4=-3

The tangent is vertical at (-2,-3) and (2,-3).

The tangent is horizontal, when \frac{dy}{dx}=0 or  \frac{dy}{dt}=0

\implies 2t=0

\implies t=0

When t=0,

x=0^3-3(0)=0 and y=0^2-4=-4

There is a horizontal tangent at (0,-4)

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