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anzhelika [568]
2 years ago
14

Find an equation for the line tangent to the curve at the point defined by the given value of t.​ Also, find the value of d^2y /

dx^2 at this point.
x = 3t^2 + 7, y = t^6, t = -1
Mathematics
1 answer:
Ilya [14]2 years ago
5 0

The tangent line has slope \frac{\mathrm dy}{\mathrm dx}, which we can find using the chain rule:

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

We have

x(t)=3t^2+7\implies\dfrac{\mathrm dx}{\mathrm dt}=6t

y(t)=t^6\implies\dfrac{\mathrm dy}{\mathrm dt}=6t^5

so that

\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{6t^5}{6t}=t^4

When t=-1, the tangent line has slope 1.

To compute the second derivative \frac{\mathrm d^2y}{\mathrm dx^2}, first notice that the first derivative \frac{\mathrm dy}{\mathrm dx} turns out to be a function of t. Denote this first derivative by f(t). Then by the chain rule, we find

\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\mathrm df}{\mathrm dx}=\dfrac{\frac{\mathrm df}{\mathrm dt}}{\frac{\mathrm dx}{\mathrm dt}}=\dfrac{4t^3}{6t}=\dfrac23t^2

and at t=1, we get a value of 2/3.

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