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Ivahew [28]
3 years ago
7

The curve with equation y2 = 5x4 − x2 is called a kampyle of eudoxus. find an equation of the tangent line to this curve at the

point (−1, 2).
Mathematics
1 answer:
Vesnalui [34]3 years ago
3 0
To find the tangent line we will need the slope of the tangent at x=-1 (the x-coordinate of the point given). We find the slope by using the derivative of the curve.

Th curve given is y^{2} =5 x^{4} - x^{2} which can be solved for y by taking the root of both sides. We obtain y=(5 x^{4}- x^{2} ) ^{1/2}

We find the derivative using the chain rule. Bring down the exponent, keep the expression in the parenthesis, raise it to 1/2 - 1 and then take the derivative of what is inside.

y’=(1/2)(5 x^{4} - x^{2} )^{-1/2}(20 x^{3} -2x)

Next we evaluate this expression for x=-1 and obtain:
\frac{20(-1)^{3} -(2)(-1)}{2 \sqrt{5(-1 ^{4}-(-1 ^{2})  } }= \frac{-20+2}{2 \sqrt{5-1} }=-18/4=-9/2

So we are looking for a line through (-1,2) with slope equal to -9/2. We use y=mx+b with m=-9/2, x=-1 and y=2 to find b.

2=(-9/2)(-1)+b
2-(9/2)=b
b=-5/2

So the tangent line is given by y=(-9/2)x+(-5/2)

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Step-by-step explanation:

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