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Find the equation of the tangent line to the curve y=x^5 at x=2. Give the equation of the tangent line.

Sagot :

Answer:

(a)

The equation of the given curve is  y=x2−2x+7.

On differentiating with respect to x, we get: 

dxdy=2x−2

The equation of the line is 2x−y+9=0.⇒y=2x+9

This is of the form y=mx+c.

Slope of the line =2

If a tangent is parallel to the line 2x−y+9=0, then the slope of the tangent is equal to the slope of the line.

Therefore, we have: 2=2x−2

⇒2x=4⇒x=2

Now, at x=2

⇒y=22−2×2+7=7

Thus, the equation of the tangent passing through (2,7) is given by, 

y−7=2(x−2)

⇒y−2x−3=0

Hence, the equation of the tangent line to the given curve (which is parallel to line (2x−y+9=0) is y−2x−3=0.

(b)

The equation of the line is 5y−15x=13.

Slope of the line =3

If a tangent is perpendicular to the line 5y−15x=13, 

then the slope of the tangent is Slope of the line−1=3−1.

⇒dxdy=2x−2=3−1

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