   Chapter 3.P, Problem 4P

Chapter
Section
Textbook Problem

# Find the point on the parabola y = 1 − x 2 at which the tangent line cuts from the first quadrant the triangle with the smallest area.

To determine

To find:

The point on the parabola y=1-x2 at which the tangent line cuts from the first quadrant the triangle with the smallest area.

Explanation

1) Concept:

The point on the parabola y=1-x2 at which the tangent line cuts has to find tangent line at that point and then with the help of x and y-intercepts find area of triangle. Then find points at which parabola is intersec.t

2) Formula:

i. Differentiation rule:ddxuv=vdudx-udvdxv2

ii. Power rule:ddxxn=nxn-1

3) Given:

The parabola y=1+x2.

4) Calculation:

Let a point p(a,b) is on the curve where a tangent line drawn.

Given that

y=1-x2

Since this point is on the curve, it satisfy the equation of parabola.

Put y=b and x=a in the above equation.

b=1-a2

To get the slope of tangent on that point, we have to differentiate the curve y=1-x2 on that point.

Differentiate y=1-x2 by using power rule.

dydx=-2x

That is

dydx=-2a

Write an equation of tangent line which is having a slope of -2a and passing through (a,b)

y-b=(-2a)·(x-a).

Simplify.

y-b=-2ax+2a2

Put x=0 in above equation to get y intercept.

y-b=-2a0+2a2

Simplify.

y-b=2a2

y=b+2a2

So, y- intercept is b+2a2.

Put y=0 in above equation to get x intercept.

(0)-b=-2ax+2a2

-b+2ax=-2ax+2a2+2ax

Simplify.

-b+2ax=2a2

-b+b+2ax=b+2a2

Simplify.

2ax=b+2a2

Divide by 2a on both sides.

2ax2a=b+2a22a

Simplify.

x- intercept is a+b2a

In the triangle, x- intercept will represent base and y- intercept will represent height. So area can be written as A.

A=12·base·height

Put x- intercept and y- intercept values in above equation.

A=12·(a+b2a)·(b+2a2)

Simplify.

A=12·(2a2+b2a)·(b+2a2)

Simplify.

A=b+2a224a

Now put b=1-a2 in the above equation.

A=1-a2+2a224a

Simplify the bracket.

A=1+a224a

For the smallest area differentiate A with respect to a then make it zero to find the value of a.

Differentiate A with respect to a by using power rule.

A'=4a·dda1+a22-1+a22·dda4a4a2

Take derivatives

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