Question

The graph of any quadratic function \(f\left( x \right) = a{x^2} + bx + c\) is a parabola. Prove that the average of the slopes of the tangent lines to the parabola at the endpoints of any interval \(\left( {p,q} \right)\) equals the slope of the tangent line at the midpoint of the interval.

Short answer

It has been proved that the average of the slopes of the tangent lines to the parabola at the endpoints of any interval \(\left( {p,q} \right)\) equals the slope of the tangent line at the midpoint of the interval.

Step-by-step solution

Tangent to a Parabola

A tangent to a parabola can be defined as a straight line curve that basically passes through the only point on the parabola which can be expressible in the form of equation of the line.

Evaluating equation of parabola.

Thegiven equation is:

\(f\left( x \right) = a{x^2} + bx + c\)

Differentiating with respect to\(x\), we have:

\(f'\left( x \right) = \;2ax + b\)

Now, slope of tangent line at\(x = p{\rm{ and }}x = q\), we have:

\(2ap + b\;\;{\rm{and}}\;\;\;2aq + b\)

Evaluating the average of the slope.

Now, the average of the slopes will be:

\(\frac{{\left( {2ap + b} \right) + \left( {2aq + b} \right)}}{2} = a\left( {p + q} \right) + b\)

Now, mid-point of interval\(\left( {p,q} \right)\)will be:\(\frac{{p + q}}{2}\)

Then, slope of tangent line for this point will be:

\(\;2a\left( {\frac{{p + q}}{2}} \right) + b = a\left( {p + q} \right) + b\)

This is equal to the equation of average of the slopes of tangents.

Hence proved the average of the slopes of the tangent lines to the parabola at the endpoints of any interval \(\left( {p,q} \right)\) equals the slope of the tangent line at the midpoint of the interval.