Q. 12 - Calculus

Question

Restate the Mean Value Theorem so that its conclusion has to do with tangent lines.

Step-by-Step Solution

Answer

The restated theorem in terms of tangent lines is:

"If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists at least one value $c\in (a,b)$ such that, the tangent line of the curve drawn at $(c,f(c\left)\right)$ is parallel to the line joining the points $(a,f(a\left)\right)$ and $(b,f(b\left)\right)$"

Step 1. Given information.

The actual Mean value theorem is:

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists at least one value $c\in (a,b)$ such that, $f\text{'}\left(c\right) = \frac{f\left(b\right) - f\left(a\right)}{b - a}$

Step 2. Remember the geometrical meaning of derivative.

Geometrically, first derivative $f\text{'}\left(x\right)$ of a function $f$ at a point $x$ means the slope of the tangent drawn at that point $(x,f(x\left)\right)$.

we have, for some point $c\in (a,b)$ the first derivative $f\text{'}\left(c\right) = \frac{f\left(b\right) - f\left(a\right)}{b - a}$

$\Rightarrow$ slope of the tangent line drawn at $c$ is $\frac{f\left(b\right) - f\left(a\right)}{b - a}$.

$\Rightarrow$ slope of the line joining the points $(a,f(a\left)\right)$ and $(b,f(b\left)\right)$ is also $\frac{f\left(b\right) - f\left(a\right)}{b - a}$.

$\Rightarrow$ we have "if the slopes of two lines are equal, then those lines are parallel"

      that is, the tangent line at $c$ and the line joining the points $(a,f(a\left)\right)$ and $(b,f(b\left)\right)$ are parallel.