Q. 12 Restate the Mean Value Theorem s... [FREE SOLUTION]

Chapter 3: Q. 12 (page 248)

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

Short Answer

Expert verified

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 $c 鈭� (a, b)$ such that, the tangent line of the curve drawn at $(c, f (c))$ is parallel to the line joining the points $(a, f (a))$ and $(b, f (b))$ "

Step by step solution

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 $c 鈭� (a, b)$ such that, $f' (c) = \frac{f (b) - f (a)}{b - a}$

Step 2. Remember the geometrical meaning of derivative.

Geometrically, first derivative $f' (x)$ of a function $f$ at a point $x$ means the slope of the tangent drawn at that point $(x, f (x))$ .

we have, for some point $c 鈭� (a, b)$ the first derivative $f' (c) = \frac{f (b) - f (a)}{b - a}$

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

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

$鈬�$ 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 $(a, f (a))$ and $(b, f (b))$ are parallel.

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91影视

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Remember the geometrical meaning of derivative.

One App. One Place for Learning.

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