# OpenAlgebra.com

*Free Algebra Help: Math simplified open tutorials.*

Showing posts with label

**derivative**. Show all postsShowing posts with label

**derivative**. Show all posts### Interactive: Mean Value Theorem

**Mean Value Theorem:**Let \(f\) be a function that satisfies the following:

- \(f\) is continuous on a closed interval \([a, b]\).
- \(f\) is differentiable on the open interval \((a,b)\).

\[f'\left( p \right) = \frac{{f\left( b \right) - f\left( a \right)}}{{b - a}}\]

**Instructions:**With the mouse, move points A and B along the function to see the Mean Value Theorem in action. Refresh browser to start over.

**YouTube Video Lectures**by Rob Shone

### Interactive: Tangent Line at a Point

**Definition:** The tangent line to the curve \(y = f(x)\) at the point \(P(a, f(a))\) is the line through \(P\) with slope

\[m = \mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right) - f\left( a \right)}}{{x - a}}\]

provided that this limit exists.

**Instructions:**With the mouse, move the

*x*-value toward

*a*to see that the tangent line is the limiting position of the secant line shown dashed. You can also move the points on the function. Refresh browser to start over.

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