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Showing posts with label derivative. Show all posts
Showing posts with label derivative. Show all posts
Interactive: Mean Value Theorem
Mean Value Theorem: Let \(f\) be a function that satisfies the following:
\[f'\left( p \right) = \frac{{f\left( b \right) - f\left( a \right)}}{{b - a}}\]
- \(f\) 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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