Showing posts with label derivative. Show all posts
Showing posts with label derivative. Show all posts

Saturday, May 4, 2013

Interactive: Derivative of Cosine

Derivative of the Sine Function: \[\frac{d}{{dx}}\left( {\cos x} \right) = - \sin x\]


Instructions: Drag P along \(f(x) = cos(x)\) to see that the slope of the tangent line through it traces out \(f'(x) = -sin(x)\).



Interactive: Derivative of Sine

Derivative of the Sine Function: \[\frac{d}{{dx}}\left( {\sin x} \right) = \cos x\]


Instructions: Drag P along \(f(x) = sin(x)\) to see that the slope of the tangent line through it traces out \(f'(x) = cos(x)\).



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 \((a,b)\).
Then there is a number \(p\) in \((a,b)\) such that
\[f'\left( p \right) = \frac{{f\left( b \right) - f\left( a \right)}}{{b - a}}\]

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: Derivative of a Function

Derivative of a Function at \(a\), if it exists:

\[f'\left( a \right) = \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {a + h} \right) - f\left( a \right)}}{h}\]


Instructions: Drag the green dot representing a+h toward a to simulate h tending toward zero.



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.