## Algebra

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

### 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.