Showing posts with label interactive. Show all posts
Showing posts with label interactive. Show all posts

Friday, May 10, 2013

Interactive: Hyperbolas


Interactive Instructions: Move any of the green points. Also, you can change the orientation below.

Hyperbola: Horizontal Vertical


[ NOTESHyperbolas ]


Saturday, May 4, 2013

Interactive: Mean Value Theorem for Integrals

Mean Value Theorem for Integrals: If \(f\) is continuous on \([a,b]\), then there exists a number \(c\) in \([a,b]\) such that \[\int_a^b {f\left( x \right)dx = f\left( c \right)\left( {b - a} \right)} \]


Instructions: Drag a and b to see c calculated dynamically. The area under the curve from a to b is the same as the area of the pink rectangle formed by b - a and f(c).



Interactive: Net Area

Net Area: If the function f takes on both positive and negative values, then the Riemann sum is the result of the area under the curve above the x-axis less the area under the x-axis above the curve.


Instructions: Drag a and b to see net area change sign. Purple indicates positive net area and pink indicates negative net area.



Interactive: Definite Integral

Definite Integral: If \(f\) is a continuous function defined on the interval \([a, b]\) that is divided into \(n\) equal subintervals \(\Delta x = \frac{{b - a}}{n}\) where \(x_i^*\) lies in the ith subinterval then the definite integral of \(f\) from \(a\) to \(b\) follows:

\[\int_a^b {f\left( x \right)dx} = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {f\left( {x_i^*} \right)} \,\Delta x\]


Instructions: Drag the slider to see that the area under the curve is the limit of Riemann sums. You can also change the sums to left, middle, right, and trapezoidal below the graph.

Riemann Sum: Left Middle Right Trapezoidal


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: Fermat's Theorem

Fermat's Theorem: If \(f\) is a local maximum or minimum at \(c\), and if \(f'(c)\) exists, then \(f'(c)=0\).

Critical Number: A number \(c\) in the domain of \(f\) such that either \(f'(c)=0\) or \(f'(c)\) does not exist.


Instructions: With the mouse, move point P along the function and you will see it's derivative traced in green. Here c and d are the critical numbers for the function graphed in blue. Refresh browser to start over.



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.



Sunday, April 28, 2013

Interactive: Circumcenter of a Triangle


Interactive Instructions: Move the green dots to see the circumcenter of the triangle change.





Saturday, April 27, 2013

Interactive: Ellipses


Interactive Instructions: Move any of the green points.



[ NOTESEllipses ]


Friday, April 26, 2013

Interactive: Circles


Interactive Instructions: Move the green points C and P. You can also drag the circle around.



[ NOTESCircles ]


Thursday, April 25, 2013

Interactive: Perpendicular Lines


Interactive Instructions: Move the green points to change m and see that perpendicular lines have opposite reciprocal slopes.



[ NOTESPerpendicular Lines ]


Interactive: Parallel Lines


Interactive Instructions: Move the green points to change m and see that parallel lines have the same slope. You can also move the orange line.


[ NOTESParallel Lines ]


Tuesday, April 23, 2013

Interactive: Slope-Intercept Form



Interactive Instructions: Slope-intercept form: y = mx + b. Move the green points to change m and b.





Monday, April 22, 2013

Interactive: Slope


Interactive Instructions:
Move the green points A and B.



Interactive: Distance and Midpoint

Interactive Instructions: Move the green points A and B. Hover over the points to see the coordinates. You can also drag the line segment.




Interactive: Rectangular Coordinate System


Interactive Instructions: Move the green point.