LIMITS AND DERIVATIVES

LIMITS, DERIVATIVES, DIFFERENTIALS AND TANGENTS

Epsilon-Delta Definition of Limit

These animations relate to applying the epsilon-delta definition of limit to

as discussed in Example 8 on page 54 in your text. The animation on the left zooms in on the point (2,4). For the epsilon-delta definition of limit to be satisfied for the ever decreasing positive values for epsilon it is necessary for the blue graph of f(x) = x² to remain between the horizontal green lines (between 4 minus epsilon and 4 plus epsilon) whenever the blue graph is between the vertical green lines (between 2 minus delta and 2 plus delta).

This animation is intended as a demonstration of the definition of derivative applied to the function f(x) = 4 - x² at the point (-1,3). The green triangle represents taking the limit from the right and the length of the green tangent line segment corresponds to the changing value of the difference quotient. The red triangle represents taking the limit from the left and the length of the red tangent line segment corresponds to the changing value of the difference quotient. Both the red and green tangent line segments are approaching a limiting length of 2. Quicktime animation

Click on the picture at the right to see an animation of the tangent line moving across the graph. Click here to see an animation with both the tangent line and the normal line moving across the graph.

Click on the picture on the left to see an animation.

The green line is the tangent to the graph of

y = 4 - x² at the point (1,3).

Click here to see an animation of the Mean Value Theorem being applied to a quadratic function and click here to see it applied to a cubic function. Click on the image on the left to see an animation of that image.

Click here for an animation comparing the graph of u_n(x) = (1 + x/n)ⁿ as n goes from 1 to 100 with the graph of f(x) = e^x.

Here is an animation relating to differentials that looks at the graph of y = x² starting out with a delta x value of 1/2 at x = 1/4 as in the picture at the right and animating as delta x approaches 0. Click here or on the picture to see the animation. Click here to see a pair of animations, one zooming, one not zooming.

Here is an animation that zooms in on the point (1.5,0) in graphically analyzing the following limit:

This animation zooms in on the point (2,2) in graphically analyzing the limit as x approaches 2 of

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Lane Vosbury, Mathematics, Seminole State College email: vosburyl@seminolestate.edu