EXAMPLES FOR SECTIONS 12.1 AND 12.2

EXAMPLES FOR EXAM II Sections 12.1 and 12.2

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Limit and Continuity Example

The graph of the position function is pictured at the right.

Click on the picture above to see an animation.

See the picture at the right.

See the picture at the right.

See the picture at the right.

Note: The answer above is a VECTOR. Even though this limit exists, we can observe that r(t) is not defined at t = 1 because both x(t) and z(t) are not defined at t = 1. Thus r(t) is not continuous at t = 1. For r(t) to be continuous at t = 1 all three of its component functions, x(t), y(t), and z(t) must be continuous at t =1. Recall that the definition of continuity for a function of one variable, say f(x), requires that for f to be continuous at x = a it must be true that

This means that the limit must exist at a, the function must be defined at a, and the two values must be equal. This boils down to saying that you must be able to compute the limit by direct substitution.

Section 12.1 #64

DPGraphPicture of the surfaces

Another view with higher resolution

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Example--the Definite Integral of a Vector Valued Function

I'll say that again. The answer is a VECTOR.

Section 12.2 #66 Similar (one changed initial condition)

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Powerpoint Presentation

Finding Where a Graph Is Not Smooth
The velocity vector is in green and the acceleration vector is in red. Click here or on the picture at the right to view the animation. One loop animation Quicktime Version

Note that the values along the horizontal axis are x-values, not t-values. When t = 2, x = 2pi and when t = 4, x = 4pi.

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