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What you may not do is use them for anything other than that. If you haven't signed my sheet (and I'm missing a couple of names), please do.
I'd like to make a few comments on individual problems....
We were discussing limits such as
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Notice that we approach
When we write
We say that ``the limit of
Questions:
infinite limits for
Similarly we can define
In any of these cases, we say that
Desmos can help us investigate that somewhat ugly function....
The best case scenario is when the limit exists at a point, the
function exists at that point, and the limit is the same as the
function value:
This graph shows us that the function
But this graph also illustrates that there are three things that can go wrong:
Why is continuity in a graph important? We interpret the derivative as the slope of the tangent line (the limit of secant lines!) at a point. You can frequently look at a graph and see where things go awry. For example, if there's no tangent line at a point, then there's no instantaneous velocity! And at any point of discontinuity, it's impossible to create a tangent line (so there will be no derivative).
Which leads us into our next topic...
The most important definition in calculus is the definition of
the derivative (here is the derivative of
This was defined as the slope of the tangent line to a curve (provided the tangent line exists).
In particular, we're only going to have indeterminacy in our
derivative if the numerator goes to 0 when the denominator goes
to 0. But that only happens if
We see, therefore, that a derivative exists at
We've already seen how the secant lines approach the tangent line for a smooth curve. It's one of the first important problems we'll want to address in calculus. It's why we're interested in limits of things at the outset.
We usually find the equation of a line using two points, or a point and a slope. The secant line method approaches the tangent line at a point by using a succession of nearby points that are ever closer to the point of tangency:
The tangent ("touching") line osculates ("kisses") the curve at this point.
Notice the focus on linear functions: linear functions are the most important functions in calculus.
Once we have the derivative at