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The key is connectedness. Continuous things don't have to be smooth, although we often think of them that way. But we certainly don't want to encounter cliffs....
Note: three things have to happen:
Otherwise $f$ is discontinuous at $a$.
There are various kinds of discontinuity (which we've already seen):
Open | Filled (and continuous): $f(0)\equiv -0.5$ |
This function has a limit at zero (-.5), but is not defined there. If $f$ is not defined at $x=0$, then it cannot be continuous there. We can fix this, by the way.... Just define $f(0)$ to be the limit (the closed dot in the graph above).
If $g$ is continuous at $x=a$ and $f$ is continuous at $g(a)$ then
$F(x)$ is continuous at $x=a$: \[ \lim\limits_{x\to a}F(x)=\lim\limits_{x\to a}f(g(x))=f(\lim\limits_{x\to a}g(x))=f(g(a))=F(a) \]
This theorem is often used to show that a function has a root (that is, a value $x=c$ where $f(c)=0$): we show that it's negative at point $a$, positive at point $b$, and that it's continuous between -- hence there must be a point $c$ at which $f(c)=0$, by the IVT.
These are tangent lines (places where a line osculates a curve):
And the slopes of these tangent lines are instantaneous "rates of change" of the function f at the point of tangency. What does that mean?
The thing that tells you how fast a function is changing is its slope, isn't it?
We often represent change in mathematics by a "delta", $\Delta$. So the slope is a change in $y=f(x)$, which we might call $\Delta y$, divided by a change in $x$, or $\Delta x$: \[ slope=\frac{\Delta y}{\Delta x}=\frac{rise}{run} \] So this quantity tells how how fast $y$ is changing given a change in $x$.
The rate of change is dictated by the slope. So it should come as no surprise that the derivative of a function, which is the rate of change of a function at a point, is the same as the slope of the tangent line at a point:
We can approximate tangent lines with secant lines:
The slope m of the tangent line at P(a,f(a)) is approximated by
the slope of the blue line segment (the slope of a secant line),
This is an average rate of change in f over a finite interval. In the limit, this average rate of change becomes an instantaneous rate of change: |
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Here's an alternative notation for the slope:
The slope m of the tangent line at P(a,f(a)) is approximated by the slope of the blue line segment, In the limit, this is which I call the most important definition in calculus. This is the formula for the derivative at a point: I've already shared with you this definition of the derivative, at any value of $x$. |
$f^\prime(a) = \lim\limits_{h \to 0} \frac{f(a+h)-f(a)}{h} = \lim\limits_{x \to a} \frac{f(x)-f(a)}{x-a}$
The most important definition in calculus! (I just can't say it enough!)
Now let's look at some problems, and see how this concept is connected to real-world problems.
First, however, a word on the examples in this section: you see that, for the most part, we're using the definition(s) of a derivative to compute a slope of a tangent line. The tricky part is often writing the composition correctly -- i.e. just properly using the definition. Then you have to use a little algebra to remove the indeterminacy.