Day 7, Mat129: Calculus I

• A new administration in government?
• A sidewalk?
• Coffee poured into a cup?

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:

1. The limit of $f$ exists at $a$,
2. The function value $f(a)$ exists at $a$, and
3. The two values are equal.

Otherwise $f$ is discontinuous at $a$.

There are various kinds of discontinuity (which we've already seen):

• Removable discontinuities (e.g. holes)
• Jump discontinuities
• Infinite discontinuities
• "Crazy discontinuities" like the function of #59, p. 71: $x^2$ for rational values of $x$, and 0 for irrational ones.

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).

• We define one-sided continuity
• we have sum laws, product, and quotient laws. We've already used these to demonstrate that polynomials are continuous on all real numbers: \[ \lim_{x\to{a}}P(x)=P(a) \]

• Polynomials
• Rational functions: (ratios of polynomials)
• Trigonometric functions: sine, cosine, tangent
• Root functions: $f(x)=x^{1/n}$
• Exponential functions: $f(x)=b^x$

• Roughly: the composition of continuous functions is continuous.

• Let $F(x)=f(g(x))$ be a composite function.

  If $g$ is continuous at $x=a$ and $f$ is continuous at $g(a)$ then

  $F(x)$ is continuous at $x=a$: \[ \lim\limits_{x\to a}F(x)=\lim\limits_{x\to a}f(g(x))=f(\lim\limits_{x\to a}g(x))=f(g(a))=F(a) \]

• pp. 90-93, #3, 10, 12, 19, 23, 36, 46 (with Mathematica)
• That crazy function which was $x^2$ for rational values of $x$, and 0 for irrational ones, has an interesting property: it is continuous at only a single point! It satisfies the definition only at $x=0$: it's discontinuous everywhere else! That's pretty weird.

• Theorem: Suppose that f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b), where $f(a)\ne{f(b)}$. Then there exists a number $c$ in $(a,b)$ such that $f(c)=N$.

  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.

• Example: pp. 93, #69

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 $x$.

• If a function is constant, then it's not changing at all ($\Delta y = 0$).
• If the slope is steep (either up or down), then the function's values are changing dramatically and quickly ($\Delta y$ large).

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), $\frac{f(x)-f(a)}{x-a}=\frac{\Delta y}{\Delta x}$ 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: $m=\lim\limits_{x\to{a}}\frac{f(x)-f(a)}{x-a}$

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, $\frac{f(a+h)-f(a)}{h}$ In the limit, this is $m=\lim\limits_{h\to 0}\frac{f(a+h)-f(a)}{h}$ 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.

• #3, pp. 110, etc.
• #11, pp. 111
• #13, pp. 111
• #29, pp. 112
• #45, pp. 112
• #51, pp. 113