- Homework due today
- Your quizzes are returned.
- The First Major/Minor Lunch of the 2015/2016 Academic year!
Friday 09/18 from 11:30a.m. to 1:30 p.m.
MEP 4th Floor Atrium
All Mathematics and Statistics Majors and Minors are invited (or even if you're thinking about it!).
Friday's menu: Reuben Sandwiches, Brats, Mets, a little sauerkraut for the Brats and Mets, a few vegetarian hotdogs, Potato pancakes, hot slaw, baked beans, and very chunky applesauce. Cookies for dessert.
This is a great chance to meet your faculty and fellow Mathematics and Statistics Majors and Minors
- Definition: continuity at the point $x=a$:
\[
\lim_{x\rightarrow{a}}f(x)=f(a)
\]
Note: three things have to happen:
- The limit of $f$ exists at $a$,
- The function value $f(a)$ exists at $a$, and
- The two values are equal.
Otherwise $f$ is discontinuous at $a$.
- The Intermediate Value Theorem is an important consequence of continuity:
- 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 $f(c)=N$.
- Example: pp. 93, #69
- 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 $f(c)=N$.
- We lead into the derivatives by discussing secants, tangents, limits.
These are tangent lines (places where a line osculates a curve):
These are "rates of change" of the function f. What does that mean? The thing that tells you how fast a function is changing is its slope, isn't it? If a function is constant, then it's not changing at all. If the slope is steep (either up or down), then the function's values are changing dramatically and quickly.
The rate of change is dictated by the slope. So it should come as no surprise that the derivative 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), $\frac{f(x)-f(a)}{x-a}$ 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:
$m=\lim_{x\to{a}}\frac{f(x)-f(a)}{x-a}$ 
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
$m=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$ 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 the definition of the derivative function, at any value of $x$.
- Definition of the derivative at a point: well, it's just a slope, isn't it:
$f^\prime(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h} = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}$
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.
- #3, pp. 110
- #11, pp. 111
- #13, pp. 111
- #45, pp. 112
- #51, pp. 113