Day 13, Mat128: Calculus A

• Homework returned: graded 4, 21, 47ab
  • #4: we're having interval problems, and some trouble with infinity....
  • #21: Graph better. You should be able to sketch this quickly.
  • #47: you need to justify your answer

• 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

• 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
  • #29, pp. 112
  • #51, pp. 113

Now: let's begin by noting the difference between the derivative at a point, and the derivative on an interval (i.e. the derivative as function).

We see the derivative (thought of as a slope) at four separate points (A, B, C, and P). At three of these points we see that the slope (and hence the derivative) is zero. These are especially important points, since, as you can see, the function achieves its extreme values on an interval at these points. That's something we are often interested in -- when is a function at its peak, or valley?

We can see, however, that the slope is perfectly well defined everywhere for this function (with the possible exception of the end points, where the graph disappears, and we won't have limits from left and right). Now let's see how to draw a rough approximation to the derivative function in this case.

How well can you recognize derivatives and the functions that give rise to them? Let's try #3, p. 122.

Now I'd like to use the definition of the derivative function,

\[ f^\prime(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \]

to determine the derivative functions of these important functions:

1. Constant functions: $f(x)=c$
2. Linear functions: $f(x)=mx+b$
3. Quadratic functions: $f(x)=ax^2+bx+c$

4. An important application of the tangent line

If a function's derivative is another function, does that function have a derivative?

The derivative of the derivative of a function is called the second derivatives of the function. And how do we interpret these "higher derivatives" in the context of a motion?

Let's think of a quadratic motion, e.g. the motion of an eraser thrown across the room. Let s be the height of the eraser:

$s(t)=at^2+bt+c$

Each of the coefficients has an important, intuitive role to play:

• $c$ is the height at time $t=0$;
• $b$ is the rate of change in height at time $t=0$;
• $2a$ is the acceleration (due to gravity) of the height.

\[ f^\prime(x) = \frac{dy}{dx} = \frac{d}{dx}\left(f(x)\right)=\left(f(x)\right)'=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \]

There are several different ways of writing the derivative, and you need to get used to them. The third form reminds us that differentiation is itself a function: it takes a function in its domain and returns another function -- the derivative.

• #13, p. 136
• #22, p. 136
• #48, p. 137
• #56, p. 137

• Rules that help us avoid having to use the definition each time (with Proofs):

  • The sum rule

    works as we'd hope: "The derivative of a sum is the sum of the derivatives."

  • The constant multiple rule (makes sense)

  • The power rule (via dominoes -- i.e., mathematical induction)

  • NOTE: at this point we have all the rules necessary to differentiate all polynomials, without needing to resort to the definition!

    The derivative of the monomial $a_nx^n$ is $na_nx^{n-1}$.

    And a polynomial is just a sum of these. So

    $s'(t)=(at^2+bt+c)'=2at+b$

    and

  • $s''(t)=(2at+b)'=2a$

• Other rules:
  • The product rule
  • The quotient rule