Day 14, Mat128: Calculus A

• #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 derivative 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.

• #2, p. 122
• #25
• #49

• 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 product rule

  • 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 quotient rule!