Day 36, Mat128: Calculus A

This is the relationship that we explore in this section.

Let's look at the graph of #5, p. 220. This is a graph of the derivative:

Questions:

1. On what intervals is $f$ increasing or decreasing?
2. At what values of $x$ does $f$ have a local max or min?

• 1st: When a function is differentiable at a point x=a and has zero derivative (tangent line slope) there, then if the first derivative changes sign from
  • positive to negative, there's a max at x=a;
  • negative to positive, there's a min at x=a;
  • no change in sign, there's neither max nor min at x=a.

• 2nd: When a function is differentiable at a point x=a and has zero derivative (tangent line slope) there, then if the second derivative is
  • positive at x=a, there's a min at x=a;
  • negative at x=a, there's a max at x=a;
  • zero, we're not sure what's going on....
    • 
    • 

1. A graph is concave up if it's "bowlish"; concave down if it's "umbrellaish" (see Figures 5, 6, p. 216)
   

2. inflection point: Point P on a curve is an inflection point if the curve changes concavity at P.

   Concavity test:

   • If $f^{\prime\prime}>0$ for all $x$ in interval $I$, then $f$ is concave up on $I$;
   • if $f^{\prime\prime}<0$ for all $x$ in interval $I$, then $f$ is concave down on $I$.

3. ``There is a point of inflection at any point where the second derivative changes sign.'' (p. 218)
   

   You might notice that the function itself looks cubic, and hence think to yourself that the derivative probably looks quadratic....

• #6, p. 220
  

• #27
  

• #23
• #39
• #50
• #51

• non-constant polynomials (go to infinity, in the positive or negative sense)
• the usual trig functions (while sine and cosine are bounded, but don't settle down to some nice limiting value)

Another example with a horizontal asymptote is knowledge as a function of time -- #51, p. 222. We might guess that accumulated knowledge in studying for an exam looks something like this:

We might imagine that this physical process becomes less productive from hour to hour as the evening wears on (the law of diminishing returns).

Other Examples:

• hyperbola: $f(x)=\frac{1}{x}$

  More generally, If $r>0$ is a rational number, then

  $\lim_{x \to \infty} \frac{1}{x^r}=0$ If, furthermore, $x^r$ is defined for negative numbers, then $\lim_{x \to -\infty} \frac{1}{x^r}=0$

• The most general form of an hyperbola is this: $f(x)=\frac{ax+b}{cx+d}$
  • What is its horizontal asymptote?
  • Where is its vertical asymptote?

• A rational function will have a horizontal asymptote any time that the degree of the denominator equals or exceeds that of the numerator: if
  $r(x)=\frac{p(x)}{q(x)}$

  then if the degree of q exceeds that of p, there is a horizontal asymptote, and the value of the asymptote is given by examining the approximating function given by the ratio of leading terms alone.

• The usual rules of limits apply, with the caveat that one must be careful when dealing with $\infty$. It is not a number, and cannot be added to anything else (e.g. it is not valid to write that $\infty=0$).

• #7, 8, p. 234
• #44, p. 235
• #49