Day 27, Mat129: Calculus I

• Your 3.4 homework is due today.
• Your 3.5 homework is due Friday.

• We discovered that just rewriting the form of a function can expose its shape to us. We need to know in our guts a few important function: e.g.
  • The parabola: $f(x)=x^2$
  • The fourth power: $f(x)=x^4$
    The hyperbola: $f(x)=\frac{1}{x}$
  and we have to know how to transform them -- e.g. shifts left and right, or reflections: so we looked at transformed parabolas and hyperbolas, such as

  The hyperbola: $f(x)=1+\frac{1}{x-1}$	The parabola: $f(x)=1-x^2$
  shifted right 1, up 1	reflected over the x-axis, shifted up 1

• You will notice that this section includes the "new" technique of the calculation of "slant asymptotes" (which we just discussed in section 3.4).

  This is just one type of asymptotic behavior that is sometimes useful. In fact, as mentioned in the previous section, every rational function approaches a polynomial in its end behavior, so that we're interested here in those rational functions (and some others) which approach linear functions as x gets large.

  E.g.,

  $f(x)=\frac{\displaystyle x^2-4x+17}{\displaystyle x-3}$

  Long-division allows us to show that its end behavior is like $x-1$. Actually we can re-write $f$ as

  $f(x)=x-1+\frac{\displaystyle 14}{\displaystyle x-3}$

  using either long-division or completing the square.

  However non-rational functions may also have polynomial asymptotic behavior, such as

  $f(x)=3x-6 + \frac{\sin(x)}{x}$

• So here's the upshot on the graphing of functions, "the strategy":
  1. Look at the expression of the function: can you simplify it in any way?
  2. Domain -- find it.
  3. Symmetry -- hope for it!
     • even and odd functions;
     • functions that have symmetry about some displaced point;
     • periodicity.

     These help to reduce the amount of work we have to do.

  4. Intercepts - crossings of the axes. The y-intercept is the point $(0,f(0))$ (if 0 is in the domain); the x-crossings are the roots of the equation $f(x)=0$: The special values of x that make this so.
  5. Asymptotes - vertical, horizontal, slant
  6. Intervals of increase or decrease - consult the sign of the first derivative.
  7. Local maxima and minima - use the first or second derivative tests.
  8. Concavity and points of inflection - use the second derivative and the concavity test.
  9. Compute some points on the curve, especially any that are trivial to calculate, or especially interesting.
  10. Sketch the curve - sketch asymptotes as dashed lines; plot any known points on the curve (e.g. intercepts); finish by connecting the points in accord with the information above.

      Use anything that you know about the form of the function to help inform your graph (e.g. if you know it's cubic, then you already have a good idea what it's going to look like).

      Maybe it's made up of functions you recognize.

      If your function is periodic, then you only need to sketch it on one period.

• Examples:
  • #9, p. 242 (use known curves and translations, etc. to help you)
  • #12 (use symmetry)
  • #23 (use completion of the square to get the asymptotic behaviour)
  • Let's do one more together: #51

1. Come up with the appropriate function, and then
2. find extrema in the ordinary ways.

• Understand
• Plan
• Carry out
• Evaluate

Examples:

• #15, p. 257
• #27
• #9