Day 21, Mat129: Calculus I

• Your homework is returned (2.5). Graded 17, 58, and 78.

• Your quiz is returned.

• Please put your rectangles away.

I've talked a bit about how the approximation $\sin(x) \approx x$ when $x$ is small is important in physics. That is mentioned in this section, which is one reason why I wanted to have a look at 2.9:

The simple pendulum

More generally, we frequently use the linear function representing the tangent line at a point to make certain approximations.

That linear function is called the linearization of $f$ at $a$:

• For example, suppose you need to compute $\sqrt{15.96}$. Now the easy way is to use your calculator, but the cowboy way would be to use the linearization! Let's have a go at that one.
  • We know the square root of 16, which is close: our answer should be near 4.

  • Draw a picture! The picture will demonstrate two different types of quantities that we discuss in this section: differentials $dy$, and increments $\Delta y$.

    Differentials versus increments

    
    The increment is the true change in the function value; the differential approximates the true change: One way to remember the difference:
    • Differentials $dy$ follow the direction suggested by the derivative
    • Increments $\Delta y$ are actual changes in the function value.

    $dy \approx \Delta y$

    We want the increment, but may settle for the easily computed differential.

  • We use the tangent line for our approximation, and, when we're done, we can ask "is our answer an over- or an under-estimate?"

• Examples
  • #2, p. 187
  • #5
  • #38

• Example application of extrema:
  • The Chickens of Manitoba
  • How was the flock egg production over time?
  • How long should we let the flock live to maximize the average daily return?

• Maxima and minima are both types of extrema. Our author begins by distinguishing between global (or absolute) extrema and local extrema:
  • Local is the biggest thing in the neighborhood;
  • Global is the biggest thing anywhere!

• Take a moment and sketch a continuous function on the interval [-2,2]. Make it look kind of interesting -- maybe even ugly.
  1. Does your function have a global maximum and a global minimum?
  2. Does your function have local extrema which are not global extrema?
  3. If your function does have extrema, how would you characterize the derivative of the function at those extrema?

• By the way, our author requires that the function be defined in an open interval about a local extremum. So in this figure from p. 199, the right endpoint is not considered a local maximum:
  

• Now I want you to sketch a function defined on the interval [-2,2] that does not have any global extrema. What is true about your function?

• Now I want you to sketch a smooth function on [-2,2] that has a value of x=a where $f'(a)=0$ but no extremum there. What is true about your function?

• By now we should have a good idea of what these two theorems say:
  • Extreme Value Theorem
  • Fermat's Theorem

  We have hopefully deduced these:

  • The Extreme Value Theorem (EVT): If $f$ is continuous on a closed interval $[a,b]$, then $f$ attains a global maximum and global minimum on $[a,b]$.
  • Fermat's Theorem: If $f$ has a local max or min at $c$ and $f^\prime(c)$ exists, then $f^\prime(c)=0$.

    Rephrased: If $f$ has a local max or min at $c$, then $c$ is a critical number of $f$.

  We might also have deduced the closed interval method, to find the extrema of a function on a closed interval [a,b]:
  1. Find the critical numbers for f (that is, those values of x at which the derivative is zero or doesn't exist).
  2. Find the values of f at the critical numbers of f in (a,b).
  3. Find the values of f at the endpoints of the interval (at a and b).
  4. The largest and smallest of the values from steps 1 and 2 are the absolute or global extrema.

• Now let's look at some standard problems:
  • #8, p. 204
  • #22
  • #31, p. 205 (and let's decide whether they're local maxes or mins)
  • #45 (closed interval method)
  • #63