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04. Graph Sketching

$\gdef \N{\mathbb{N}} \gdef \Z{\mathbb{Z}} \gdef \Q{\mathbb{Q}} \gdef \R{\mathbb{R}} \gdef \C{\mathbb{C}} \gdef \setcomp#1{\overline{#1}} \gdef \sseq{\subseteq} \gdef \pset#1{\mathcal{P}(#1)} \gdef \covariant{\operatorname{Cov}} \gdef \of{\circ} \gdef \p{^{\prime}} \gdef \pp{^{\prime\prime}} \gdef \ppp{^{\prime\prime\prime}} \gdef \pn#1{^{\prime\times{#1}}} $

May 22 Lecture. Links to the board and the recording.

Definition:

\(x_0\) is an extreme point of \(f\) if there exists a neighborhood of \(x_0\), \(I=(x_0-\epsilon, x_0+\epsilon)\) such that either \(f(x_0) > f(x)\;\forall x\in I\) or \(f(x_0) < f(x)\;\forall x\in I\) \([x\not=x_0]\)
In the first case, \(x_0\) is a local maximum; in the second case, \(x_0\) is a local minimum.

Fermat’s Theorem (#36)

Theorem

If \(f\) is differentiable at \(x_0\), and \(x_0\) is an extreme point of \(f\), then \(f^\prime(x_0)=0\)

Caveats:
  1. Note the condition \(f\) is differentiable at \(x_0\):
    • \(f\) can have an extreme point where f is not differentiable.
      Ex.: \(f(x)=|x|\)
  2. Not every point where \(f^\prime=0\) is an extreme point. (i.e., the converse of Fermat’s theorem is not true.)
Critical points

By Fermat’s Theorem, the only candidates for extreme points of \(f\) are points where

  1. \(f^\prime(x)=0\)
  2. \(f\) is not differentiable at \(x_0\)
    If \(x_0\) satisfies either of these conditions, then \(x_0\) is a critical point of \(f\).
Rolle’s Theorem

Theorem

Suppose \(f\) is continuous in \([a,b]\) and differentiable in \((a,b)\), with \(f(a)=f(b)\). Then \(\exists c\in(a,b)\mid f^\prime(c)=0\)

Mean Value Theorem (MVT)

Theorem

Suppose \(f\) is continuous in \([a,b]\) and differentiable in \((a,b)\). Then, \(\exists c\in(a,b)\mid f^\prime(c)=\frac{f(b)-f(a)}{b-a}\)
Practically speaking, there exists at least one point on \(f\) between \((a,b)\) where the slope of the tangent is exactly equal to the slope of the line between \(a\) and \(b\).
You can think of this as a “sideways” version of Rolle’s Theorem.

Corollaries of MVT:
  1. If \(f^\prime(x)=0\) for all \(x\in(a,b)\), then \(f\) is constant.
  2. Let \(f,g\) be continuous on \([a,b]\), and differentiable on \((a, b)\). If \(f^\prime(x)=g^\prime(x)\mid\forall x\in(a,b)\), then they differ by a constant (i.e. \(g(x)=f(x)+k\) for some constant k on all of \((a,b)\))
  3. If \(f\) is differentiable on \((a,b)\) and \(f^\prime(x)>0\mid\forall x\in(a,b)\) then f is increasing on \((a,b)\).
    • If \(f\) is differentiable on \((a,b)\) and \(f^\prime(x)<0\mid\forall x\in(a,b)\) then f is decreasing on \((a,b)\).
    • \(f\) is increasing \(\iff f^\prime(x)>0\)
    • \(f\) is decreasing \(\iff f^\prime(x)<0\)
  4. Linear Approximation:
    For h small, \(f(x_0+h)\approx f(x_0)+h\cdot f^\prime (x_0)\)
    See: Explanation
Sketching graph using information about \(f^\prime\)

Ex.: \(f(x)=x^3-6x^2+9x\)
We wish to find extreme points where \(f\) is increasing and where \(f\) is decreasing.

  1. Find critical points (where \(f^\prime=0\) or \(f\) is not differentiable)
    • \(f\) is a polynomial, so it’s differentiable for all \(x\)
    • \(f^\prime(x)=3x^2-12x+9=0\) (set it to be zero to find extreme points)
      • \(f^\prime(x)=3(x^2-4x+3)=0\)
      • \(f^\prime(x)=3(x^2-4x+3)=0\)
      • \(f^\prime(x)=3(x-1)(x-3)=0\)
    • Critical points are \(1, 3\)
  2. Examine critical points
    1. First derivative test:
      1. For local maximum, \(f^\prime(x<x_0)<0<f^\prime(x>x_0)\)
        For local minimum, \(f^\prime(x<x_0)>0>f^\prime(x>x_0)\)
      2. Check the signs of the derivative function using the snake method
        • Excalidraw Excalidraw
      3. \(f^\prime>0\mid x<1,x>3\)
        \(f^\prime<0\mid1<x<3\)
      4. \(x_0=1\) is a local max
        \(x_0=3\) is a local min
    2. \(2^{nd}\) method to locate extreme points (from the critical points):
      1. Look at the second derivative \(f^{\prime\prime}(x_0)\) where \(x_0\) is a critical point
      2. If \(f^{\prime\prime}(x_0) > 0\) then \(x_0\) is a local min
      3. If \(f^{\prime\prime}(x_0)<0\) then \(x_0\) is a local max
      4. If \(f^{\prime\prime}(x_0)<0\) then this test gives no information
        - In our example, \(f^{\prime\prime}(x)=6x-12\), so we check:
        • \(f^{\prime\prime}(1)=-6\), so it’s a local maximum
        • \(f^{\prime\prime}(3)=6\), so it’s a local minimum
  3. Sketch the graph
    1. Plot the points where the graph intersect the X-axis
    2. Plot the maxima and minima
    3. connect them using information gleaned from examination of critical points.

May 22 Tirgul. Links to the board and recording.
[Incomplete]

[Incomplete until End of Unit Lecture June 5]