Lesson 24 (10/25/26)

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Lesson 24 (10/25/26)

Today: Graphing Functions

Office Hours: MWF 1:30pm – 2:30pm.

Meeting with Faculty: Tomorrow at 12pm, Windsor.

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Review: What does \( f'(x) \) & \( f''(x) \) tell us?

First Derivative:

  • \( f'(x) > 0 \implies f(x) \) increasing
  • \( f'(x) < 0 \implies f(x) \) decreasing
  • \( f'(x) = 0 \implies \) Horizontal tangent

Note: \( f'(c) = 0 \) or DNE \( \implies x=c \) is a critical point.

Note: \( x=c \) is a local max/min if \( f' \) changes sign at \( x=c \).

Second Derivative:

  • \( f''(x) > 0 \implies f(x) \) is Concave up
  • \( f''(x) < 0 \implies f(x) \) is Concave down

Note: \( x=c \) is an inflection point if \( f''(c) = 0 \) & \( f'' \) changes sign at \( x=c \).

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Example: \( f(x) = \frac{x^3}{3} - 16x \)

Find intervals where the function is increasing and decreasing.

\( f'(x) = x^2 - 16 \)

Critical points: \( x^2 - 16 = 0 \implies x = 4, x = -4 \)

Intervals of \( \uparrow \) & \( \downarrow \)

A horizontal number line marked with points at -4 and 4. The region to the left of -4 is labeled 'x^2-16 > 0'. The region between -4 and 4 is labeled 'x^2-16 < 0'. The region to the right of 4 is labeled 'x^2-16 > 0'.
Visual Description: A horizontal number line marked with points at -4 and 4. The region to the left of -4 is labeled 'x^2-16 > 0'. The region between -4 and 4 is labeled 'x^2-16 < 0'. The region to the right of 4 is labeled 'x^2-16 > 0'.

Increasing on \( (-\infty, -4), (4, \infty) \)

Decreasing on \( (-4, 4) \)

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Sketch \( \frac{x^3}{3} - 16x \) only using the information from \( f' \)

A hand-drawn sketch of a cubic function on a Cartesian plane. The horizontal axis is marked with points at -4 and 4. The curve originates in the third quadrant, passes through the x-axis to the left of -4, reaches a local maximum between -4 and 0, crosses the origin (0,0), reaches a local minimum between 0 and 4, crosses the x-axis again to the right of 4, and continues into the first quadrant.
Visual Description: A hand-drawn sketch of a cubic function on a Cartesian plane. The horizontal axis is marked with points at -4 and 4. The curve originates in the third quadrant, passes through the x-axis to the left of -4, reaches a local maximum between -4 and 0, crosses the origin (0,0), reaches a local minimum between 0 and 4, crosses the x-axis again to the right of 4, and continues into the first quadrant.

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\[ f(x) = \frac{x^3}{3} - 16x \]

Find intervals where the function is concave up and down.

\[ f'(x) = x^2 - 16 \]

\[ f''(x) = 2x \]

\[ f''(x) = 0 \implies x = 0 \]

Concave up on \( (0, \infty) \)

Concave down on \( (-\infty, 0) \)

\( x = 0 \) is inflection point

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Concavity and Slope Analysis

The upper portion of the diagram displays a horizontal number line with tick marks at \(-4\) and \(4\). Above the line, a series of vertical arrows indicate the direction of the slope of a function: upward-pointing arrows (increasing slope) appear to the left of \(-4\), downward-pointing arrows (decreasing slope) appear between \(-4\) and the midpoint, followed by a sequence of upward and downward arrows, and finally upward-pointing arrows to the right of \(4\). Below this line, there are two distinct wavy blue lines drawn: one segment is labeled "C. down" (concave down) spanning a region roughly between \(-4\) and the midpoint, and another segment is labeled "C. up" (concave up) spanning the region to the right of the midpoint.

A coordinate plane showing a periodic wave function. The x-axis includes markings for -4, 0, and 4. A local maximum occurs near x = -4, indicated by a red horizontal tangent line and a dashed vertical drop line. A local minimum occurs to the right of x = 4, also indicated by a red horizontal tangent line. The function crosses the x-axis at x = 0. The graph illustrates the relationship between the stationary points and the intervals of concavity defined in the upper section.
Visual Description: A coordinate plane showing a periodic wave function. The x-axis includes markings for -4, 0, and 4. A local maximum occurs near x = -4, indicated by a red horizontal tangent line and a dashed vertical drop line. A local minimum occurs to the right of x = 4, also indicated by a red horizontal tangent line. The function crosses the x-axis at x = 0. The graph illustrates the relationship between the stationary points and the intervals of concavity defined in the upper section.

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More information about \( f(x) = \frac{x^3}{3} - 16x \)

Domain: \( (-\infty, \infty) \)

Intercepts:

X-intercept: \( \frac{x^3}{3} - 16x = 0 \)

\( \frac{x}{3} [x^2 - 48] = 0 \implies x = 0, x = \sqrt{48} = 4\sqrt{3} \)

\( x = -\sqrt{48} = -4\sqrt{3} \)

Y-intercept: \( x = 0 \implies \frac{x^3}{3} - 16x = 0 \)

No Vertical Asymptotes:

\( \lim_{x \to \pm\infty} f(x) = \pm\infty \implies \) No Horizontal Asymptotes

\( f(4) = \frac{64}{3} - 64 = -64 \times \frac{2}{3} \), \( f(-4) = \frac{2}{3} \times 64 \)

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Sketch of \( f(x) = \frac{x^3}{3} - 16x \)

The following diagram provides a graphical representation of the cubic function \( f(x) = \frac{x^3}{3} - 16x \).

A hand-drawn sketch of the cubic function f(x) = x^3/3 - 16x on a Cartesian coordinate system. The graph is a standard cubic shape, starting from the third quadrant, increasing to a local maximum, crossing the origin (0,0), decreasing to a local minimum, and then increasing into the first quadrant. Key features marked on the graph include x-intercepts at -4√3, 0, and 4√3. The local maximum is marked at y = (2 * 64) / 3, and the local minimum is marked at y = -(2 * 64) / 3. There are auxiliary marks along the x-axis at -4 and 4 for scale.
Visual Description: A hand-drawn sketch of the cubic function f(x) = x^3/3 - 16x on a Cartesian coordinate system. The graph is a standard cubic shape, starting from the third quadrant, increasing to a local maximum, crossing the origin (0,0), decreasing to a local minimum, and then increasing into the first quadrant. Key features marked on the graph include x-intercepts at -4√3, 0, and 4√3. The local maximum is marked at y = (2 * 64) / 3, and the local minimum is marked at y = -(2 * 64) / 3. There are auxiliary marks along the x-axis at -4 and 4 for scale.

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Sketching Guidelines

  • What is the Domain?
  • Are there any Vertical Asymptotes / Holes
  • What is the end behaviour \( \lim_{x \to \infty} f(x) \), \( \lim_{x \to -\infty} f(x) \)? Note: Horizontal or slant Asymptotes.
  • What are \( x \) & \( y \) intercepts
  • critical points & intervals where \( f \) is \( \uparrow \) & \( \downarrow \)
  • inflection points & intervals where \( f \) is concave up & down

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Example: Sketch \( f(x) = \frac{x^3}{x^2 - 1} \)

Domain: everything except \( x=1 \) & \( x=-1 \)

\( (-\infty, -1) \cup (-1, 1) \cup (1, \infty) \)

Vertical Asymptotes/Holes: \( \lim_{x \to 1} f(x) \) & \( \lim_{x \to -1} f(x) \)

\[ \lim_{x \to 1^+} \frac{x^3}{x^2 - 1} = \frac{1}{\text{+ve, small}} = +\infty \]

\[ \lim_{x \to 1^-} \frac{x^3}{x^2 - 1} = \frac{1}{\text{-ve, small}} = -\infty \]

\[ \lim_{x \to -1^+} \frac{x^3}{x^2 - 1} = \frac{-1}{\text{-ve small}} = +\infty \]

\[ \lim_{x \to -1^-} \frac{x^3}{x^2 - 1} = \frac{-1}{\text{+ve small}} = -\infty \]

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End Behaviour!

\[ \lim_{x \to \pm \infty} \frac{x^3}{x^2-1} \]

If \( f(x) = \frac{p(x)}{q(x)} \), if degree of \( p(x) = \) degree of \( q(x) + 1 \), \( f(x) \) has a slant asymptote.

Do Long Division to find Slant Asymptote

A long division setup showing (x^2 - 1) dividing x^3, resulting in x with a remainder of x. The steps shown are: x^2 - 1 long division symbol enclosing x^3, with x as the quotient. Under x^3, there is a (x^3 - x) term being subtracted, resulting in x.
Visual Description: A long division setup showing (x^2 - 1) dividing x^3, resulting in x with a remainder of x. The steps shown are: x^2 - 1 long division symbol enclosing x^3, with x as the quotient. Under x^3, there is a (x^3 - x) term being subtracted, resulting in x.

\[ \frac{x^3}{x^2-1} = \frac{x}{x^2-1} + x \]

Note: The term \( \frac{x}{x^2-1} \) has a limit of 0 as \( x \to \infty \).

\( y=x \) is a slant asymptote

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\[ f(x) = \frac{x^3}{x^2 - 1} \]

x-intercept: \( f(x) = 0 \Rightarrow \frac{x^3}{x^2 - 1} = 0 \Rightarrow x = 0 \)

y-intercept: \( y = f(0) = \frac{0^3}{0^2 - 1} = 0 \)

Conclusion: The graph passes through \((0, 0)\) and does not intersect axes anywhere else.

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Use \( f'(x) \) to find intervals of \( \uparrow \) and \( \downarrow \).

\[ f(x) = \frac{x^3}{x^2-1} \]

\[ f'(x) = \frac{(3x^2)(x^2-1) - x^3(2x)}{(x^2-1)^2} = \frac{3x^4 - 3x^2 - 2x^4}{(x^2-1)^2} = \frac{x^4 - 3x^2}{(x^2-1)^2} \]

Note: \( (x^2-1)^2 > 0 \) for \( x \neq \pm 1 \)

Numerator \( = x^4 - 3x^2 = x^2(x^2-3) = x^2(x-\sqrt{3})(x+\sqrt{3}) \)

A sign chart diagram on a horizontal number line. The line is marked with critical points at negative square root of 3, 0, and positive square root of 3. Intervals are labeled as follows: to the left of negative square root of 3, f'(x) is greater than 0, indicating an increasing function (upward arrow); between negative square root of 3 and 0, f'(x) is less than 0, indicating a decreasing function (downward arrow); between 0 and positive square root of 3, f'(x) is less than 0, indicating a decreasing function (downward arrow); to the right of positive square root of 3, f'(x) is greater than 0, indicating an increasing function (upward arrow).
Visual Description: A sign chart diagram on a horizontal number line. The line is marked with critical points at negative square root of 3, 0, and positive square root of 3. Intervals are labeled as follows: to the left of negative square root of 3, f'(x) is greater than 0, indicating an increasing function (upward arrow); between negative square root of 3 and 0, f'(x) is less than 0, indicating a decreasing function (downward arrow); between 0 and positive square root of 3, f'(x) is less than 0, indicating a decreasing function (downward arrow); to the right of positive square root of 3, f'(x) is greater than 0, indicating an increasing function (upward arrow).

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\( f(x) = \dfrac{x^4 - 3x^2}{(x^2 - 1)^2} \)

\( f'(x) = \dfrac{(4x^3 - 6x)(x^2 - 1)^2 - (x^4 - 3x^2) \cdot (2(x^2 - 1) \cdot 2x)}{((x^2 - 1)^2)^2} \)

Note: Denominator \( > 0 \)

Numerator:

\( (x^2 - 1) [(4x^3 - 6x)(x^2 - 1) - (x^4 - 3x^2)(4x)] \)

\( (x^2 - 1) [4x^5 - 6x^3 - 4x^3 + 6x - 4x^5 + 12x^3] \)

\( (x^2 - 1)(2x^3 + 6x) \)

\( x(x^2 - 1)(2x^2 + 6) = x(x - 1)(x + 1) \cdot (2x^2 + 6) \)

Margin Note: \( \underbrace{2x^2 + 6}_{\ge 0} \)

A number line showing the sign of f''(x). The line is marked with vertical tick marks at -1, 0, and 1. Between -infinity and -1, the interval is labeled 'f'' < 0'. Between -1 and 0, the interval is labeled 'f'' >= 0'. Between 0 and 1, the interval is labeled 'f'' < 0'. Between 1 and infinity, the interval is labeled 'f'' > 0'.
Visual Description: A number line showing the sign of f''(x). The line is marked with vertical tick marks at -1, 0, and 1. Between -infinity and -1, the interval is labeled 'f'' < 0'. Between -1 and 0, the interval is labeled 'f'' >= 0'. Between 0 and 1, the interval is labeled 'f'' < 0'. Between 1 and infinity, the interval is labeled 'f'' > 0'.

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A coordinate plane grid containing a blue function curve exhibiting vertical asymptotes. A dashed diagonal line y=x passes through the origin. The horizontal axis is marked with intervals labeled 'c. down' and 'c. up' indicating concavity, with specific points labeled at -\sqrt{3} and \sqrt{3}. Purple arrows pointing up and down indicate the slope direction. Red horizontal line segments indicate local extrema on sections of the curve. The curve is concave down for x < -\sqrt{3}, concave up between -\sqrt{3} and 0, concave down between 0 and \sqrt{3}, and concave up for x > \sqrt{3}.
Visual Description: A coordinate plane grid containing a blue function curve exhibiting vertical asymptotes. A dashed diagonal line y=x passes through the origin. The horizontal axis is marked with intervals labeled 'c. down' and 'c. up' indicating concavity, with specific points labeled at -\sqrt{3} and \sqrt{3}. Purple arrows pointing up and down indicate the slope direction. Red horizontal line segments indicate local extrema on sections of the curve. The curve is concave down for x < -\sqrt{3}, concave up between -\sqrt{3} and 0, concave down between 0 and \sqrt{3}, and concave up for x > \sqrt{3}.