#Graphs of $f$, $f'$ & $f''$

#Table of Contents

1. How to Sketch the Graph of a Function
2. Using Derivatives to Sketch Graphs
3. Worked Examples
4. Practice Questions
5. Glossary
6. Summary and Key Takeaways

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#How to Sketch the Graph of a Function

You should be familiar with the general shapes of the graphs of common functions, including:

• Linear functions
• Quadratic functions
• Cubic and higher-order polynomial functions
• Trigonometric functions
• Exponential and logarithmic functions
• Reciprocals and reciprocal powers of $x$, e.g., $\frac{1}{x^2}$

When sketching a function, consider the following:

#Domain and Range

• Domain: Identify values of $x$ for which $f(x)$ is defined.
• Range: Sometimes useful to find using the candidates test for global extrema.

#Points of Undefined Values

• Identify where $f(x)$ is undefined, leading to vertical asymptotes.
  • Example: $x=0$ for $y=\frac{1}{x}$.

#Limits and Asymptotes

• Determine the limit of the function as $x \to \pm \infty$ to find horizontal asymptotes.
  • Example: $y=2$ for $y=\frac{1}{x}+2$.

#Symmetry

• Determine if the function is even ($f(-x)=f(x)$) or odd ($f(-x)=-f(x)$).

#Periodicity

• Check if the function is periodic: $f(x+a)=f(x)$ for some constant $a$.

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#Using Derivatives to Sketch Graphs

Derivatives help identify key features and properties of the graph of a function. Recall the following properties of the first and second derivatives:

#Critical Points

• Critical points occur where the first derivative is zero or does not exist.
  • Example: $x=0$ for $y=x^{\frac{1}{3}}$.

#Points of Inflection

• At a point of inflection, the second derivative changes sign.
  • It is not enough that $f''(x)=0$.

#Behavior of the Function

• Where $f'(x) \ge 0$, the graph is increasing.
• Where $f'(x) \le 0$, the graph is decreasing.
• Where $f''(x) \ge 0$, the graph is concave up (u-shaped).
• Where $f''(x) \le 0$, the graph is concave down (n-shaped).

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#Worked Examples

#Example 1

Graph of $f'$

The graph of $f'$, the derivative of the function $f$, is shown above. Which of the following statements must be true?

i. $x=0$.

ii. The graph of $f$ has a point of inflection at $x=-2$.

iii. The graph of $f$ is concave up for $-7 < x < 1$.

iv. The function $f$ is increasing for $-7 < x < 1$.

Options: (A) i and ii only (B) i, ii, and iii only (C) i, ii, and iv only (D) i, ii, iii, and iv

Answer:

Let's analyze each statement:

1. $x=0$: True, as it is a critical point.
2. Point of Inflection at $x=-2$: True, since $f''(x)$ changes sign.
3. Concave Up for $-7 < x < 1$: False, as $f''(x)$ is negative in this interval.
4. Increasing for $-7 < x < 1$: True, as $f'(x) \ge 0$.

#Example 2

Sketching $f$

Let $f$ be the function defined by $f(x)=x(x-2)^2(x+3)$. Sketch the graph and identify any points of inflection.

Solution:

1. Inspect the Function: It is a positive quartic (highest power of $x$ is 4), meaning it will have a w-shape.
2. Roots: Set $f(x)=0$:
   • Roots at $x=0$, $x=-3$, and a repeated root at $x=2$.
3. Critical Points:
   • Expand the function: $f(x)=x^4-x^3-8x^2+12x$.
   • First derivative: $f'(x)=4x^3-3x^2-16x+12=0$.
   • Factorize: $f'(x)=(x-2)(4x^2+5x-6)=0$.
   • Critical points at $x=2$, $x=\frac{3}{4}$, and $x=-2$.
4. Second Derivative:
   • $f''(x)=12x^2-6x-16$.
   • Check concavity at critical points:
     • $x=2$: Minimum.
     • $x=\frac{3}{4}$: Maximum.
     • $x=-2$: Minimum.
   • Points of Inflection: Solve $f''(x)=0$: $x=\frac{3 \pm \sqrt{201}}{12}$.
5. Sketch the Graph: Label all critical points and points of inflection.

Graph of a positive quartic with 2 real roots, one repeated root, three stationary points, and two points of inflection.

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#Practice Questions

Practice Question

Question 1: Given the function $g(x) = 3x^3 - 2x^2 + x - 5$, find all critical points and determine their nature (local maxima, minima, or points of inflection).

Question 2: Sketch the graph of $h(x) = \frac{2x}{x^2 + 1}$ and identify any asymptotes and critical points.

Question 3: Determine whether the function $k(x) = e^{-x^2}$ has any points of inflection. If so, find them.

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#Glossary

• Domain: The set of all input values for which a function is defined.
• Range: The set of all output values a function can produce.
• Asymptote: A line that a graph approaches but never touches.
• Critical Point: A point where the first derivative of a function is zero or undefined.
• Point of Inflection: A point where the graph changes concavity.

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#Summary and Key Takeaways

#Summary

• To sketch a function, understand its domain, range, asymptotes, symmetry, and periodicity.
• Derivatives provide crucial information about the function's behavior, including critical points and concavity.

#Key Takeaways

• First Derivative: Indicates increasing/decreasing behavior.
• Second Derivative: Indicates concavity and points of inflection.
• Critical Points: Where $f'(x)=0$ or $f'(x)$ is undefined.
• Points of Inflection: Where $f''(x)=0$ and changes sign.

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#Exam Strategy

• Identify Key Features: Focus on domain, range, asymptotes, and symmetry first.
• Use Derivatives: Apply the first and second derivatives to find critical points and concavity.
• Check Your Work: Ensure all key points and behaviors are accurately represented in your sketch.

Remember, practice makes perfect. Use these strategies to improve your graph-sketching skills.