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🚀 Level 3 - Topic 4: Curve Sketching Using Derivatives 🎨🖌️

Putting it All Together: Sketching Accurate Graphs Using Calculus Tools

1) Introduction: Mastering the Art of Curve Sketching

Curve sketching is the process of creating a reasonably accurate graph of a function by systematically analyzing its properties using calculus. By using derivatives, we can go beyond just plotting points and understand the key features of a function's graph: where it increases/decreases, its peaks and valleys, its concavity, and overall shape.

In this topic, we will combine all the derivative analysis techniques we've learned into a step-by-step approach for curve sketching. We'll cover:

  • Checklist for Curve Sketching: A systematic approach to analyze a function.
  • Domain and Range (Brief Review): Basic function properties to consider.
  • Intercepts: Finding x-intercepts and y-intercepts.
  • Asymptotes (if any): Identifying vertical, horizontal, and slant asymptotes.
  • Increasing/Decreasing Intervals: Using the first derivative \( f'(x) \).
  • Local Extrema: Using the First or Second Derivative Tests.
  • Concavity and Inflection Points: Using the second derivative \( f''(x) \).
  • Sketching the Graph: Putting all the information together to create the graph.
By following these steps, you'll be able to sketch accurate graphs of functions and understand their behavior in detail.

2) Checklist for Curve Sketching: A Step-by-Step Approach

To sketch a curve \( y = f(x) \) effectively, we can follow a systematic checklist:

Curve Sketching Checklist

  1. A. Domain: Determine the domain of the function \( f(x) \). Are there any restrictions on \( x \) values?
  2. B. Intercepts:
    • y-intercept: Find the y-intercept by setting \( x = 0 \) and calculating \( f(0) \) (if \( x = 0 \) is in the domain). Point is \( (0, f(0)) \).
    • x-intercepts: Find the x-intercepts by setting \( y = f(x) = 0 \) and solving for \( x \). Points are \( (x, 0) \). May be hard to find exactly, sometimes approximations are needed.
  3. C. Symmetry: (Optional but helpful)
    • Even function: \( f(-x) = f(x) \) (symmetric about y-axis).
    • Odd function: \( f(-x) = -f(x) \) (symmetric about origin).
    • Periodic function: \( f(x + P) = f(x) \) for some period \( P \).
  4. D. Asymptotes:
    • Horizontal Asymptotes: Evaluate limits \( \lim_{x \to \infty} f(x) \) and \( \lim_{x \to -\infty} f(x) \). If either limit is a finite number \( L \), then \( y = L \) is a horizontal asymptote.
    • Vertical Asymptotes: Look for values of \( x \) where the denominator of a rational function is zero, or where the function becomes undefined in a way that suggests \( \lim_{x \to c} f(x) = \pm \infty \). Check limits as \( x \) approaches these values from the left and right. \( x = c \) is a vertical asymptote if at least one of these limits is \( \pm \infty \).
    • Slant Asymptotes (Oblique): For rational functions where the degree of the numerator is exactly one more than the degree of the denominator. Perform polynomial long division to express \( f(x) = mx + b + R(x) \), where \( \lim_{x \to \pm \infty} R(x) = 0 \). Then \( y = mx + b \) is a slant asymptote.
  5. E. Increasing/Decreasing Intervals:
    • Find \( f'(x) \) and critical numbers (where \( f'(x) = 0 \) or DNE).
    • Create a sign chart for \( f'(x) \) to determine intervals where \( f'(x) > 0 \) (increasing) and \( f'(x) < 0 \) (decreasing).
  6. F. Local Extrema:
    • Use the First Derivative Test or Second Derivative Test to classify critical points as local maxima, local minima, or neither. Find the \( y \)-values of local extrema.
  7. G. Concavity and Inflection Points:
    • Find \( f''(x) \) and solve \( f''(x) = 0 \) or find where \( f''(x) \) is undefined to find potential inflection points.
    • Create a sign chart for \( f''(x) \) to determine concavity (concave up \( f''(x) > 0 \), concave down \( f''(x) < 0 \)).
    • Identify inflection points where concavity changes. Find their \( y \)-values.
  8. H. Plot Key Points and Sketch the Graph:
    • Plot intercepts, local extrema, inflection points, and any other easy-to-calculate points. Draw asymptotes as dashed lines.
    • Sketch the curve, connecting the plotted points and respecting the information about increasing/decreasing intervals, concavity, and asymptotes to get the overall shape.

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

Let's apply the checklist to sketch the graph of \( f(x) = \frac{2x^2}{x^2 - 1} \).

    A. Domain: Denominator \( x^2 - 1 = (x - 1)(x + 1) = 0 \) when \( x = \pm 1 \). Domain is \( (-\infty, -1) \cup (-1, 1) \cup (1, \infty) \).

    B. ntercepts:

    • y-intercept: \( f(0) = \frac{2(0)^2}{0^2 - 1} = \frac{0}{-1} = 0 \). y-intercept is \( (0, 0) \).
    • x-intercepts: Set \( f(x) = \frac{2x^2}{x^2 - 1} = 0 \Rightarrow 2x^2 = 0 \Rightarrow x = 0 \). x-intercept is also \( (0, 0) \). So, it passes through the origin.
    C. Symmetry: \( f(-x) = \frac{2(-x)^2}{(-x)^2 - 1} = \frac{2x^2}{x^2 - 1} = f(x) \). Function is even, symmetric about the y-axis.

    D. Asymptotes:

    • Horizontal Asymptotes: \( \lim_{x \to \pm \infty} \frac{2x^2}{x^2 - 1} = \lim_{x \to \pm \infty} \frac{2}{1 - \frac{1}{x^2}} = \frac{2}{1 - 0} = 2 \). Horizontal asymptote is \( y = 2 \).
    • Vertical Asymptotes: Vertical asymptotes at \( x = -1 \) and \( x = 1 \) (where denominator is zero and numerator is non-zero). \( \lim_{x \to -1^-} \frac{2x^2}{x^2 - 1} = +\infty \), \( \lim_{x \to -1^+} \frac{2x^2}{x^2 - 1} = -\infty \), \( \lim_{x \to 1^-} \frac{2x^2}{x^2 - 1} = -\infty \), \( \lim_{x \to 1^+} \frac{2x^2}{x^2 - 1} = +\infty \). Vertical asymptotes are \( x = -1 \) and \( x = 1 \).
    • Slant Asymptotes: None, since horizontal asymptote exists.
    E. Increasing/Decreasing Intervals:
    1. \( f'(x) = \frac{d}{dx}\left[\frac{2x^2}{x^2 - 1}\right] = \frac{(4x)(x^2 - 1) - (2x^2)(2x)}{(x^2 - 1)^2} = \frac{4x^3 - 4x - 4x^3}{(x^2 - 1)^2} = \frac{-4x}{(x^2 - 1)^2} \).
    2. Critical numbers: \( f'(x) = 0 \Rightarrow -4x = 0 \Rightarrow x = 0 \). \( f'(x) \) is undefined at \( x = \pm 1 \) (but these are not in the domain of \( f \)). Critical number is \( x = 0 \).
    3. Sign chart for \( f'(x) = \frac{-4x}{(x^2 - 1)^2} \). Denominator \( (x^2 - 1)^2 \) is always positive where defined. Sign of \( f'(x) \) is determined by \( -4x \).
      Interval Test Value \( f'(x) = \frac{-4x}{(x^2 - 1)^2} \) sign \( f(x) \) behavior
      \( (-\infty, -1) \) \( x = -2 \) \( f'(-2) = \frac{-4(-2)}{+} = + \) Increasing
      \( (-1, 0) \) \( x = -0.5 \) \( f'(-0.5) = \frac{-4(-0.5)}{+} = + \) Increasing
      \( (0, 1) \) \( x = 0.5 \) \( f'(0.5) = \frac{-4(0.5)}{+} = - \) Decreasing
      \( (1, \infty) \) \( x = 2 \) \( f'(2) = \frac{-4(2)}{+} = - \) Decreasing

      Increasing on \( (-\infty, -1) \) and \( (-1, 0] \). Decreasing on \( [0, 1) \) and \( (1, \infty) \).

    F. Local Extrema: At \( x = 0 \), \( f'(x) \) changes from positive to negative. By First Derivative Test, local maximum at \( x = 0 \). Local max value \( f(0) = 0 \). Local maximum point \( (0, 0) \). No local minima.

    G. Concavity and Inflection Points:

    1. \( f''(x) = \frac{d}{dx} \left[ \frac{-4x}{(x^2 - 1)^2} \right] = \frac{(-4)(x^2 - 1)^2 - (-4x)(2)(x^2 - 1)(2x)}{((x^2 - 1)^2)^2} \)

      2. \( = \frac{-4(x^2 - 1)^2 + 16x^2(x^2 - 1)}{(x^2 - 1)^4} = \frac{4(3x^2 + 1)}{(x^2 - 1)^3} \)

    2. \( f''(x) = 0 \Rightarrow 4(3x^2 + 1) = 0 \Rightarrow 3x^2 + 1 = 0 \Rightarrow 3x^2 = -1 \Rightarrow x^2 = -\frac{1}{3} \). No real solutions. So, \( f''(x) \neq 0 \) for any real \( x \). \( f''(x) \) undefined at \( x = \pm 1 \). Potential concavity change at \( x = -1, 1 \) (but not in domain).
    3. Sign chart for \( f''(x) = \frac{4(3x^2 + 1)}{(x^2 - 1)^3} \). Numerator \( 4(3x^2 + 1) \) is always positive. Sign of \( f''(x) \) determined by denominator \( (x^2 - 1)^3 \), which has the same sign as \( x^2 - 1 \).
      Interval Test Value \( f''(x) \) Sign Concavity
      \( (-\infty, -1) \) \( x = -2 \) \( f''(-2) = \frac{+}{(+)^3} = + \) Concave Up (CU)
      \( (-1, 1) \) \( x = 0 \) \( f''(0) = \frac{+}{(-1)^3} = - \) Concave Down (CD)
      \( (1, \infty) \) \( x = 2 \) \( f''(2) = \frac{+}{(+)^3} = + \) Concave Up (CU)

      Concave up on \( (-\infty, -1) \) and \( (1, \infty) \). Concave down on \( (-1, 1) \). No inflection points because concavity changes occur at \( x = \pm 1 \) which are not in the domain.

    H. Sketch the Graph: Plot intercepts \( (0, 0) \), asymptotes \( y = 2, x = -1, x = 1 \), local max at \( (0, 0) \). Concave up on \( (-\infty, -1) \) and \( (1, \infty) \), concave down on \( (-1, 1) \). Even function (symmetric about y-axis). Use this information to sketch the curve. The graph will approach \( y = 2 \) as \( x \to \pm \infty \), go through origin with a local maximum, and approach vertical asymptotes at \( x = \pm 1 \).

By following these steps systematically, we obtain a detailed understanding of the graph and can sketch it accurately.

4) Practice Questions - Curve Sketching

Practice your curve sketching skills using derivatives.

Fundamental Practice Questions

Instructions: For each function, use the curve sketching checklist to analyze and sketch the graph. Identify all key features (domain, intercepts, asymptotes, intervals of increase/decrease, local extrema, concavity, inflection points).

  1. Q1. \( f(x) = x^3 - 12x \)
  2. Q2. \( g(x) = x^4 - 4x^3 \)
  3. Q3. \( y = \frac{x}{x^2 + 1} \)
  4. Q4. \( f(x) = \frac{x^2}{x + 2} \)
  5. Q5. \( h(x) = x\sqrt{4 - x} \)

Challenging Practice Questions

Instructions: Sketch these more complex curves using derivative analysis.

  1. Q1. \( f(x) = \frac{x^2 - 4}{x^2 - 1} \)
  2. Q2. \( g(x) = x - 2\arctan(x) \)
  3. Q3. \( y = e^{-x} \sin(x) \) for \( x \geq 0 \) (consider behavior as \( x \to \infty \)).
  4. Q4. \( f(x) = \ln(x^2 + 4) \)
  5. Q5. \( h(x) = \frac{x^3}{x^2 - 4} \)

5) Summary & Cheat Sheet - Curve Sketching Using Derivatives

A quick guide to the curve sketching process.

5.1) Curve Sketching Checklist (Steps)

  1. A. Domain
  2. B. Intercepts (x and y)
  3. C. Symmetry (optional)
  4. D. Asymptotes (Horizontal, Vertical, Slant)
  5. E. Increasing/Decreasing Intervals (using \( f'(x) \))
  6. F. Local Extrema (using \( f'(x) \) or \( f''(x) \))
  7. G. Concavity & Inflection Points (using \( f''(x) \))
  8. H. Plot and Sketch

5.2) Derivative Tools for Curve Sketching

  • First Derivative \( f'(x) \): Increasing/Decreasing intervals, local extrema.
  • Second Derivative \( f''(x) \): Concavity, inflection points, Second Derivative Test for local extrema.
  • Limits: Asymptotes (horizontal, vertical, slant).

Congratulations! You've now learned a systematic approach to curve sketching using derivatives. By mastering these steps, you can analyze and sketch a wide variety of functions, gaining deep insights into their behavior. Next, we'll move on to the final topic of Level 3: L'Hôpital's Rule for evaluating indeterminate forms.

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