Connecting f, f', and f'' in Graph Analysis

Curve Sketching Using Calculus

By the end of this lesson, you will be able to analyze a function comprehensively using limits, derivatives, and asymptotes to produce an accurate sketch.

The Calculus Approach to Curve Sketching

Calculus lets us go far beyond a table of values. By analyzing $f$, $f'$, and $f''$, we can determine the complete behavior of the curve.

The Checklist

1. Domain: where is $f(x)$ defined?

2. Intercepts:

• x-intercepts: solve $f(x) = 0$
• y-intercept: $f(0)$ (if in domain)

3. Symmetry:

• Even ($f(-x) = f(x)$) → symmetric about y-axis
• Odd ($f(-x) = -f(x)$) → symmetric about origin

4. Asymptotes:

• Vertical: $\lim_{x\to a^\pm} f(x) = \pm\infty$ (denominator = 0, numerator ≠ 0)
• Horizontal: $\lim_{x\to\pm\infty} f(x)$
• Oblique: if degree of numerator = degree of denominator + 1, perform polynomial division

5. Increasing/Decreasing: Find $f'(x)$; set = 0 or find where undefined; sign chart.

6. Local Extrema: At critical points: first derivative test (sign change of $f'$) or second derivative test ($f''(c)$ sign).

7. Concavity and Inflection Points: Find $f''(x)$; sign chart; inflection where concavity changes.

8. Sketch: use all the above information.

Example: $f(x) = \frac{x^2}{x^2 - 4}$

1. Domain: $x \neq \pm 2$ → $(-\infty,-2) \cup (-2,2) \cup (2,\infty)$

2. Intercepts: $f(0) = 0/(-4) = 0$ (y-intercept and x-intercept)

3. Symmetry: $f(-x) = (-x)^2/((-x)^2-4) = x^2/(x^2-4) = f(x)$ → even; symmetric about y-axis

4. Asymptotes:

• Vertical: $x = \pm 2$ (denominator = 0, numerator = 4 ≠ 0)
• Horizontal: $\lim_{x\to\pm\infty}\frac{x^2}{x^2-4} = 1$ → $y = 1$

5. $f'(x)$: Using quotient rule:

$f'(x) = \frac{2x(x^2-4) - x^2(2x)}{(x^2-4)^2} = \frac{-8x}{(x^2-4)^2}$