Solved Math Exams

What Is Curve Sketching?

Curve sketching is the process of drawing a rough but accurate graph of a function using calculus tools — without relying on a graphing calculator.

You'll combine everything you've learned:

Domain and intercepts
Asymptotes
First derivative (increasing/decreasing, local extrema)
Second derivative (concavity, inflection points)

The Curve Sketching Checklist

Use this template for any curve sketching problem:

1. Domain

Ask: Where is $f(x)$ defined?

Look for:

Division by zero → exclude those $x$-values
Square roots of negatives → exclude where expression is negative
Logarithms of non-positives → exclude where $\ln(\text{stuff}) \leq 0$

2. Intercepts

$y$-intercept: Set $x = 0$, find $f(0)$

$x$-intercepts (zeros): Set $f(x) = 0$, solve for $x$

3. Symmetry (Optional but Helpful)

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

4. Asymptotes

Vertical asymptotes: Where denominator = 0 (and numerator ≠ 0)

Horizontal asymptotes: $\lim_{x \to \pm\infty} f(x)$

If limit is a number $L$, then $y = L$ is a horizontal asymptote

Slant asymptotes: If degree of numerator = degree of denominator + 1, do polynomial division

5. First Derivative Analysis

Find $f'(x)$ and determine:

Critical points: Where $f'(x) = 0$ or $f'(x)$ DNE

Sign chart for $f'$:

$f'(x)$	$f(x)$ is...
$+$	Increasing ↗
$-$	Decreasing ↘

Local extrema:

$f'$ changes $+$ to $-$ → local max
$f'$ changes $-$ to $+$ → local min

6. Second Derivative Analysis

Find $f''(x)$ and determine:

Possible inflection points: Where $f''(x) = 0$ or $f''(x)$ DNE

Sign chart for $f''$:

$f''(x)$	$f(x)$ is...
$+$	Concave up ∪
$-$	Concave down ∩

Inflection points: Where $f''$ actually changes sign

7. Plot Key Points and Sketch

Plot: intercepts, critical points, inflection points, asymptotes

Connect with a smooth curve that respects increasing/decreasing and concavity.

Quick Reference: Derivative Sign Charts

First Derivative $f'(x)$

$f'$	Meaning	Graph
$f' > 0$	Increasing	Goes up left to right
$f' < 0$	Decreasing	Goes down left to right
$f' = 0$	Critical point	Possible max/min

Second Derivative $f''(x)$

$f''$	Meaning	Graph Shape
$f'' > 0$	Concave up	Holds water ∪
$f'' < 0$	Concave down	Spills water ∩
$f''$ changes sign	Inflection point	Concavity switches

Combining First and Second Derivatives

$f'$	$f''$	Behavior
$+$	$+$	Increasing, concave up ↗∪
$+$	$-$	Increasing, concave down ↗∩
$-$	$+$	Decreasing, concave up ↘∪
$-$	$-$	Decreasing, concave down ↘∩

Example 1: Polynomial

Sketch $f(x) = x^3 - 3x^2$.

Step 1: Domain

All real numbers (polynomial).

Step 2: Intercepts

$y$-intercept: $f(0) = 0$ → $(0, 0)$

$x$-intercepts: $x^3 - 3x^2 = x^2(x - 3) = 0$ → $x = 0, 3$

Step 3: First derivative

$$f'(x) = 3x^2 - 6x = 3x(x - 2)$$

Critical points: $x = 0$ and $x = 2$

Sign chart for $f'$:

Interval	$f'(x)$	$f(x)$
$x < 0$	$+$	Increasing
$0 < x < 2$	$-$	Decreasing
$x > 2$	$+$	Increasing

Local max at $x = 0$: $f(0) = 0$

Local min at $x = 2$: $f(2) = 8 - 12 = -4$

Step 4: Second derivative

$$f''(x) = 6x - 6 = 6(x - 1)$$

$f''(x) = 0$ when $x = 1$

Sign chart for $f''$:

Interval	$f''(x)$	Concavity
$x < 1$	$-$	Concave down
$x > 1$	$+$	Concave up

Inflection point at $x = 1$: $f(1) = 1 - 3 = -2$ → $(1, -2)$

Step 5: Sketch

Key points: $(0, 0)$ local max, $(2, -4)$ local min, $(1, -2)$ inflection, $(3, 0)$ zero

The curve goes through these points, changing from concave down to concave up at $x = 1$.

Example 2: Rational Function

Sketch $f(x) = \frac{x}{x^2 - 1}$.

Step 1: Domain

$x^2 - 1 = 0$ when $x = \pm 1$

Domain: all $x \neq \pm 1$

Step 2: Intercepts

$y$-intercept: $f(0) = 0$ → $(0, 0)$

$x$-intercept: numerator = 0 → $x = 0$

Step 3: Symmetry

$f(-x) = \frac{-x}{x^2 - 1} = -f(x)$ → Odd function (symmetric about origin)

Step 4: Asymptotes

Vertical: $x = 1$ and $x = -1$

Horizontal: $\lim_{x \to \pm\infty} \frac{x}{x^2 - 1} = 0$ → $y = 0$

Step 5: First derivative

Using quotient rule: $$f'(x) = \frac{(1)(x^2-1) - x(2x)}{(x^2-1)^2} = \frac{-x^2 - 1}{(x^2-1)^2}$$

Since $-x^2 - 1 < 0$ always, and $(x^2-1)^2 > 0$ always (where defined):

$f'(x) < 0$ everywhere → always decreasing (on each piece of domain)

Step 6: Sketch

Three pieces (separated by vertical asymptotes), all decreasing, approaching $y = 0$ as $x \to \pm\infty$.

Tips for Efficient Curve Sketching

Start with easy info: Domain and intercepts take seconds
Factor everything: Makes finding zeros and analyzing signs much easier
Use symmetry: If the function is even or odd, you only need to analyze half
Asymptotes frame the picture: Draw them first as guidelines
Sign charts are your friend: Organize $f'$ and $f''$ analysis clearly
Plot key points: Intercepts, extrema, inflection points
Connect the dots: Respect the increasing/decreasing and concavity info

Common Mistakes and Misunderstandings

❌ Mistake: Confusing $f'$ and $f''$ information

Wrong: "$f''(x) > 0$, so the function is increasing."

Why it's wrong: $f''$ tells you about concavity, not increasing/decreasing.

Correct: $f' > 0$ → increasing. $f'' > 0$ → concave up.

❌ Mistake: Saying inflection point without checking sign change

Wrong: "$f''(2) = 0$, so there's an inflection point at $x = 2$."

Why it's wrong: $f''$ must actually change sign at that point.

Example: $f(x) = x^4$ has $f''(0) = 0$, but $f'' > 0$ on both sides — no inflection point.

❌ Mistake: Forgetting vertical asymptotes when finding domain

Wrong: Sketching through $x = 1$ when $f(x) = \frac{1}{x-1}$.

Why it's wrong: The function doesn't exist at $x = 1$ — there's a vertical asymptote!

Correct: Always identify where the denominator is zero first.

❌ Mistake: Assuming critical point = extremum

Wrong: "$f'(c) = 0$, so there's a local max or min at $c$."

Why it's wrong: The derivative can be zero at an inflection point too.

Example: $f(x) = x^3$ has $f'(0) = 0$, but no extremum there.

Correct: Check if $f'$ changes sign (or use second derivative test).

Curve Sketching for MATH 122

Exam Relevance for MATH 122

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