Section 4.5

The Shape of a Graph I

The derivative is a compass. It tells us which way is "up." By analyzing the sign of , we can reconstruct the rising and falling intervals of the original function without plotting points.

1

Sign Maps & Direction

Increasing Function

Tangent slopes are positive (uphill).

Decreasing Function

Tangent slopes are negative (downhill).

The First Derivative Test allows us to classify critical points:

  • If changes from Positive to Negative, it's a Local Max.
  • If changes from Negative to Positive, it's a Local Min.
2

Worked Example

Analyzing f(x) = x³ - 3x

Find the intervals of increase/decrease and local extrema.

Derivative Sign Map

Top: Function f(x) | Bottom: Derivative f'(x)

Notice: When the bottom graph (derivative) is Above its axis, the top graph is Rising.
1. Critical Points

CPs at .

2. Sign Map (Test Intervals)
(-∞, -1)
Pos (+)
Inc ↗
(-1, 1)
Neg (-)
Dec ↘
(1, ∞)
Pos (+)
Inc ↗
3. Conclusion
  • Local Max at (Up to Down)
  • Local Min at (Down to Up)
3

Level Up Examples

Example A: Rational Function

Analyze .

1. Differentiate:
2. Critical Points & Asymptotes:

at .
undef at (VA).

3. Signs of f':

(Denominator is always positive!)

  • : Pos (Inc)
  • : Neg (Dec) → Max at 0
  • : Neg (Dec)
  • : Pos (Inc) → Min at 2

Example B: Exponential Product

Find extrema of .

1. Product Rule:
2. Critical Point:
is never 0.
.
3. Test:
  • (e.g., 0): (Inc)
  • (e.g., 2): (Dec)

Local Max at .

5

Practice Quiz

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