Welcome back to Professor Baker's Math Class! In Section 4-3, we are bridging the gap between algebraic calculus and geometric graphs. Until now, we’ve calculated derivatives as abstract rates of change. Now, we are going to use them to become detectives, deducing exactly what a function looks like without needing a graphing calculator.

The First Derivative: Slope and Direction

The first derivative, $f'(x)$, tells us about the direction of the curve. It answers the question: Is the function going uphill or downhill?

  • Critical Numbers: These are the $x$-values where interesting things happen. A critical number $c$ exists where $f'(c) = 0$ (horizontal tangent) or where $f'(c)$ does not exist (like a sharp corner).
  • Increasing/Decreasing Test:
    • If $f'(x) > 0$ on an interval, the function is increasing.
    • If $f'(x) < 0$ on an interval, the function is decreasing.

The First Derivative Test

Once we identify critical numbers, we can test the intervals between them to find local maximums and minimums. Think of walking along the graph:

  • If $f'$ changes from positive ($+$) to negative ($-$), you've walked up a hill and are now going down. That peak is a Local Maximum.
  • If $f'$ changes from negative ($-$) to positive ($+$), you've walked down a valley and are starting to climb. That bottom is a Local Minimum.

Worked Example from Class

Let's look at the polynomial function from our notes:

$$f(x) = 3x^4 - 4x^3 - 12x^2 + 5$$

To find where it is increasing or decreasing, we take the derivative:

$$f'(x) = 12x^3 - 12x^2 - 24x$$

Factoring out $12x$, we get:

$$f'(x) = 12x(x-2)(x+1)$$

Setting this to zero gives us our critical numbers: $x = 0$, $x = 2$, and $x = -1$.

By testing values between these points (creating a sign chart), we found:

  • $x = -1$ is a Local Minimum (slope went $-$ to $+$).
  • $x = 0$ is a Local Maximum (slope went $+$ to $-$).
  • $x = 2$ is a Local Minimum (slope went $-$ to $+$).

The Second Derivative: Concavity

While $f'$ tells us direction, the second derivative, $f''(x)$, tells us about the curvature or "concavity" of the graph.

  • Concave Up ($f''(x) > 0$): The graph is shaped like a cup (holds water). Slopes are increasing.
  • Concave Down ($f''(x) < 0$): The graph is shaped like a frown (spills water). Slopes are decreasing.

We can even use the Second Derivative Test as a shortcut! If you have a critical number $c$ where $f'(c)=0$:

  • If $f''(c) > 0$, the graph is concave up, so $c$ is a Local Minimum.
  • If $f''(c) < 0$, the graph is concave down, so $c$ is a Local Maximum.

Putting It All Together

Curve sketching is about combining these clues. By looking at limits (asymptotes), the first derivative (increase/decrease), and the second derivative (concavity), you can draw highly accurate graphs of complex functions—even ones involving trigonometric or exponential terms like $xe^{bx^2}$.

Keep practicing those sign charts, and remember: the derivative is your map to the shape of the function!