Calc 1: Section 4.3 - What Derivatives Tell Us

Welcome back to Professor Baker's Math Class! In this section, we're diving deep into the power of derivatives. Specifically, we'll uncover how the first derivative, $f'(x)$, and the second derivative, $f''(x)$, can tell us a great deal about the original function, $f(x)$. Get ready to analyze increasing/decreasing intervals, concavity, and local extrema!

Increasing and Decreasing Intervals

The sign of the first derivative, $f'(x)$, provides immediate insight into whether a function is increasing or decreasing. Remember these key relationships:

  • If $f'(x) > 0$ on an interval, then $f(x)$ is increasing on that interval.
  • If $f'(x) < 0$ on an interval, then $f(x)$ is decreasing on that interval.

To find these intervals, we first need to identify the critical numbers of $f(x)$. A critical number, $c$, is a value in the domain of $f$ where either $f'(c) = 0$ or $f'(c)$ does not exist. These critical numbers divide the domain into intervals that we can test to determine where the function is increasing or decreasing.

Example Find where the function $f(x) = 3x^4 - 4x^3 - 12x^2 + 5$ is increasing and decreasing.

First, find the derivative: $f'(x) = 12x^3 - 12x^2 - 24x$. Then, set the derivative equal to zero and solve for x:

$$0 = 12x(x^2 - x - 2) = 12x(x-2)(x+1)$$

This gives us critical points at $x = -1, 0, 2$. Now, we can test values in the intervals $(-\infty, -1)$, $(-1, 0)$, $(0, 2)$, and $(2, \infty)$ to determine the sign of $f'(x)$ in each interval.

The First Derivative Test

The First Derivative Test helps us classify critical points as local maximums or local minimums. Consider a critical number, $c$, of a continuous function, $f$.

  • If $f'(x)$ changes from positive to negative at $c$, then $f$ has a local maximum at $c$.
  • If $f'(x)$ changes from negative to positive at $c$, then $f$ has a local minimum at $c$.
  • If $f'(x)$ does not change sign at $c$, then $f$ has neither a local maximum nor a local minimum at $c$.

Concavity and the Second Derivative

The second derivative, $f''(x)$, tells us about the concavity of the function's graph:

  • If $f''(x) > 0$ on an interval, then the graph of $f$ is concave upward on that interval. This means the function is curving upwards.
  • If $f''(x) < 0$ on an interval, then the graph of $f$ is concave downward on that interval. This means the function is curving downwards.

The Second Derivative Test

The Second Derivative Test offers another way to find local maximums and minimums. Suppose $f''$ is continuous near $c$ and $f'(c) = 0$:

  • If $f''(c) > 0$, then $f$ has a local minimum at $c$.
  • If $f''(c) < 0$, then $f$ has a local maximum at $c$.

Keep practicing, and you'll become a master of derivatives in no time!