Question

Explain how the first derivative of a function determines where the function is increasing or decreasing.

Short answer

Answer: The first derivative of a function, denoted as f'(x), helps us understand the behavior of the function. When f'(x) > 0, the function is increasing, and when f'(x) < 0, the function is decreasing. To determine where the function is increasing or decreasing, follow these steps: 1) Find the first derivative, f'(x), 2) Equate f'(x) to zero and solve for x to find the critical points, 3) Determine the intervals between critical points, and 4) Test the intervals using any value within the interval in f'(x). By analyzing the intervals, you can determine whether the function is increasing or decreasing in each interval.

Step-by-step solution

Understand the Concept of Derivative

A derivative of a function represents the rate of change of the function with respect to its independent variable. In other words, it measures how fast the function is changing at any point. Mathematically, the derivative of a function f(x) is denoted as f'(x) or $\frac{d}{dx}$(f(x)).

Know the Relation between a First Derivative and the Function's Behavior

The first derivative of a function, f'(x), can help us understand how the function behaves. It can do this by: - If f'(x) > 0, it means the function is increasing at that point, as the rate of change is positive. - If f'(x) < 0, it means the function is decreasing at that point, as the rate of change is negative. - If f'(x) = 0, the point might be a critical point like a maximum, minimum, or an inflection point.

Steps to Determine Where a Function is Increasing or Decreasing Using its First Derivative

1. Find the first derivative, f'(x), of the given function f(x). 2. Equate f'(x) to zero and solve for x to find the critical points. 3. Determine the intervals between critical points. 4. Test the intervals using any value within the interval in f'(x) to know if the function is increasing or decreasing. Let's see the process through an example.

Example

Suppose we have a function f(x) = $x^3-3x^2$. Step 1: Find the first derivative, f'(x) f'(x) = $\frac{d}{dx}$(x^3 - 3x^2) = 3x^2 - 6x. Step 2: Equate f'(x) to zero and solve for x 3x^2 - 6x = 0 x(3x - 6) = 0 x = 0 or x = 2 are the critical points. Step 3: Determine the intervals The intervals are (-∞, 0), (0, 2), and (2, +∞). Step 4: Test the intervals using any value within the interval in f'(x) - Interval (-∞, 0), let's take x = -1, f'(-1) = 3(1) - 6(-1) = 9 > 0, so the function is increasing. - Interval (0, 2), let's take x = 1, f'(1) = 3(1) - 6(1) = -3 < 0, so the function is decreasing. - Interval (2, +∞), let's take x = 3, f'(3) = 3(9) - 6(3) = 9 > 0, so the function is increasing. So, the function f(x) = $x^3-3x^2$ is increasing in the intervals (-∞, 0) and (2, +∞), and decreasing in the interval (0, 2).

Key concepts

Increasing and Decreasing Functions

To unravel the secret of increasing and decreasing functions, we must court the first derivative like an insightful partner. The first derivative swiftly tells us whether our function is scaling the uphill trek of increase or sauntering downwards in decrementation.

Imagine you're trekking and your pace, which is akin to the function's rate of change, determines if you're ascending or descending. A positive pace (derivative greater than zero) insinuates an uphill journey — the function is increasing. Conversely, when your pace reflects a negative value, it's a tell-tale sign of a downward path — the function is decreasing.

Let's ally this with the example provided. The first derivative of the function, when positive, signifies the intervals where our function cleaves through the y-axis upwards. Thus, in our intervals (-∞, 0) and (2, +∞), our mathematical hiker is indeed climbing. On the flip side, the decrepit negative pace within the (0, 2) interval suggests a downhill descent for our function.

Critical Points

Think of critical points as the crossroads or milestones of the function's journey. These are the potent points where the function's increasing or decreasing trend might just take a dramatic turn — and it all transpires when the first derivative takes a breather and hits zero or doesn't exist.

At these junctures, the function could be at a peak (a local maximum), a trough (a local minimum), or at the peculiar flatness of an inflection point. The critical points are the x-values we found by setting the first derivative to zero, such as x = 0 or x = 2 for our function. At these points, our function's graph might pivot, twist, or continue its course in an altered demeanor. It's these critical points that often hold the keys to the function's extremities and can make for exciting discussions in calculus.

Rate of Change

Like a speedometer gauges velocity, the rate of change in mathematics measures how swiftly a function's value is altering at any one point. This is the realm of the first derivative. It serves as a sentinel, marking the speed — fast or slow — at which the function is changing with respect to the input variable.

In the example, the derivative 3x^2 - 6x alters its sign as x transits through the critical points. It's an immediate whisper of whether the function hastens (increases) or slackens (decreases). By observing the sign of the function's derivative, we glean a narrative of its velocity — an allegory for its instantaneous rate of change across different intervals.

Function Intervals

Our function's journey is segmented into function intervals — sections between critical points where the function's behaviour is consistent in nature; it's either uniformly increasing or decreasing. These intervals are the corridors of certainty in calculus where the function's behavior is predictable. By specifying these intervals, we delineate the function's increasing and decreasing territories.

In the illustrative function, the intervals emerge from its cocoon of criticality at x = 0 and x = 2. Each interval is tested independently. For instance, using a test value of x = 1 in the interval (0, 2), we deduced the decreasing nature of our function. These intervals, when sketched on a graph, provide a panoramic view of the function's landscape and are like chapters in the function's epic story of ascents and descents.