Calculus Question: Determining intervals

In summary, the intervals of increase and decrease for this function are: decreasing at x < -1 and x > 3, and increasing at -1 < x < 3.
  • #1
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Greetings,

I have a question about determining intervals of increase or decrease on a given function. For example...

f(x) = (X-1)/(X^2+3)

The next step is to get the derivative, which is...

f'(x) = ((X^2 + 3) - 2X^2 + 2X))/(X^2+3)^2

Then, you set that equal to zero and solve for X, and f'(x) = 0 at -1 and 3. So now, my interval is -1 to 3. According to my textbook, I should then choose a numer (c) on the numberline, plug it into f'(c'), and if it is less than zero than it is decreasing, and if is greater than zero it is increasing. However, according to the answer key for this particular problem, the interval is descreasing at x < -1 and x > 3, and increasing at -1 < x < 3. What am I not understanding? Thanks for any help.

Edit: I show that it is increasing when x < -1, increasing when x is between -1 and 3, and increasing when x > 3

Edit^2: Nevermind... I tried to simplify f'(x) into (-X^2 + 2X + 3)/(X^2+3)^2 and that's what was given me problems.
 
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  • #2
What you are not understanding is that the derivative of your function tells you what is happening to the function in the intervals around each point. For example, when x < -1, the derivative is negative, which means the function is decreasing in that interval. Similarly, when x is between -1 and 3, the derivative is positive, so the function is increasing in that interval. Finally, when x > 3, the derivative is negative again, so the function is decreasing in that interval.
 

1. What is the purpose of determining intervals in calculus?

The purpose of determining intervals in calculus is to understand the behavior of a function and its rate of change over a specific range of values. This allows us to identify important features of the function, such as its maximum and minimum values, and to analyze its behavior and trends.

2. How do you determine intervals in calculus?

To determine intervals in calculus, we first find the critical points of the function by setting its derivative equal to zero. Then, we use these critical points to create a number line and test points in each interval to see if the function is increasing or decreasing. The intervals where the function is increasing are called "increasing intervals" and the intervals where the function is decreasing are called "decreasing intervals".

3. Can you give an example of determining intervals in calculus?

Sure, let's say we have the function f(x) = x^2 - 3x + 2. To determine intervals, we first find the critical points by setting the derivative equal to zero: f'(x) = 2x - 3 = 0. This gives us x = 3/2 as the only critical point. Then, we test points on either side of this critical point (e.g. x = 0, x = 2) and see if the function is increasing or decreasing in those intervals. In this case, we find that the function is increasing on the interval (-∞, 3/2) and decreasing on the interval (3/2, ∞).

4. Why is it important to determine intervals in calculus?

Determining intervals in calculus is important because it allows us to understand the behavior of a function and to make predictions about its future values. This is particularly useful in real-world applications such as economics, physics, and engineering, where we need to analyze and optimize functions to solve problems.

5. Are there any shortcuts or tricks for determining intervals in calculus?

While there are no shortcuts for determining intervals in calculus, there are some strategies that can make the process easier. These include using a graphing calculator or software to visualize the function, and understanding the behavior of common functions (e.g. polynomials, exponential and logarithmic functions) which can help in identifying the intervals. Additionally, with practice and familiarity, the process of determining intervals can become quicker and more intuitive.

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