How To Find Critical Numbers Of A Fraction

How To Find Critical Numbers Of A Fraction

Fractions are essential concepts in mathematics that we encounter regularly in our lives. We use fractions in various applications and calculations, such as measuring ingredients for cooking, calculating discounts, and solving equations. Finding the critical numbers of a fraction is an important concept that many students struggle with. In this article, we will discuss the steps to find the critical numbers of a fraction.

What Are Critical Numbers Of A Fraction?

Critical numbers of a fraction are the values of x that make the derivative of the numerator or denominator of the fraction equal to zero. These critical numbers are essential in determining the behavior and characteristics of the fraction, such as its local maxima and minima, inflection points, and asymptotes. By finding the critical numbers of a fraction, we can easily solve complex equations and evaluate integrals.

Steps To Find Critical Numbers Of A Fraction

The following are the steps to find the critical numbers of a fraction:

Step 1: Find The Derivative Of The Numerator And Denominator

The first step in finding the critical numbers of a fraction is to find the derivative of the numerator and denominator separately. To differentiate a fraction, we use the quotient rule, which states that the derivative of a fraction is equal to (the denominator multiplied by the derivative of the numerator minus the numerator multiplied by the derivative of the denominator) divided by the square of the denominator.

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For example, suppose we have the fraction f(x) = (x^2 – 1) / (x – 2). The derivative of the numerator, which is (x^2 – 1), is 2x, and the derivative of the denominator, which is (x – 2), is 1. Thus, the derivative of the fraction f(x) is:

f'(x) = [(x – 2) * 2x – (x^2 – 1) * 1] / (x – 2)^2

f'(x) = [2x^2 – 4x – x^2 + 1] / (x – 2)^2

f'(x) = (x^2 – 4x + 1) / (x – 2)^2

Step 2: Set The Derivatives Equal To Zero

The next step is to set the derivative of the fraction equal to zero and solve for x. The critical numbers are the values of x that make the derivative equal to zero. To solve for x, we can use factoring or the quadratic formula.

For our example, we have:

f ‘(x) = (x^2 – 4x + 1) / (x – 2)^2 = 0

x^2 – 4x + 1 = 0

Using the quadratic formula, we have:

x = [4 +/- sqrt(4^2 – 4(1)(1))] / 2

x = 2 +/- sqrt(3)

Thus, the critical numbers of the fraction f(x) = (x^2 – 1) / (x – 2) are x = 2 + sqrt(3) and x = 2 – sqrt(3).

Step 3: Check For Non-Removable Discontinuities

In some cases, the critical numbers may not be valid values for the fraction because they result in non-removable discontinuities, such as vertical asymptotes. To check for non-removable discontinuities, we can evaluate the limit of the fraction as x approaches the critical number.

For example, suppose we have the fraction g(x) = (x^2 + 3x + 2) / (x + 2). Using the derivative, we find that the critical number is x = -2. To check for non-removable discontinuities, we evaluate the limit of the fraction as x approaches -2:

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lim x->-2 g(x) = (-2) + 6

lim x->-2 g(x) = 2

Since the limit exists, there is no non-removable discontinuity at x = -2.

Step 4: Evaluate The Fraction At The Critical Numbers

After finding the critical numbers, we evaluate the fraction at these values to determine their behavior and characteristics. We can use a number line or a table to evaluate the fraction.

For example, using the fraction f(x) = (x^2 – 1) / (x – 2), we have the critical numbers x = 2 + sqrt(3) and x = 2 – sqrt(3). We evaluate the behavior of the fraction using a number line:

x < 2 - sqrt(3) | 2 - sqrt(3) < x < 2 | x > 2 + sqrt(3)
———————————————————————————–
f(x) + – + | – + | +
———————————————————————————–

From the number line, we can see that the fraction is negative when x < 2 - sqrt(3), positive when 2 + sqrt(3) < x, and undefined when x = 2. The fraction has a local minimum at x = 2 - sqrt(3) and a local maximum at x = 2 + sqrt(3).

FAQs

Q: What is the significance of finding the critical numbers of a fraction?

A: Finding the critical numbers of a fraction is significant in determining its behavior and characteristics, such as its local maxima and minima, inflection points, and asymptotes. By finding the critical numbers, we can easily solve complex equations and evaluate integrals.

Q: Can there be multiple critical numbers for a fraction?

A: Yes, a fraction can have multiple critical numbers. These critical numbers are the values of x that make the derivative of the numerator or denominator of the fraction equal to zero.

Q: How do I check for non-removable discontinuities in a fraction?

A: To check for non-removable discontinuities in a fraction, we can evaluate the limit of the fraction as x approaches a critical number. If the limit does not exist or is infinite, then there is a non-removable discontinuity at that critical number.

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In conclusion, finding the critical numbers of a fraction is an important skill that is essential in determining its behavior and characteristics. By following the steps outlined in this article, you can easily find the critical numbers of any fraction and evaluate its properties. Remember to also check for non-removable discontinuities and evaluate the function at its critical numbers to determine its behavior accurately.

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