Problem 31 The given limit is a derivative,... [FREE SOLUTION]

Chapter 2: Problem 31

The given limit is a derivative, but of what function and at what point? \(\lim _{x \rightarrow 3} \frac{x^{3}+x-30}{x-3}\)

Short Answer

Expert verified

The limit is the derivative of the function \(f(x) = x^3 + x - 30\) at \(x = 3\). The derivative value is 28.

Step by step solution

Analyze the Limit Expression

The given limit expression \(\lim _{x \rightarrow 3} \frac{x^{3}+x-30}{x-3}\) involves a fraction where the numerator is \(x^3 + x - 30\) and the denominator is \(x - 3\). This type of expression resembles the definition of a derivative, where the function \(f(x)\) would typically be in the numerator and \(x - c\) in the denominator.

Recognize the Structure of a Derivative

The limit \(\lim_{x \to c} \frac{f(x) - f(c)}{x-c}\) equates to \(f'(c)\), the derivative of the function \(f(x)\) at the point \(x = c\). Here, \(c = 3\) and \(f(x) = x^3 + x - 30\). We need to find \(f(c)\) which corresponds to evaluating \(f(3)\).

Calculate \(f(3)\)

To find \(f(3)\), substitute \(x = 3\) into \(f(x) = x^3 + x - 30\): \(f(3) = 3^3 + 3 - 30 = 27 + 3 - 30 = 0.\) Hence, \(f(3) = 0\).

Interpret the Limit

Thus, the limit \(\lim _{x \to 3} \frac{x^3 + x - 30}{x-3}\) simplifies to the derivative of \(f(x) = x^3 + x - 30\) at \(x = 3\). Therefore, this limit is the derivative \(f'(3)\).

Find the Derivative \(f'(x)\)

Find the derivative of \(f(x) = x^3 + x - 30\) using the power rule: \(f'(x) = 3x^2 + 1\).

Evaluate \(f'(3)\)

Substitute \(x = 3\) into \(f'(x) = 3x^2 + 1\): \(f'(3) = 3(3)^2 + 1 = 27 + 1 = 28\). Thus, \(f'(3) = 28\).

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limits

In calculus, limits help us understand the behavior of a function as it approaches a certain value. Specifically, they provide a foundation for defining continuity, derivatives, and integrals. In simpler terms, a limit tells us where a function is headed as the input gets closer to a particular point.
For example, the expression \( \lim_{x \rightarrow 3} \frac{x^3+x-30}{x-3} \) explores what happens to the fraction as \( x \) approaches 3 from either direction.

When working with limits, one often encounters indeterminate forms like \( \frac{0}{0} \), which requires manipulation to solve correctly, such as factoring, rationalization, or using l'Hopital's rule.
In this particular exercise, recognizing the limit as a derivative in disguise helps simplify the analysis, as we manipulate it to calculate \( f'(3) \).

Derivatives

Derivatives are a core concept in calculus, capturing the idea of the rate at which a function changes. We often think of derivatives as the slope of a tangent line to the curve at a given point. The derivative at a specific point tells us the instantaneous rate of change of the function.
Using the limit definition, the derivative \( f'(c) \) is represented as \( \lim_{x \to c} \frac{f(x)-f(c)}{x-c} \), which reveals its dependence on limits.

In practice, recognizing this form in expressions can quickly identify what function and point are involved, as seen in the solution where \( f(x) = x^3 + x - 30 \) and the point \( c = 3 \).
To find the derivative, we analyze the rate of change using algebraic manipulation or calculus methods like the power rule.

Power Rule

The power rule is a straightforward technique for finding the derivative of powers of \( x \). It's an efficient shortcut that makes differentiating polynomials particularly easy. The rule states that if you have a function \( f(x) = x^n \), its derivative \( f'(x) \) is given by \( nx^{n-1} \).
In the exercise, the function \( f(x) = x^3 + x - 30 \) utilizes the power rule to calculate the derivative quickly:

For the term \( x^3 \), apply the rule: the derivative is \( 3x^2 \).
The derivative of \( x \) using the power rule is simply \( 1 \), since \((1)x^{1-1} = 1\).
Constant terms, like \(-30\), have a derivative of zero as they do not change.

Using these steps, the derivative \( f'(x) \) becomes \( 3x^2 + 1 \), which is then evaluated at \( x = 3 \) to yield \( f'(3) = 28 \).
This method is not only helpful for simple functions but also serves as a building block for understanding more complex derivatives.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

The given limit is a derivative, but of what function and at what point? \(\lim _{x \rightarrow 3} \frac{x^{3}+x-30}{x-3}\)

Short Answer

Step by step solution

Analyze the Limit Expression

Recognize the Structure of a Derivative

Calculate \(f(3)\)

Interpret the Limit

Find the Derivative \(f'(x)\)

Evaluate \(f'(3)\)

Key Concepts

Limits

Derivatives

Power Rule

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Mechanics Maths

Applied Mathematics

Decision Maths

Discrete Mathematics

Geometry

Study anywhere. Anytime. Across all devices.