Problem 8 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use \(f^{\prime}(x)=\lim _{h \rightarrow 0}[f(x+h)-f(x)] / h\) to find the derivative at \(x\). $$ f(x)=x^{2}+x+1 $$

Step-by-Step Solution

Answer

The derivative of \(f(x) = x^2 + x + 1\) is \(f'(x) = 2x + 1\).

Step 1: Define the Difference Quotient

Start by substituting the given function \(f(x) = x^2 + x + 1\) into the difference quotient:\[f(x+h) - f(x) = [(x+h)^2 + (x+h) + 1] - [x^2 + x + 1].\]

Step 2: Expand \((x+h)^2\)

Expand the expression \((x+h)^2\):\[(x+h)^2 = x^2 + 2xh + h^2.\]This will be used in the difference quotient.

Step 3: Substitute and Simplify

Substitute the expansion from Step 2 into the difference quotient:\[f(x+h) - f(x) = [x^2 + 2xh + h^2 + x + h + 1] - [x^2 + x + 1].\]Simplify by canceling out common terms:\[f(x+h) - f(x) = 2xh + h^2 + h.\]

Step 4: Formulate the Difference Quotient

Substitute the simplified expression into the difference quotient formula:\[\frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 + h}{h}.\]Simplify by dividing every term by \(h\):\[\frac{2xh}{h} + \frac{h^2}{h} + \frac{h}{h} = 2x + h + 1.\]

Step 5: Take the Limit as \(h \rightarrow 0\)

Apply the limit to the simplified difference quotient:\[\lim_{h \rightarrow 0} (2x + h + 1) = 2x + 0 + 1.\]Thus, \[f^{\prime}(x) = 2x + 1.\]

Key Concepts

Understanding the Difference Quotient

To find the derivative of a function, we first need to understand the difference quotient. This quotient helps us evaluate how a function changes as the input changes. Picture it as finding the slope of the line that connects two points on a curve. These points are very close to each other. We start with our function:

• Given function: \( f(x) = x^2 + x + 1 \)
• Difference quotient formula: \( \frac{f(x+h) - f(x)}{h} \)

Plugging in our given function into this formula, we get:\[\frac{[(x+h)^2 + (x+h) + 1] - [x^2 + x + 1]}{h}\]The difference quotient thus prepares us to explore the behavior of the function as the variable \( h \) approaches zero.

Limit Process and Its Role

The limit process is a cornerstone in calculus, especially in the context of derivatives. It helps us understand what happens as we zoom very close to a specific point. Here, we use the limit to comprehend the behavior of the difference quotient as \( h \) approaches zero. When we calculate the derivative, this process ensures we're looking at an instantaneous rate of change rather than an average over a distance.
For instance, once we simplify the difference quotient for our function \( f(x) = x^2 + x + 1 \), it is expressed as:\[\frac{2xh + h^2 + h}{h} = 2x + h + 1\]Using the limit, we then compute:\[\lim_{h \to 0} (2x + h + 1) = 2x + 1\]This result is the derivative, a measure of how \( f(x) \) changes at any given point \( x \).

Diving Deeper into Differentiation

Differentiation is the process of finding a derivative, which tells us how a function changes at any point. By using it, we get insights into the function's behavior, like its slope or how it might curve. It all starts with the difference quotient and ends with applying the limit process to find the derivative.

• Formula: \( f^{\prime}(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \)
• Result for our function: \( f^{\prime}(x) = 2x + 1 \)

Here, \( 2x + 1 \) is the derivative of \( x^2 + x + 1 \). It signifies how steep the function's graph is at every point. Differentiation is like a magnifying glass, showing minute changes over infinitesimal intervals.

Exploring Polynomial Functions

Polynomial functions are equations made up of terms that consist of variables raised to whole number powers and their coefficients. Our function, \( f(x) = x^2 + x + 1 \), is a polynomial. Understanding the structure of polynomials is crucial since they often come up in calculus, and many functions can be approximated by polynomials.
Polynomials have straightforward rules when it comes to differentiation:

• The derivative of \( x^n \) is \( nx^{n-1} \).
• Simpler rules for the addition/subtraction of differentiable functions.

In the case of our polynomial, the differentiation resulted in \( f^{\prime}(x) = 2x + 1 \), which was arrived at by using these basic polynomial differentiation rules. Polynomials are generally smooth, continuous functions, making differentiation an effective tool in analyzing their changes.