Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use \(f^{\prime}(x)=\lim _{t \rightarrow x}[f(t)-f(x)] /[t-x]\) to find \(f^{\prime}(x)\) (see Example 5). $$ f(x)=x^{2}-3 x $$

Step-by-Step Solution

Answer

The derivative is \( f^{\prime}(x) = 2x - 3 \).

Step 1: Identify the Definition

The derivative of a function \( f(x) \) at any point \( x \) is given by the limit: \[ f^{\prime}(x) = \lim_{t \to x} \frac{f(t) - f(x)}{t - x} \] Here, \( f(x) = x^2 - 3x \). Our task is to find \( f^{\prime}(x) \).

Step 2: Substitute the Function into the Limit Definition

Substitute \( f(t) = t^2 - 3t \) and \( f(x) = x^2 - 3x \) into the derivative definition: \[ f^{\prime}(x) = \lim_{t \to x} \frac{(t^2 - 3t) - (x^2 - 3x)}{t - x} \]

Step 3: Simplify the Numerator

Simplify the expression in the numerator: \[ t^2 - 3t - (x^2 - 3x) = t^2 - 3t - x^2 + 3x \] This can be rewritten as: \[ (t^2 - x^2) + (-3t + 3x) \] Using the identity \( a^2 - b^2 = (a-b)(a+b) \), we rewrite \( t^2 - x^2 \) as \((t-x)(t+x)\).

Step 4: Factor the Expression

Continue by factoring the expression in the numerator: \[ t^2 - x^2 = (t-x)(t+x) \] Combine this with the other terms: \[ (t-x)(t+x) + (-3)(t-x) = (t-x)(t+x-3) \]

Step 5: Simplify the Fraction and Take the Limit

Substitute the factorized form back into the limit definition and simplify: \[ f^{\prime}(x) = \lim_{t \to x} \frac{(t-x)(t+x-3)}{t-x} \] The \( (t-x) \) terms cancel out: \[ f^{\prime}(x) = \lim_{t \to x} (t+x-3) \]

Step 6: Evaluate the Limit

Since the terms \( (t-x) \) have canceled, plug \( t = x \) into the remaining expression: \[ f^{\prime}(x) = x + x - 3 = 2x - 3 \]

Step 7: Conclusion: Derivative Function

Therefore, the derivative function is \( f^{\prime}(x) = 2x - 3 \).

Key Concepts

Limit Definition

The limit definition of a derivative is a foundational concept in calculus. It allows us to find the rate at which a function changes at any given point, otherwise known as the derivative. The derivative of a function \( f(x) \) at a specific point \( x \) is given by:

\[ f^{\prime}(x) = \lim_{t \to x} \frac{f(t) - f(x)}{t-x} \]

This formula captures the idea of the slope of the tangent line of the function at the point \( x \). The expression \( f(t) - f(x) \) represents the change in the function values, and \( t-x \) is the change in the inputs. The limit examines what happens to this ratio as \( t \) approaches \( x \). This approach is essential because it handles cases where a simple difference quotient would become undefined.

• The limit helps to "zoom in" on the point and find the instantaneous rate of change.
• Understanding and applying limit definition is crucial for finding derivatives in calculus.

Polynomial Function

A polynomial function is a mathematical expression involving sums and powers of variables with non-negative integer exponents. The general form of a polynomial is \( a_nx^n + a_{n-1}x^{n-1} + \, \ldots \, + a_1x + a_0 \), where \( a_n, a_{n-1}, \, \ldots \, , a_1, a_0 \) are constants, and \( n \) is a non-negative integer. In our original exercise, the function \( f(x) = x^2 - 3x \) is an example of a polynomial function.Polynomial functions have several characteristics:

• They are continuous and smooth curves.
• The degree of the polynomial determines its number of roots and the number of times it can cross the x-axis.
• They are easy to differentiate compared to other types of functions.

Understanding polynomial functions is important because they frequently appear in calculus problems, especially when applying the limit definition to find derivatives.

Simplifying Expressions

Simplifying expressions involves rewriting them in a more efficient or recognizable form. In calculus, simplifying expressions is an important skill that makes it easier to solve problems, especially when finding derivatives using the limit definition.In the given problem:

• We begin by expressing \((t^2 - 3t) - (x^2 - 3x)\).
• This simplifies first to \(t^2 - 3t - x^2 + 3x\) which can further be broken down to \((t^2 - x^2) + (-3t + 3x)\).

By breaking down the components, we make it easier to see how factoring can be applied, such as recognizing a difference of squares, which then leads to canceling of terms that make the limit solvable.

Factoring Expressions

Factoring expressions is a key technique in algebra and calculus that entails breaking down larger expressions into simpler, multiplied components. This is particularly useful when simplifying fractions or expressions in complex calculations.In our derivative calculation:

• The expression \( t^2 - x^2 \) was rewritten as \((t-x)(t+x)\) using the difference of squares.
• The terms combined as \((t-x)(t+x-3)\) by factoring out \(t-x\).

Factoring allows us to cancel out terms in the numerator and denominator, simplifying the limit process. Once terms are canceled, the expression's limit becomes straightforward, which is essential in deriving the derivative efficiently. Factoring helps avoid complexities and reduces the equation to a point where direct substitution is possible, allowing us to smoothly evaluate the limit and find the derivative.