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Chapter 2: Problem 47

Simplify the difference quotient \(\frac{f(2+h)-f(2)}{h}\) if \(h \neq 0\). $$f(x)=x^{2}-3 x$$

Short Answer

Expert verified
The simplified difference quotient is \(h + 1\).

Step by step solution

01

Substitute x with (2 + h) in f(x)

First, substitute \(x\) with \(2+h\) in the function \(f(x) = x^2 - 3x\). This means calculating \(f(2+h)\):\[ f(2+h) = (2+h)^2 - 3(2+h) \]. Expand the expressions to simplify.
02

Simplify f(2+h)

Now, simplify the expression that we formulated in the previous step: \[ (2+h)^2 = 4 + 4h + h^2 \ \text{and} \ 3(2+h) = 6 + 3h \]. Substitute these back, we have: \[ f(2+h) = 4 + 4h + h^2 - (6 + 3h) \]. Simplify further by combining like terms: \[ f(2+h) = h^2 + h - 2 \].
03

Calculate f(2)

Next, determine \(f(2)\) by substituting \(x\) with \(2\) in \(f(x) = x^2 - 3x\). Thus: \[ f(2) = 2^2 - 3 \times 2 = 4 - 6 = -2 \].
04

Form the difference quotient

Now, form the difference quotient: \(\frac{f(2+h)-f(2)}{h}\). Substitute \(f(2+h) = h^2 + h - 2\) and \(f(2) = -2\) into the quotient expression: \[ \frac{f(2+h) - f(2)}{h} = \frac{(h^2 + h - 2) - (-2)}{h} \]. Simplify the expression in the numerator.
05

Simplify the difference quotient

Simplify the numerator: \((h^2 + h - 2) - (-2) = h^2 + h - 2 + 2 = h^2 + h\). Now the difference quotient simplifies to: \[ \frac{h^2 + h}{h} \]. Factor out \(h\) from the numerator: \[ \frac{h(h+1)}{h} \]. Since \(h eq 0\), \(h\) cancels out: \[ h+1 \].

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Key Concepts

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

Function Simplification
Function simplification is a critical skill in mathematics that helps make complex expressions more manageable. In this exercise, the expression \( f(2+h) = (2+h)^2 - 3(2+h) \) was simplified step-by-step.
  • Substitute: Replace variables in the function to determine values like \( f(2+h) \).
  • Expand: Spread multiplied terms, such as \((2+h)^2\) becomes \(4 + 4h + h^2\).
  • Combine like terms: Gather similar terms to reduce and simplify, leading us to \( h^2 + h - 2 \).
Breaking expressions down into these steps reduces complexity and confusion, making mathematical problems easier to solve.
Polynomial Functions
Polynomial functions consist of terms with variables raised to whole-number exponents, like \(x^2 - 3x\). Recognizing the structure of polynomial functions makes them easier to manipulate.
  • Terms: The separate parts like \(x^2\) or \(-3x\) are terms. Here, \(x^2\) is quadratic, and \(-3x\) is linear.
  • Degrees: The highest degree (power of x) in the polynomial guides the function's nature. In our case, it's 2 for \(x^2\).
  • Operations: Addition, subtraction, and multiplication are used to combine terms as seen in simplifying \((2 + h)^2\).
By understanding terms and degrees, we can predict the behavior of polynomial functions, such as symmetry and growth rates.
Limit Process
A crucial calculus concept, the limit process often evaluates functions as variables approach certain values. For difference quotients, it underpins derivative definitions.When we compute the difference quotient \( \frac{f(2+h)-f(2)}{h} \), we're exploring how \( f(x) \) changes at \( x=2 \).
  • The quotient represents an average rate of change over \(h\), which we aim to shrink towards zero for exact change.
  • You simplify the quotient and cancel variables like \(h\) (given \(h eq 0\)) to resolve indeterminate forms.
  • The final expression \(h+1\) is what remains as \(h\) approaches zero, critical for calculating derivatives.
Understanding limits allows you to handle instantaneous change, leading to deeper calculus insights.

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