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Chapter 1: Problem 89

Find \(\frac{f(x+h)-f(x)}{h}\) for the indicated functions. $$ f(x)=3 x+1 $$

Short Answer

Expert verified
The answer is 3.

Step by step solution

01

Understand the Given Function

The given function is a linear function defined as \( f(x) = 3x + 1 \). We are asked to find the expression \( \frac{f(x+h)-f(x)}{h} \), which is the difference quotient and often used to find the derivative.
02

Substitute in the Difference Quotient Formula

First substitute \( f(x) \) and \( f(x+h) \) into the difference quotient formula. We compute \( f(x+h) = 3(x+h) + 1 \). The difference quotient is given by:\[\frac{f(x+h)-f(x)}{h} = \frac{3(x+h) + 1 - (3x + 1)}{h}\]
03

Simplify the Inside of the Fraction

Simplify the numerator of the expression by expanding and combining like terms:\[3(x+h) + 1 = 3x + 3h + 1\]Substitute back into the expression:\[\frac{3x + 3h + 1 - 3x - 1}{h}\]
04

Cancel Out Terms

In the numerator, the \(3x\) terms and the constant \(1\) cancel out:\[\frac{3x + 3h + 1 - 3x - 1}{h} = \frac{3h}{h}\]
05

Simplify the Fraction

Simplify the fraction by canceling \(h\) in the numerator and the denominator:\[\frac{3h}{h} = 3\]
06

Conclusion

The value of \( \frac{f(x+h)-f(x)}{h} \) as derived from the steps is 3. This represents the derivative of the function \(f(x) = 3x + 1\).

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Key Concepts

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

Linear Function
A linear function is perhaps the simplest type of function. It is a polynomial function of degree one, and its graph is a straight line. In the function equation, the highest power of the variable is one. Our linear function here is defined as \( f(x) = 3x + 1 \). This gives us a slope of 3 and a y-intercept of 1.
Understanding the basics of a linear function helps in predicting and analyzing its graph.
  • The coefficient of \( x \) in the function, here 3, is the slope, which indicates how steeply the line rises or falls.
  • The constant term, 1, is the y-intercept, the point where the line crosses the y-axis.
Linear functions are fundamental in many mathematical concepts due to their simplicity and structure.
Derivative
A derivative represents the rate of change of a function with respect to its variable. The difference quotient, \( \frac{f(x+h)-f(x)}{h} \), is critical for finding this rate of change. It describes how a function's output value changes as the input changes.
In our case, the function \( f(x) = 3x + 1 \) is linear. Interestingly, the derivative of a linear function is constant. This means its rate of change does not vary across its domain.
  • For \( f(x) = 3x + 1 \), the derivative \( f'(x) = 3 \) reflects a constant rate of change or slope of 3.
  • The concept of derivatives is used widely in calculus to explore changes within various functions.
The derivative from the difference quotient is particularly useful because it provides a tangent line's slope at a point, giving insight into the function's curve behavior at that point.
Function Simplification
Function simplification is an essential step in solving mathematical problems and understanding expressions. It involves reducing complex expressions into simpler forms.
This simplification allows easier manipulation and calculation. In our problem, we simplified the difference quotient to find the derivative of the function.
  • Initially, we expanded \( f(x+h) = 3(x+h) + 1 \) to \( 3x + 3h + 1 \).
  • Next, we subtracted \( f(x) = 3x + 1 \) from this, leading to 3h.
  • Finally, by dividing by \( h \), we simplified the expression to just 3.
Simplification not only makes the function easier to work with but also aids in concluding problems effectively. Recognizing patterns for simplification is crucial in mathematics.

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