Question

\(\triangle\) 52. Find the difference quotient of \(f: f(x)=3 x^{2}-5 x\) \(\Delta\)

Short answer

The difference quotient is \(6x - 5\).

Step-by-step solution

- State the Definition

The difference quotient of a function is given by the formula: \[\frac{f(x+h) - f(x)}{h}\] where \(f(x)\) is the original function and \(h\) is a small difference in \(x\).

- Find \(f(x+h)\)

Substitute \(x+h\) into the function \(f(x) = 3x^2 - 5x\) to find \(f(x+h)\): \[f(x+h) = 3(x+h)^2 - 5(x+h)\]

- Expand \(f(x+h)\)

Expand the expression: \[f(x+h) = 3(x^2 + 2xh + h^2) - 5(x + h) = 3x^2 + 6xh + 3h^2 - 5x - 5h\]

- Form the Difference Quotient

Substitute \(f(x)\) and \(f(x+h)\) into the difference quotient formula: \[\frac{f(x+h) - f(x)}{h} = \frac{(3x^2 + 6xh + 3h^2 - 5x - 5h) - (3x^2 - 5x)}{h}\]

- Simplify the Expression

Simplify the expression in the numerator: \[\frac{3x^2 + 6xh + 3h^2 - 5x - 5h - 3x^2 + 5x}{h} = \frac{6xh + 3h^2 - 5h}{h}\]

- Factor and Final Simplification

Factor out \(h\) from the numerator and then cancel \(h\) from the denominator: \[\frac{h(6x + 3h - 5)}{h} = 6x + 3h - 5\]

- Conclusion

As \(h\) approaches 0, the difference quotient becomes: \[6x - 5\]

Key concepts

Algebra

Algebra lays the groundwork for understanding the difference quotient. It involves manipulating expressions and solving for variables. In our problem with the function \(f(x) = 3x^2 - 5x\), we apply some fundamental algebraic operations.
First, we substitute \(x + h\) into the function to get \(f(x + h)\).
This part involves expanding the expression \(3(x + h)^2 - 5(x + h)\) using the distributive property:

• Step 2 shows how we substitute \(x + h\) into the function\(
  \)
• Step 3 breaks down how to expand \(3(x + h)^2 - 5(x + h)\), using basic algebraic manipulation.
  

These algebraic steps are crucial because they allow us to simplify complex expressions. Remember, understanding how to distribute and combine like terms is key in making progress toward solving the difference quotient.

Calculus Fundamentals

The difference quotient is a central concept in calculus.
It's the starting point for understanding derivatives, which measure how a function changes. Essentially, the difference quotient \(\frac{f(x+h) - f(x)}{h}\) represents the average rate of change of the function over the interval \[x, x+h\].

• Step 4 moves us closer to understanding derivatives by substituting \(f(x)\) and \(f(x+h)\) into the difference quotient formula.
• Step 5 simplifies the expression, which involves canceling like terms and factoring out \(h\).
  

The final result shows the expression \(6x + 3h - 5\) and, as \(h\) approaches 0, the difference quotient simplifies to \(6x - 5\), linking directly to the concept of the derivative. Knowing these steps will help you grasp the fundamental principles of calculus.

Function Analysis

Analyzing functions is another critical aspect of understanding the difference quotient. By studying how a function behaves, we can predict how changes in \(x\) will affect \(f(x)\).
In our exercise, the function \(f(x) = 3x^2 - 5x\) is quadratic and behaves predictably.

• When evaluating \(f(x+h)\), we notice it expands into terms involving both \(x\) and \(h\). This step requires us to consider each term's contribution and its simplification.
• The final step, which results in \(6x - 5\), shows how small changes in \(x\) affect the value of \(f(x)\). This directly ties into understanding the slope of the function at any given point.
  

Function analysis allows us to visualize and comprehend the graphical behavior of the function, making the link between the difference quotient and the derivative clearer.