Understanding the Calculation of f(3 h) - f(3) / h with f(x) 2x3

When working with functions, one of the fundamental concepts we often encounter is the difference quotient. This expression, f(3 h) - f(3) / h, is indicative of how a function changes as its input changes. For the function f(x) 2x3, let's explore how to calculate this specific difference quotient step-by-step, and understand its significance.

Step 1: Evaluate f(3)

To start, we need to evaluate the function at x 3. Given the function f(x) 2x3, we substitute x 3 into the equation.

Calculation: f(3) 2(3)3 2(27) 54

Step 2: Evaluate f(3 h)

Now, we need to evaluate the function at a slightly different point, x 3 h. Let's substitute x 3 h into the function.

Calculation: f(3 h) 2(3 h)3 2(27 27h 9h2 h3) 54 54h 18h2 2h3

Step 3: Compute f(3 h) - f(3)

Next, we subtract f(3) from f(3 h) to find the difference between these two values.

Calculation: f(3 h) - f(3) (54 54h 18h2 2h3) - 54 54h 18h2 2h3

Step 4: Divide by h

To find the difference quotient, we then divide the result from Step 3 by h.

Calculation: (f(3 h) - f(3)) / h (54h 18h2 2h3) / h 54 18h 2h2

Simplification

As h approaches 0, the terms involving h (18h and 2h2) will approach 0. Therefore, the simplified form of the expression (f(3 h) - f(3)) / h as h approaches 0 is 54.

Understanding the Limit as h Approaches 0

The limit as h approaches 0 of (f(3 h) - f(3)) / h gives us the instantaneous rate of change of the function f(x) 2x3 at x 3. This is the derivative of the function evaluated at x 3.

Derivative of f(x) 2x3

The derivative of f(x) 2x3 is given by the power rule, which states that if f(x) axn, then f'(x) anxn-1. Applying this rule, we get:

Calculation: f'(x) 6x2

Evaluating this derivative at x 3, we get:

Calculation: f'(3) 6(3)2 6(9) 54

Conclusion: Direct Computation

Alternatively, you might be tempted to think of it as a simpler approach involving basic arithmetic. Considering:

Calculation: (f(3 h) - f(3)) / h (2(3 h)3 - 2(3)32 2h3 - 54) / h (54h 18h2 2h3) / h 54 18h 2h2

As h approaches 0, this simplifies to 54, which is equivalent to f'(3).

Key Concepts and Further Exploration

The calculation of (f(3 h) - f(3)) / h introduces us to the concept of derivatives and the fundamental limit definition of derivatives. It is also closely related to the concept of instantaneous rate of change and the slope of a tangent line.

Relevant Keywords and Terms:

derivative calculation function evaluation limit definition

Conclusion

In summary, the calculation of (f(3 h) - f(3)) / h for the function f(x) 2x3 leads us to a deeper understanding of the derivative concept. This process not only reinforces the understanding of limits but also provides insights into the behavior of functions at specific points. By examining these calculations and concepts, you can better appreciate the foundational principles of calculus and their applications.