Problem 51 - Calculus

Question

Estimate the value of \(f^{\prime}(1)\) by substitutang small values for \(h .\) Then use algebra to find \(f^{\prime}(1)\) exactly.$$f(x)=3 x+1$$

Step-by-Step Solution

Answer

The derivative of the function at \(x=1\) is 3.

Step 1: Understand the Problem

We need to find the derivative of the function \(f(x) = 3x + 1\) at the point \(x = 1\). This involves using the definition of the derivative, \(f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h}\).

Step 2: Substitute Small Values into the Derivative Formula

First, we substitute small values for \(h\) into the definition of the derivative to estimate \(f'(1)\). For a small \(h\), evaluate \(f(1+h)\) and \(f(1)\).

Step 3: Calculate f(1+h)

Use the function \(f(x) = 3x + 1\) to find \(f(1+h) = 3(1+h) + 1 = 3 + 3h + 1 = 4 + 3h\).

Step 4: Apply Derivative Definition

Substitute \(f(1+h)\) and \(f(1) = 4\) into the derivative formula: \(f'(1) \approx \frac{{(4 + 3h) - 4}}{h} = \frac{{3h}}{h} = 3\).

Step 5: Algebraically Calculate the Exact Derivative

We calculate the derivative of \(f(x)\) directly. The derivative of \(f(x) = 3x + 1\) with respect to \(x\) is \(f'(x) = 3\), so \(f'(1) = 3\).

Step 6: Verify the Result

Both the estimation with small \(h\) values and the algebraic calculation show that \(f'(1) = 3\).

Key Concepts

Calculus

Calculus is a branch of mathematics dedicated to studying how things change. One of its most important concepts is the derivative, which tells us how a function changes at any given point. Essentially, it describes the rate of change or the slope of the curve at any specific point. In the context of the given exercise, we are asked to find the derivative of the function of a linear equation, which in this case is

• Calculating how the output of the function changes as the input changes infinitesimally.
• Offering a detailed understanding of instantaneous rates of change.

For linear functions, calculus provides a straightforward way to find the derivative, which remains constant. This means, the rate of change is the same at any point on a linear graph, making calculus simpler for linear equations.
However, calculus becomes essential for exploring more complex functions, where the derivative might change along the curve. This involves not only derivatives but also integrals, giving a comprehensive tool to analyze variations in natural and abstract phenomena.

Limit Definition

When finding the derivative of a function, the limit definition is fundamental. This approach allows us to derive the function's rate of change at a specific point by observing how the function's output changes as the input changes slightly. The limit definition of the derivative for a function \(f(x)\) is given by:
\[f'(x) = \lim_{{h \to 0}} \frac{{f(x + h) - f(x)}}{h}\]This equation explains:

• What happens to the difference between \(f(x + h)\) and \(f(x)\) as \(h\), a small increment, approaches zero.
• How small changes in the \(x\) value affect the output of \(f(x)\).

For the function \(f(x) = 3x + 1\), using the limit definition allows us to test very small values of \(h\) and find an approximate slope. In this example, the slope remains constant at 3 because of the linear nature of the function. This demonstrates the practical application of the limit definition in verifying derivative calculations with precision.

Linear Functions

Linear functions take the form \(f(x) = ax + b\), where \(a\) and \(b\) are constants. These create straight-line graphs. A vital characteristic of linear functions is that their slopes are constant, meaning the derivative is the same across all points on the line

• In our exercise, the linear function \(f(x) = 3x + 1\) represents a line with a slope of 3. This slope is assessed by calculating the derivative.
• The derivative of a linear function is simply the coefficient of \(x\), in this case, 3.

Linear functions deliver consistent and predictable changes in \(f(x)\) when \(x\) changes. This property simplifies derivative calculations since the rate of change doesn't vary along the line. When estimating derivatives using the limit, the consistency of the slope helps confirm the accuracy of our calculations. The direct application to find the exact value, \(f'(1) = 3\), illustrates how linear functions keep calculus straightforward, especially when verifying estimation with algebraic methods.