Problem 9 - Calculus

Question

Estimate \(f^{\prime}(2)\) for \(f(x)=3^{x} .\) Explain your reasoning.

Step-by-Step Solution

Answer

The estimated value of \( f'(2) \) is approximately 9.8874.

Step 1: Understand the concept of the derivative

The derivative of a function at a point gives us the rate of change or the slope of the tangent line to the function at that point. For the function \( f(x) = 3^x \), we need to find the slope of the tangent line at \( x = 2 \).

Step 2: Recall the derivative formula

The derivative of the exponential function \( a^x \) with respect to \( x \) is given by \( f'(x) = a^x \ln(a) \). Thus, for \( f(x) = 3^x \), the derivative is \( f'(x) = 3^x \ln(3) \).

Step 3: Substitute the value into the derivative

Now that we have \( f'(x) = 3^x \ln(3) \), substitute \( x = 2 \) to find the derivative at this point: \( f'(2) = 3^2 \ln(3) \). Calculate to get \( f'(2) = 9 \ln(3) \).

Step 4: Calculate the numerical value

Using a calculator, find \( 9 \ln(3) \). Approximating \( \ln(3) \approx 1.0986 \), we have: \( f'(2) = 9 \times 1.0986 \approx 9.8874 \).

Step 5: State the estimated value of the derivative

Therefore, the estimated value of \( f'(2) \) is approximately 9.8874. This represents the slope of the tangent line to the curve \( y = 3^x \) at the point where \( x = 2 \).

Key Concepts

Exponential Function

An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. It is represented as \( f(x) = a^x \), where \( a \) is a positive real number, not equal to one. The function \( 3^x \) is an example, where 3 is the base and \( x \) is the exponent.
An important feature of exponential functions is their rapid growth or decay, depending on whether \( a > 1 \) or \( 0 < a < 1 \), respectively. In this case, because the base is 3, \( 3^x \) grows as \( x \) increases.
This type of function often appears in real-world applications, such as population growth, radioactive decay, and interest calculations in finances, making them crucial in various fields. Understanding the behavior of exponential functions helps us model and predict these phenomena.

Rate of Change

The rate of change refers to how a quantity changes in relation to another. In mathematics, it is often expressed as the derivative of a function.
In the context of the function \( f(x) = 3^x \), finding \( f'(2) \) involves determining how fast \( f(x) \) changes with respect to \( x \) at \( x = 2 \).
This rate of change is essentially the slope of the tangent line at the specific point \( x = 2 \) on the graph of \( y = 3^x \). It gives us a precise rate at which the function value changes at that point, which can be useful in predictions and analyses of dynamic systems in science and engineering.

Tangent Line

A tangent line to a curve at a given point is the straight line that just touches the curve at that point, matching its slope without crossing it immediately in the vicinity.
For \( y = 3^x \) at \( x = 2 \), the derivative \( f'(2) \) tells us the slope of this tangent line. A tangent gives us a linear approximation to the curve near that point.
Tangent lines play a fundamental role in calculus and geometry. They help in understanding the motion and rate of change, which is why derivatives, slopes, and tangents are tightly interlinked concepts. Being linear, they offer simpler representations of the curve for small sections, significantly aiding in computations and visualization.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is the logarithm with base \( e \), where \( e \approx 2.71828 \).
It is a fundamental mathematical function akin to the exponential function and is crucial in calculus, particularly in interpreting the derivatives of exponential functions.
For a general exponential function \( f(x) = a^x \), its derivative involves the natural logarithm, expressed as \( f'(x) = a^x \ln(a) \). In our example, \( 3^x \), the derivative becomes \( 3^x \ln(3) \), demonstrating how the natural logarithm facilitates these calculations.
The importance of \( \ln(x) \) goes beyond derivatives; it is extensively used in solving exponential equations, integration, and modeling logarithmic growth or decay processes in diverse scientific disciplines.