Question

\(\mathrm{T} / \mathrm{F}: \frac{d}{d x}\left(3^{x}\right) \approx 1.1 \cdot 3^{x}\).

Short answer

True, the derivative \( \frac{d}{dx}(3^x) \approx 1.1 \cdot 3^x \).

Step-by-step solution

Understand the Derivative

To determine if the statement is true or false, we need to calculate the derivative of the function \( 3^x \). The given claim is that \( \frac{d}{dx}(3^x) \approx 1.1 \cdot 3^x \).

Use the Exponential Derivative Formula

The general formula for the derivative of \( a^x \) is \( \frac{d}{dx}(a^x) = a^x \ln a \). Here, \( a = 3 \), so we find \( \frac{d}{dx}(3^x) = 3^x \ln 3 \).

Calculate the Value of \( \,\ln 3 \,\)

To understand if \( 3^x \ln 3 \approx 1.1 \cdot 3^x \), calculate \( \ln 3 \). Using a calculator, \( \ln 3 \approx 1.0986 \).

Compare with the Given Expression

Compare \( 3^x \ln 3 \) with \( 1.1 \cdot 3^x \). Since \( \ln 3 \approx 1.0986 \) is very close to \( 1.1 \), the two expressions are approximately equal.

Conclusion

Since \( 3^x \ln 3 \approx 1.1 \cdot 3^x \) is a reasonably valid approximation, the statement is approximately true. The approximation is based on the fact that \( 1.0986 \) is close to \( 1.1 \).

Key concepts

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In the expression \( 3^x \), "3" is the base and "x" is the exponent. This type of function is crucial in various fields such as science, finance, and statistics. Exponential functions exhibit rapid growth or decay, depending on whether the base is greater than or less than one. Understanding how these functions behave and how to manipulate them is vital for solving complex problems.

• Base of the Exponential: The number that is raised to the power of "x". For example, in \( 3^x \), the base is 3.
• Exponent: Often denoted by a variable like "x", it determines the power to which the base is raised. This variable can represent time, distance, or any other quantifiable factor.

These functions grow faster than linear or quadratic functions, making them useful for modeling things that change rapidly, such as populations or interest rates.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is the logarithm to the base "e", where "e" is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm is widely used in calculus and other branches of mathematics due to its unique properties and its ability to simplify complex mathematical expressions. For example, when differentiating exponential functions like \( 3^x \), the natural logarithm of the base appears in the derivative formula.

• The Relationship with Exponents: The natural logarithm function is essentially the inverse of the exponential function. This means if \( y = e^x \), then \( x = \ln y \).
• Logarithm of Numbers: Natural logarithms often simplify the process of working with exponents in calculus and beyond. Calculating \( \ln 3 \) as in this problem gives us approximately 1.0986, a key step in verifying the derivative approximation.

The importance of the natural logarithm comes from its appearance in various growth and decay processes, as well as its utility in calculus when working with exponential functions.

Approximation in Calculus

Approximation in calculus involves finding a close estimate of an exact mathematical value or expression. This is particularly helpful when exact calculations are complex or impossible to do manually. In the context of the problem we are examining, we approximate the derivative of \(3^x\) by determining whether it is approximately equal to \(1.1 \cdot 3^x\). The use of approximation reveals how calculus allows us to simplify complex derivatives.

• Derivative Approximations: By approximating \(\ln 3 \approx 1.0986\) as \(1.1\), we make calculations more manageable while understanding how close they mirror exact values.
• Uses in Real Life: Approximations help us make fast and effective predictions or decisions in practical scenarios like engineering and physics. They allow us to handle real-world problems where precise information may be unavailable.

In calculus, approximations are essential tools that accelerate problem-solving and enhance our comprehension of mathematical concepts, especially when dealing with intricate or abstract functions.