Problem 41 - Precalculus with Calculus Previews

Question

Use a calculator to approximate the value \(m(b)=\lim _{h \rightarrow 0} \frac{b^{h}-1}{h}\) for \(b=1.5, b=2\), \(b=3\), and \(b=5\) by filling out the given table. $$ \begin{array}{|l|l|l|l|l|l|l|} \hline h \rightarrow 0 & 0.1 & 0.01 & 0.001 & 0.0001 & 0.00001 & 0.000001 \\ \hline \frac{3^{h}-1}{h} & & & & & & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Calculate \(\frac{b^{h}-1}{h}\) for each \(b\) and \(h\); observe convergence as \(h\) approaches 0.

Step 1: Understand the Limit Expression

The expression to approximate is \(m(b)=\lim _{h \rightarrow 0} \frac{b^{h}-1}{h}\). This is a derivative-like expression found in calculus, related to the function \(b^x\), specifically at the point where \(x=0\). The limit represents the instantaneous rate of change or the slope of the tangent of the curve at that point.

Step 2: Approximate Using a Calculator

Start with small values of \(h\) approaching 0, and calculate \(\frac{b^{h}-1}{h}\) for each given \(b\): 1.5, 2, 3, and 5. We can use a calculator to do this. Compute these values for each given \(h = 0.1, 0.01, 0.001, 0.0001, 0.00001, 0.000001\).

Step 3: Calculate Values for \(b=3\)

Let's fill in the table for \(b=3\) as an example:- For \(h = 0.1\), compute \(\frac{3^{0.1}-1}{0.1}\).- For \(h = 0.01\), compute \(\frac{3^{0.01}-1}{0.01}\).- For \(h = 0.001\), compute \(\frac{3^{0.001}-1}{0.001}\).- Continue this process until \(h = 0.000001\). Use a calculator for precise results.Place the computed values in the table.

Step 4: Interpret the Results

Observe how the values obtained in Step 3 change as \(h\) gets smaller. These values should converge towards a constant value, which is the approximate limit.

Step 5: Repeat for Other Values of \(b\)

Repeat the calculations for \(b=1.5\), \(b=2\), and \(b=5\). Use a similar method as in Step 3 and observe the convergence to determine \(m(b)\) for each parameter.

Key Concepts

Approximating Limits

In calculus, the concept of limits helps us understand the behavior of functions as they approach a specific point or value. For exponential functions like \(b^x\), finding the limit \( \lim_{h \to 0} \frac{b^h - 1}{h} \) essentially involves understanding how the function behaves at very small increments.When approximating limits, we use small values of \(h\) to see how the expression changes as \(h\) approaches 0.

• Start with progressively smaller values of \(h\).
• Evaluate the expression \( \frac{b^h - 1}{h} \) for each \(h\).
• Observe how the results converge to a specific value.

This value is the limit. Approximating limits numerically using a calculator is a powerful tool to gain insights, especially when analytical methods are challenging.

Instantaneous Rate of Change

The instantaneous rate of change is a fundamental idea in calculus. It represents how quickly a function is changing at a particular point. This is closely tied to the concept of a derivative, which is the mathematical representation of that rate of change.In the limit definition \( \lim_{h \to 0} \frac{b^h - 1}{h} \), the rate of change is calculated as \(h\) approaches zero.
This gives us the slope of the tangent line to the curve of \(b^x\) at \(x = 0\).

• Helps us predict the behavior of the function at a single point.
• Gives crucial information about the function's trend and direction at that point.

Understanding this concept allows students to solve complex problems involving rates of change in real-world scenarios.

Exponential Functions

Exponential functions are mathematical functions of the form \(f(x) = b^x\), where \(b\) is a positive constant.These functions grow or decay at a rate proportional to their current value. Exponential functions are commonly used in contexts like population growth, radioactive decay, and finance.Key characteristics of exponential functions include:

• Always positive for any real number \(x\) when \(b > 0\).
• Rapidly increasing or decreasing based on the base \(b\).
• The base \(b = e\) (approximately 2.718) is special, often used in natural exponential functions.

Studying limits of exponential functions can help students better grasp continuous growth and decay processes.

Tangent Line Slopes

Tangent lines are lines that touch a curve at a single point without crossing it. The slope of the tangent line represents how steep the curve is at that particular point.In calculus, finding the slope of this tangent line involves calculating the derivative at the given point, often using the definition of a limit.For the function \(b^x\), the slope of the tangent line at \(x = 0\) can be found using the expression \( \lim_{h \to 0} \frac{b^h - 1}{h} \).

• It is a crucial part of understanding the behavior of curves and predicting changes.
• Helps solve optimization problems or when determining maximum and minimum values.

Comprehending tangent line slopes equips students with the ability to analyze and interpret the rates of change in various functions.