Problem 22 - APEX Calculus

Question

A function \(f\) and \(a\) value \(a\) are given. Approximate the limit of the difference quotient, \(\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h},\) using \(h=\pm 0.1, \pm 0.01 .\) $$ f(x)=\ln x, \quad a=5 $$

Step-by-Step Solution

Answer

The approximated limit of the difference quotient is 0.2.

Step 1: Define the Difference Quotient

We want to approximate the limit of the difference quotient for the function. The difference quotient is given by \( \frac{f(a+h)-f(a)}{h} \). In this case, the function is \( f(x) = \ln x \) and \( a = 5 \). Thus, the difference quotient becomes \( \frac{\ln(5+h) - \ln 5}{h} \).

Step 2: Calculate the Difference Quotient for \(h=0.1\)

Substitute \( h = 0.1 \) into the difference quotient: \( \frac{\ln(5+0.1) - \ln 5}{0.1} \). Calculating inside the logarithms, we have \( \ln 5.1 \approx 1.62924 \) and \( \ln 5 \approx 1.60944 \). The quotient becomes \( \frac{1.62924 - 1.60944}{0.1} = 0.198 \).

Step 3: Calculate the Difference Quotient for \(h=-0.1\)

Substitute \( h = -0.1 \): \( \frac{\ln(5-0.1) - \ln 5}{-0.1} \). This gives us \( \ln 4.9 \approx 1.58924 \). The quotient is \( \frac{1.58924 - 1.60944}{-0.1} = 0.202 \).

Step 4: Calculate the Difference Quotient for \(h=0.01\)

Substitute \( h = 0.01 \): \( \frac{\ln(5+0.01) - \ln 5}{0.01} \). Calculate \( \ln 5.01 \approx 1.61144 \). The quotient becomes \( \frac{1.61144 - 1.60944}{0.01} = 0.2 \).

Step 5: Calculate the Difference Quotient for \(h=-0.01\)

Substitute \( h = -0.01 \): \( \frac{\ln(5-0.01) - \ln 5}{-0.01} \). Calculate \( \ln 4.99 \approx 1.60744 \). The quotient becomes \( \frac{1.60744 - 1.60944}{-0.01} = 0.2 \).

Step 6: Approximate the Limit

The approximate values of the difference quotient as \( h \) approaches zero are around 0.2 for both positive and negative small \( h \). Therefore, the approximated limit of the difference quotient as \( h \rightarrow 0 \) is 0.2.

Key Concepts

Difference Quotient

The difference quotient is a fundamental concept in calculus used to approximate the derivative of a function at a particular point. It's expressed as:

• \( \frac{f(a+h)-f(a)}{h} \) where \( h eq 0 \)

For the function \( f(x) = \ln x \) and \( a = 5 \), the difference quotient expresses the average rate of change of the function over an interval \( h \).
As \( h \) approaches zero, this quotient gives us a clear picture of how the function \( \ln x \) behaves near \( x = 5 \).
This technique is particularly useful when you're working with natural logarithms, as we see how \( \ln(5+h) \) changes with very small increments or decrements in \( h \).
To compute this, you replace \( f(x) \) in the quotient with the natural logarithmic function and solve for different small values of \( h \) like 0.1, -0.1, 0.01 and -0.01, driving the quotient to show a trend toward the derivative.

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is a mathematical function that represents the logarithm to the base \( e \), where \( e \approx 2.71828 \). It is specifically significant in continuous growth processes, popping up in places such as compound interest, population growth, and, naturally, calculus.

• The key property of natural logarithms is that \( \ln(e^x) = x \).
• Another property is that \( \ln(ab) = \ln a + \ln b \). This property comes handy in simplifying expressions.

In the context of the given problem, the function \( f(x) = \ln x \) uses these properties to simplify the differentiation process.
When evaluating the difference quotient at \( x = 5 \), you use these logarithm laws to compute values of \( \ln(5+h) \) with tiny strides in \( h \).
This iterative process helps explore the changing rate of \( \ln x \) close to 5, ultimately allowing you to approximate how the function slopes at that exact point.

Limit Approximation

The concept of limit approximation is deeply rooted in calculus and underpins the calculation of derivatives. It involves finding the value that a function approaches as the input approaches some point. For the difference quotient of a given function, we observe:

• \( \lim _{h \to 0} \frac{f(a+h)-f(a)}{h} \)

The primary aim is establishing what happens to the difference quotient as \( h \) dwindles towards zero, thus hovering at the slope of the tangent line to the function at point \( a \).
This exploration of limits provides insight into function behavior, as seen in the example where the quotient approximates 0.2 when \( h \) is extremely close to zero, suggesting that this is the slope of \( \ln x \) at \( x = 5 \).
By trying different values like \( h=0.1 \) and \( h=-0.1 \) and then smaller values, you can discern a pattern that approaches the same limit, underscoring the smooth transition within the logarithmic function, and emphasizing the precision and elegance of limits in approximating derivatives.