Problem 4 - Calculus Concepts

Question

Numerically estimate the slope of the line tangent to the graph of the function \(f\) at the given input value. Show the numerical estimation table with at least four estimates. \(f(x)=5 \ln x ; x=5,\) estimate to the nearest hundredth

Step-by-Step Solution

Answer

The slope of the tangent line at \( x = 5 \) is approximately 1.61.

Step 1: Understanding the Problem

We need to estimate the slope of the tangent line to the curve of the function \( f(x) = 5 \ln x \) at \( x = 5 \). This requires finding the derivative \( f'(x) \) using numerical techniques at \( x = 5 \).

Step 2: Set Up Numerical Estimation

We'll use numerical differentiation to estimate the derivative at \( x = 5 \). The formula for numerical differentiation is given by \( f'(a) \approx \frac{f(a+h) - f(a)}{h} \), where \( h \) is a small number. We'll choose different values of \( h \) and calculate estimates for \( f'(5) \).

Step 3: Calculate Numerical Estimates

Calculate \( f'(5) \approximately \frac{f(5+h) - f(5)}{h} \) for various small values of \( h \) such as \( h = 0.1, 0.01, 0.001, 0.0001 \). Record each result:- For \( h = 0.1 \): \( \frac{f(5.1) - f(5)}{0.1} \)- For \( h = 0.01 \): \( \frac{f(5.01) - f(5)}{0.01} \)- For \( h = 0.001 \): \( \frac{f(5.001) - f(5)}{0.001} \)- For \( h = 0.0001 \): \( \frac{f(5.0001) - f(5)}{0.0001} \)

Step 4: Evaluate Function Values

Compute the function values needed for each estimation:- \( f(5) = 5 \ln 5 \)- \( f(5.1) = 5 \ln 5.1 \)- \( f(5.01) = 5 \ln 5.01 \)- \( f(5.001) = 5 \ln 5.001 \)- \( f(5.0001) = 5 \ln 5.0001 \)Evaluate these values using a calculator.

Step 5: Complete Numerical Calculations

Now calculate each derivative estimate using the function values found:- For \( h = 0.1 \): \( \approx 1.6095 \)- For \( h = 0.01 \): \( \approx 1.6094 \)- For \( h = 0.001 \): \( \approx 1.6094 \)- For \( h = 0.0001 \): \( \approx 1.6094 \)These results converge as \( h \) approaches zero, suggesting our estimation is accurate.

Step 6: Converge to Final Estimate

The estimates stabilize around 1.6094 as \( h \) decreases, suggesting the slope of the tangent line is about 1.6094, to the nearest hundredth.

Key Concepts

Tangent Line Estimation

When we talk about estimating the tangent line to a curve at a certain point, we refer to a straight line that just barely touches the curve, matching its direction exactly at that single point. This line gives us an idea of how the curve is behaving exactly at that point. To gain insight, we use numerical differentiation, which helps find this tangent line's slope without needing to solve calculus equations directly.
Numerical estimation is particularly helpful when dealing with complex functions or when an analytical derivative is difficult to determine. By making small adjustments via a variable known as "h" in our calculations, we can observe how the curve might behave more finely, getting a tangible understanding of its local behavior.

Slope of a Function

The slope of a function at a particular point tells us how steep the function is at that point. When dealing with a straightforward linear equation, finding the slope is plain sailing—it's just the ratio of the 'rise' over 'run,' or how much the y-value changes for a corresponding change in the x-value.
However, for more complicated functions, like the natural logarithm function, determining the slope involves calculus concepts, notably differentiation. This slope is the rate at which the function's value is changing concerning its input. In our exercise, we estimated this slope numerically for a clearer picture.

Naturallogarithm Function

The natural logarithm function, typically represented as \( \ln(x) \), is a special function in mathematics representing the inverse of the exponential function. One primary reason the natural logarithm is so pivotal is its connection to the concept of growth and compound interest, as well as its prominence in calculus.
Natural logarithms have properties such as \( \ln(1) = 0 \) and \( \ln(e) = 1 \), where \( e \approximates 2.718 \), making it a universal base for natural logs. When functions are transformed by logarithms, it helps simplify multiplicative processes into additive ones. In this problem, we explored the function \( f(x) = 5 \ln x \), applying numerical techniques to determine how this relationship changes at a specific value.

Numerical Techniques in Calculus

Numerical techniques in calculus, such as the numerical differentiation utilized in this problem, involve using approximations to find solutions to otherwise challenging calculus problems. These methods are beneficial when functions do not lend themselves to regular derivative calculations.
By using the formula \( f'(a) \approx \frac{f(a+h) - f(a)}{h} \), we aim to approximate the derivative of a function using small values for \( h \). The idea is that as \( h \) becomes smaller, the approximation gets closer to the factual derivative, providing an accurate insight into the function's behavior near the chosen point. We numerically differentiated \( f(x) = 5 \ln x \), around \( x = 5 \), using several small \( h \) values.