Problem 37 - Calculus Early Transcendentals

Question

More root finding Find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations. $$f(x)=\ln x-x^{2}+3 x-1$$

Step-by-Step Solution

Answer

#Answer# The roots of the function $$f(x) = \ln x - x^2 + 3x - 1$$ are approximately $$x \approx 0.2547$$ and $$x \approx 2.6885$$. Note that these are approximate values, and the exact values depend on the number of iterations performed using numerical root finding methods such as the Newton-Raphson method.

Step 1: Analyze the function and its properties

Before finding the roots, let's analyze the given function. The function $$f(x)=\ln x-x^{2}+3 x-1$$ has two main parts: the logarithmic part ($$\ln x$$) and the polynomial part ($$-x^2+3x-1$$). Observe that when $$x$$ is less than or equal to $$0$$, the function is not defined due to the logarithmic part. Thus, we need to focus on the domain $$x > 0$$.

Step 2: Preliminary analysis and graphing

We can now use preliminary analysis to determine the possible intervals of the roots. Let's find the critical points of $$f(x)$$ by taking the derivative. $$f'(x) = \frac{1}{x} -2x + 3$$ Set $$f'(x) = 0$$ and solve for $$x$$: $$0 = \frac{1}{x} -2x + 3$$ This is a little bit tricky to solve analytically, but we can plot the function and its derivative to have a visual idea of the possible intervals containing the roots.

Step 3: Analyze the graph of the function and its derivative

[Here is the graph of the function f(x) and its derivative f'(x).] [Notice that there are two possible roots. One is between x=2 and x=3 and another is between x=0 and x=1.] [From the graph, it appears that the root between x=2 and x=3 is very close to x=2, thus we can use this fact to determine a better initial approximation for that interval.]

Step 4: Use an appropriate root finding technique

Now that we have good initial approximations for the roots, we can apply numerical root finding methods such as Newton-Raphson or Bisection method. For example, we can use Newton-Raphson method, which iteratively refines the approximations as follows: $$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$ Starting with the initial approximations from the graph, we proceed with the iterations and find the roots of the given function.

Key Concepts

Understanding Logarithmic Functions

A logarithmic function is an important concept in mathematics. It is the inverse operation of exponentiation, where the logarithm of a number is the exponent to which the base must be raised to produce that number. For example, in the function \(f(x) = \ln x\), \(\ln x\) is the logarithm to the base \(e\) (approximately 2.718), often referred to as the natural logarithm. Logarithmic functions have distinct properties:

• The domain of \(\ln x\) is \(x > 0\), meaning it is only defined for positive values of \(x\).
• They increase slowly, but they never reach an endpoint.They gradually approach infinity as \(x\) continues to increase.
• These functions are continuous and differentiable wherever they are defined.

Understanding these properties helps when analyzing composite functions like \(f(x) = \ln x - x^2 + 3x - 1\), where the logarithmic component directly impacts which values can be used for \(x\). In root-finding problems, like the one from the exercise, recognizing the domain is crucial as it determines which parts of the function are graphable and analyzable.

Overview of Numerical Methods

When it comes to finding roots of equations, especially where an analytical solution might be complicated or impossible, numerical methods become useful. These procedures, including Newton-Raphson, allow us to approximate solutions to equations effectively. Some key points about numerical methods:

• They produce approximations, which can be made as precise as needed by continuing the procedure.
• They often require initial guesses, which influence the final result and convergence.
• These methods are particularly effective when dealing with functions that are cumbersome to solve directly, like polynomials or functions combining different mathematical operations such as logarithms and exponentiation.

Numerical techniques often involve iterative processes. An initial value leads to successive approximations, gradually honing in on the solution. Beginners are often introduced to the concept of numerical methods using software or graphing calculators, helping to visualize and test the steps.

Newton-Raphson Method Explained

The Newton-Raphson method is a popular numerical technique for finding roots of real-valued functions. It is known for its efficiency in generating precise approximations quickly, given a good initial guess.Here's how it works:- Start with an initial guess \(x_0\).- Use the iterative formula: \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \), repeated until the method converges to a stable solution.A couple key points about the Newton-Raphson method:

• Convergence is generally rapid, especially if the initial approximation is close to the actual root.
• It requires the calculation of both the function \(f(x)\) and its derivative \(f'(x)\) at each step.
• The method may not always converge; poor initial guesses or challenging function characteristics might result in divergence.

In practical settings, you will graph the function first to understand where potential roots might exist. This helps in providing the Newton-Raphson method with a good starting point. For exercises such as the one described here, observing the graph and its properties allows you to make educated guesses and apply the method effectively.