Problem 55 - College Algebra and Calculus: An Applied Approach

Question

Find the equation of the function \(f\) whose graph passes through the point. $$ f^{\prime}(x)=\frac{x^{2}+4 x+3}{x-1} ; \quad(2,4) $$

Step-by-Step Solution

Answer

The function \(f(x)\) whose graph passes through the point (2,4) and whose derivative is \(\frac{x^{2}+4 x+3}{x-1}\) is \(f(x) =\frac{x^{2}}{2} + 5x - 2\ln|{x-1}| - 8\).

Step 1: Integration of the Derivative

First, integrate the given derivative to find the original function. The integral of \(\frac{x^{2}+4 x+3}{x-1}\) is more easily found by expressing the derivative as a polynomial. This can be done by performing polynomial division or by using the method of partial fractions. If we use polynomial division, we find that \(\frac{x^{2}+4 x+3}{x-1} = x + 5 - \frac{2}{x-1}\). So, the integral is \(\int (x + 5 - \frac{2}{x-1}) \, dx = \frac{x^{2}}{2} + 5x - 2\ln|{x-1}| + C\).

Step 2: Substitute the Point into the Function

Now substitute the point \((2,4)\) into the function to find the value of \(C\). Plug \(x = 2\) into \(f(x) =\frac{x^{2}}{2} + 5x - 2\ln|{x-1}| + C\) and set \(f(2) = 4\), so \(4 = 2 + 10 - 2\ln|{1}| + C\). Simplifying gives \(C = -8\).

Step 3: Write the Final Equation of the Function

Now substitute \(C = -8\) into the original integral to get the final equation of the function: \(f(x) =\frac{x^{2}}{2} + 5x - 2\ln|{x-1}| - 8\).

Key Concepts

Polynomial Division

Polynomial division is similar to long division performed with numbers. This method is helpful when you have a complex fraction where the numerator is a higher degree polynomial than the denominator. In our given problem, you have the expression \(\frac{x^{2}+4x+3}{x-1}\). Here the goal is to simplify the fraction by dividing the numerator by the denominator.

• First, set up the division, treating \(x^{2}+4x+3\) as the dividend and \(x-1\) as the divisor.
• Look at the leading term in the dividend \(x^{2}\) and divide it by the leading term in the divisor \(x\), which gives you \(x\).
• Multiply \(x\) by \(x-1\) and subtract the result from the original dividend.
• This gives a new polynomial of lesser degree. Repeat the process with this remainder until you can't divide anymore.

By following these steps, you can simplify the expression to \(x + 5 - \frac{2}{x-1}\). Breaking down the polynomial helps in turning a complex fraction into much simpler terms, making calculations and further integration more manageable.

Partial Fractions

The method of partial fractions is useful to break down complex rational expressions into simpler fractions, which can easily be integrated or otherwise manipulated. In context, partial fractions can be applied to an expression like \(\frac{x^{2}+4x+3}{x-1}\) after it has been simplified using polynomial division. When the rational function is already in its simplest form after division, use partial fractions to separate into distinct parts:

• For simple linear factors in the denominator (like \(x-1\)), write each as a separate fraction with unknown constants.
• Set up an equation where the original expression equals your sum of simpler fractions.
• Multiply through by the original denominator to clear the fractions, and find constants by equating coefficients of like terms from both sides of the equation.

In this exercise, after dividing, you are left with a simple fraction \(-\frac{2}{x-1}\), which means that the partial fraction decomposition is already complete, simplifying the process and leaving you with easier integrals to solve.

Integration

Once you have simplified the expression using polynomial division, integrating is the next step. Integration reverses differentiation. Given the derivative \(f'(x) = x + 5 - \frac{2}{x-1}\), the task is to find the original function, \(f(x)\), whose derivative produced this result.Integration works as follows:

• Integate each term separately: \(\int x \, dx = \frac{x^2}{2}\), \(\int 5 \, dx = 5x\), and \(\int - \frac{2}{x-1} \, dx = -2 \ln |x-1|\).
• Add the results of each integral as a sum, including an integration constant \(C\), which accounts for any constant value lost during differentiation.

The function \(f(x)\) becomes \(\frac{x^{2}}{2} + 5x - 2 \ln|x-1| + C\). Calculating an antiderivative by integrating each term provides you with the original function before it was derived.

Equation of a Function

Finding the equation of a function from its derivative involves not only integrating but also solving for constants to meet specific conditions. Here, after integrating \(f'(x)\) to get \(f(x)\), an unknown constant \(C\) is part of the equation. To find \(C\):

• Use the given point \((2,4)\) through which the graph of the function passes.
• Substitute \(x = 2\) and \(f(x) = 4\) into your integrated function \(f(x) = \frac{x^2}{2} + 5x - 2 \ln|x-1| + C\).
• Solve for \(C\) using algebra. Substitute all known values, simplify and rearrange.

This process ensures that the function does not just meet the derivative condition but also passes through the desired point, thereby fully defining the function. Ultimately, substituting \(C = -8\) back establishes that \(f(x) = \frac{x^{2}}{2} + 5x - 2 \ln|x-1| - 8\) is your final equation.