Problem 117 - Calculus and its applications

Question

Graph fand \(f^{\prime}\) and then determine \(f^{\prime}(1) .\) $$ f(x)=\frac{5 x^{2}+8 x-3}{3 x^{2}+2} $$

Step-by-Step Solution

Answer

Calculate the derivative using the quotient rule, then evaluate it at \( x = 1 \).

Step 1: Understand the function

The function given is \( f(x) = \frac{5x^2 + 8x - 3}{3x^2 + 2} \). It is a rational function, meaning it is the ratio of two polynomials.

Step 2: Find the derivative of the function

To find \( f'(x) \), apply the quotient rule. The quotient rule states that if \( f(x) = \frac{u(x)}{v(x)} \), then \( f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \). Here, \( u(x) = 5x^2 + 8x - 3 \) and \( v(x) = 3x^2 + 2 \).

Step 3: Compute u'(x) and v'(x)

Differentiate \( u(x) = 5x^2 + 8x - 3 \) with respect to \( x \): \( u'(x) = 10x + 8 \). Differentiate \( v(x) = 3x^2 + 2 \): \( v'(x) = 6x \).

Step 4: Apply the quotient rule

Substitute into the quotient rule: \[ f'(x) = \frac{(10x + 8)(3x^2 + 2) - (5x^2 + 8x - 3)(6x)}{(3x^2 + 2)^2} \]. Simplify the expression. First, expand the terms in both the numerator and denominator.

Step 5: Simplify the expression for f'(x)

First, expand \((10x + 8)(3x^2 + 2)\) to get \(30x^3 + 20x + 24x^2 + 16\). Then, expand \((5x^2 + 8x - 3)(6x)\) to get \(30x^3 + 48x^2 - 18x\). Simplify the entire numerator to consolidate like terms.

Step 6: Evaluate f'(x) at x=1

Substitute \( x = 1 \) into the simplified derivative expression. While solving, cancel out terms appropriately to find \( f'(1) \). Perform the arithmetic to arrive at the solution.

Key Concepts

Quotient Rule

In calculus, the Quotient Rule is a method used to differentiate functions that are the ratio of two differentiable functions. It's especially useful when you encounter a rational function, which is the fraction of two polynomials. The quotient rule is given by:

• If you have a function of the form \( f(x) = \frac{u(x)}{v(x)} \), then \( f'(x) \) is calculated as:
• \[ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \]

Here, \( u(x) \) and \( v(x) \) are the numerator and denominator of the function, respectively, and \( u'(x) \) and \( v'(x) \) are their derivatives. To apply it effectively:

• Differentiate the numerator function, \( u(x) \).
• Differentiate the denominator function, \( v(x) \).
• Plug these derivatives into the formula, simplify where necessary.

The quotient rule is vital when the split application of basic derivative rules just won't cut it, especially for integrating more complex polynomial functions into derivatives.

Rational Functions

A rational function is a ratio of two polynomials, that is, it is of the form \( f(x) = \frac{p(x)}{q(x)} \), where both \( p(x) \) and \( q(x) \) are polynomials and \( q(x) eq 0 \). Rational functions can exhibit interesting behaviors, such as asymptotes and intercepts. Here's what to consider:

• The **degree** of the polynomial: This helps determine the behavior at the ends or for extreme values of \( x \).
• **Vertical asymptotes**: These occur when the denominator is zero but the numerator is not zero. To find them, set \( q(x) = 0 \) and solve for \( x \).
• **Horizontal asymptotes**: They hint at the behavior of \( f(x) \) as \( x \) approaches infinity. Compare the degrees of the numerator and denominator.

Understanding these characteristics helps analyze the function's behavior and is foundational for understanding derivatives relevant to rational functions.

Polynomial Differentiation

Differentiating polynomials is one of the most basic tasks in calculus. It involves applying the power rule, which makes finding the derivative straightforward. For a polynomial function \( p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \), the derivative is:

• \( p'(x) = a_n n x^{n-1} + a_{n-1} (n-1) x^{n-2} + ... + a_1 \)

Each term in the polynomial is differentiated separately. The power rule states that for any term \( a_n x^n \), the derivative is \( a_n n x^{n-1} \).
Steps involved in differentiating polynomials:

• Multiply the coefficient by the exponent of each term.
• Reduce the exponent by one.
• Repeat for each term in the polynomial.

This method provides the basis for tackling more complex functions, like in the quotient rule, where polynomials are only one part of the function.