Problem 18 - Calculus and its applications

Question

Differentiate two ways: first, by using the Quotient Rule; then, by dividing the expressions before differentiating. Compare your results as a check. Use a graphing calculator to check your results. \(F(x)=\frac{x^{3}+27}{x+3}\)

Step-by-Step Solution

Answer

The derivative is \(F'(x) = 2x - 3\) using both methods.

Step 1: Apply the Quotient Rule

The Quotient Rule for differentiation states that if you have a function \(F(x) = \frac{u(x)}{v(x)}\), then \(F'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\). Here, \(u(x) = x^3 + 27\) and \(v(x) = x + 3\). First, find \(u'(x) = 3x^2\) and \(v'(x) = 1\). Applying the Quotient Rule:\[ F'(x) = \frac{(3x^2)(x+3) - (x^3+27)(1)}{(x+3)^2} \]Simplify the expression:\[ F'(x) = \frac{3x^3 + 9x^2 - x^3 - 27}{(x+3)^2} = \frac{2x^3 + 9x^2 - 27}{(x+3)^2} \]

Step 2: Simplify Before Differentiating

Simplify the function by performing polynomial long division on \(x^3 + 27\) divided by \(x + 3\). Rewrite the expression:\(x^3 + 27 = (x+3)(x^2 - 3x + 9)\), thus \(F(x) = \frac{(x+3)(x^2 - 3x + 9)}{x+3}\).This simplifies to \(F(x) = x^2 - 3x + 9\) for all \(x eq -3\). Now differentiate:\[ F'(x) = 2x - 3 \]

Step 3: Compare the Results

From Step 1, using the Quotient Rule we obtained:\[ F'(x) = \frac{2x^3 + 9x^2 - 27}{(x+3)^2} \]For most practical values of \(x\), this reduces to \(2x - 3\), except for \(x = -3\) where the original expression isn't defined.From Step 2, after simplifying and then differentiating, we found:\[ F'(x) = 2x - 3 \]Both methods give the same derivative \(2x - 3\).

Step 4: Verify Using a Graphing Calculator

Using a graphing calculator, input \(F(x) = \frac{x^3 + 27}{x+3}\) and observe the derivative function graph. The calculator should confirm that the derivative is a linear function with slope 2 and y-intercept -3, which matches our derivative \(F'(x) = 2x - 3\).

Key Concepts

Quotient Rule

The Quotient Rule is a method used in calculus for finding the derivative of a quotient, which is the division of two functions. When you are dealing with a function that is the division of two expressions, such as \(F(x) = \frac{u(x)}{v(x)}\), the Quotient Rule becomes quite handy.

The rule itself is expressed as:

• \(F'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\)

Here's how it works in our exercise:

• Define the functions: \(u(x) = x^3 + 27\) and \(v(x) = x + 3\).
• Differentiate each: \(u'(x) = 3x^2\) and \(v'(x) = 1\).
• Substitute these into the Quotient Rule formula: \(F'(x) = \frac{(3x^2)(x+3) - (x^3+27)(1)}{(x+3)^2}\).
• Simplify the expression to arrive at: \(F'(x) = \frac{2x^3 + 9x^2 - 27}{(x+3)^2}\).

Through these steps, the Quotient Rule provides a structured method to differentiate functions that are quotients, allowing us to tackle otherwise complex division expressions.

Polynomial Long Division

Polynomial long division is a process similar to the long division we use with numbers. It's used to divide a polynomial by another polynomial, simplifying the expression. This can be particularly useful in differentiation.

In our example, we have \(x^3 + 27\) divided by \(x + 3\). Here's the process broken down:

• Set up the division: divide the first term of the numerator \(x^3\) by the first term of the divisor \(x\), which gives \(x^2\).
• Multiply the entire divisor \(x + 3\) by \(x^2\) and subtract the result from \(x^3 + 27\).
• Continue this division process with the remainder, which involves repeated steps of simplification until no further division is possible.
• In our exercise, after completing polynomial long division, we find \(x^3 + 27 = (x+3)(x^2 - 3x + 9)\).

Ultimately, this simplifies our original function to \(F(x) = x^2 - 3x + 9\). Hence, we derive its derivative easily: \(F'(x) = 2x - 3\).

Polynomial long division simplifies complex polynomials and makes further calculus operations more manageable.

Graphing Calculator Verification

A graphing calculator is a powerful tool that can be used to verify the results obtained from manual calculations. It allows us to visualize functions and their derivatives, offering confirmation of our work's accuracy.

To verify differentiation using a graphing calculator, input the original function that needs differentiation. For instance, enter \(F(x) = \frac{x^3 + 27}{x+3}\) into your calculator.

What the graphing calculator does:

• Plots the original function to show its overall shape and behavior.
• Calculates the derivative and plots this as well, allowing us to see the slope of the tangent to the curve at any given point.
• In our exercise, the derivative graph should match the linear function \(F'(x) = 2x - 3\), which was calculated as the derivative by hand.

Confirming manual differentiation steps with a graphing calculator bridges visual understanding and computational verification, bolstering your confidence in finding derivatives correctly.