Problem 35 - Calculus Early Transcendentals

Question

Determine whether the statement is true or false. Explain your answer. If \(f^{\prime}(2)=5,\) then $$ \left.\frac{d}{d x}\left[4 f(x)+x^{3}\right]\right|_{x=2}=\left.\frac{d}{d x}[4 f(x)+8]\right|_{x=2}=4 f^{\prime}(2)=20 $$

Step-by-Step Solution

Answer

The statement is false.

Step 1: Differentiate the First Expression

First, differentiate the function \( 4f(x) + x^3 \) with respect to \( x \). The derivative of \( 4f(x) \) is \( 4f'(x) \) and the derivative of \( x^3 \) is \( 3x^2 \). Therefore, the derivative becomes:\[\frac{d}{dx}[4f(x) + x^3] = 4f'(x) + 3x^2\]

Step 2: Evaluate at \( x = 2 \)

Substitute \( x = 2 \) into the expression \( 4f'(x) + 3x^2 \). We know that \( f'(2) = 5 \), so:\[ 4f'(2) + 3(2)^2 = 4(5) + 3(4) = 20 + 12 = 32 \]

Step 3: Differentiate the Second Expression

Next, differentiate the function \( 4f(x) + 8 \). Since the derivative of a constant is zero, the derivative becomes:\[ \frac{d}{dx}[4f(x) + 8] = 4f'(x) \]

Step 4: Evaluate at \( x = 2 \)

Substitute \( x = 2 \) into the expression \( 4f'(x) \). Again, use \( f'(2) = 5 \):\[ 4f'(2) = 4(5) = 20 \]

Step 5: Compare Results

Compare the results from Step 2 and Step 4. The first expression evaluated to 32, while the second evaluated to 20. Thus, the original statement that both derivatives are equal to 20 is false.

Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function. It's a fundamental concept in calculus that helps us understand the rate of change of one quantity concerning another. The derivative essentially represents the slope of the tangent line to the function's graph at a given point.
To differentiate a function like \(4f(x) + x^3\), we follow a set of rules:

• Constant Multiple Rule: The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. For example, the derivative of \(4f(x)\) is \(4f'(x)\).
• Power Rule: This rule tells us how to find the derivative of functions of the form \(x^n\). If \(x\) is raised to a power \(n\), the derivative is \(nx^{n-1}\). For instance, the derivative of \(x^3\) is \(3x^2\).
• Sum Rule: The derivative of a sum is the sum of the derivatives. For example, the derivative of \(4f(x) + x^3\) is \(4f'(x) + 3x^2\).

By applying these rules, we can determine the derivative of a function with respect to its variable, helping us analyze dynamic systems and predict changes.

Function Evaluation

Function evaluation involves substituting a specific value into the function or its derivative. This lets us calculate the function's output or the derivative's rate of change at that particular value.
In the exercise, once we differentiate the expressions, the next task is to evaluate these derivatives at \(x = 2\).

• For the derivative \(4f'(x) + 3x^2\), substituting \(x = 2\) yields \(4f'(2) + 3(2)^2\).
• Given \(f'(2) = 5\), it simplifies to \(4(5) + 3(4) = 20 + 12 = 32\).

This is crucial for checking the behavior of the derivative at specific points, confirming the accuracy of our calculus operations, and supporting practical applications where specific point analysis is desired.

Derivative of Constant Expression

When differentiating terms within a function, the derivative of a constant term is always zero. This is because constants do not change, and hence, their rate of change is zero.
For the function \(4f(x) + 8\), the term \(8\) is a constant, so its derivative is zero.

• Thus, the differentiation of this part results in \(4f'(x)\), since the derivative of \(8\) vanishes.
• Evaluating at \(x = 2\), we find \(4f'(2) = 4(5) = 20\).

Understanding this concept is essential in distinguishing terms that will contribute to the derivative calculation and those that will not. Through this knowledge, one can simplify derivative calculations, especially in more complex functions by filtering out constant terms early in the process.