Problem 35 - Calculus - AP Edition

Question

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

Step-by-Step Solution

Answer

False; the correct value of the derivative at \( x = 2 \) is 32, not 20.

Step 1: Differentiate the Expression

Start with the expression \( 4f(x) + x^3 \). Using differentiation rules, find the derivative with respect to \( x \). The derivative of \( 4f(x) \) is \( 4f'(x) \) and the derivative of \( x^3 \) is \( 3x^2 \). Thus, the derivative of \( 4f(x) + x^3 \) is \( 4f'(x) + 3x^2 \).

Step 2: Evaluate the Derivative at x = 2

Substitute \( x = 2 \) into the derivative \( 4f'(x) + 3x^2 \). We are told \( f'(2) = 5 \). Thus, substitute this value to get: \( 4(5) + 3(2)^2 = 20 + 12 = 32 \).

Step 3: Compare with Given Expression

The provided statement claims that the derivative evaluated at \( x = 2 \) is equal to 20. From Step 2, we calculated the derivative \( 4f'(x) + 3x^2 \) at \( x = 2 \) as 32, not 20. Also, the expression in the statement erroneously equates this with \( 4 f'(2) = 20 \), which is incorrect since \( 4 f'(2) = 4 \, \times \, 5 = 20 \) is not the correct evaluation for the derivative including \( x^3 \).

Key Concepts

Derivatives

In the context of differential calculus, derivatives are fundamental. They allow us to measure how a function changes as its input changes. The derivative of a function at a specific point gives the slope of the tangent line to the function at that point. It represents the rate of change or how fast something is changing at any given moment.

To find the derivative of a function, we use notation such as \(f'(x)\) or \(\frac{d}{dx}[f(x)]\). These notations show that we're finding the rate of change for \(f(x)\) with respect to \(x\). For instance, if we have \(f'(2) = 5\), this means at \(x = 2\), the rate of change of the function is 5.

Understanding derivatives is crucial because they form the backbone of many concepts in calculus, enabling us to solve real-world problems involving rates of change in physics, economics, and other fields.

Differentiation Rules

Differentiation rules are guidelines that help us find derivatives efficiently. Basic rules include the power rule, product rule, quotient rule, and chain rule, among others. In the original exercise, the power rule is applicable.

Let's say you have a function like \(4f(x) + x^3\). You differentiate each part using the rules:

• Constant Multiplier Rule: If you have a constant multiplied by a function, like \(4f(x)\), its derivative is \(4 \,\times \, f'(x)\).
• Power Rule: For a term like \(x^n\), the derivative is \(nx^{n-1}\). So for \(x^3\), it becomes \(3x^2\).

Combining these, the derivative of \(4f(x)+ x^3\) becomes \(4f'(x) + 3x^2\). Differentiation rules simplify the process and make finding derivatives straightforward.

Mathematical Proof

Mathematical proof is a method of demonstrating the truth or validity of a statement rigorously. In calculus, proving a derivative involves applying differentiation rules correctly to establish that an expression meets certain conditions or equals a specific value.

In our provided problem, the statement was tested by performing a series of logical steps to prove or disprove its validity. The initial claim was that the derivative equals 20 when evaluated at \(x = 2\). We proved this claim incorrect by finding the true derivative using the principles of differentiation.

Mathematical proofs require a step-by-step approach where each step logically follows the preceding one, ensuring that each conclusion is well-supported by mathematical facts and rules. This method is vital for confirming conjectures and ensuring accuracy in mathematical findings and solutions.