Problem 17 - Calculus Single Variable

Question

Find a point \(x\) where \(f^{\prime}(x)=6\). $$ f(x)=x^{3}+1 $$

Step-by-Step Solution

Answer

The points where \( f'(x) = 6 \) are \( x = \sqrt{2} \) and \( x = -\sqrt{2} \).

Step 1: Differentiate the Function

To find where the derivative equals 6, we first need to find the derivative of the function. The function given is \( f(x) = x^3 + 1 \). By using the power rule, the derivative \( f'(x) \) is calculated as follows:\[ f'(x) = \frac{d}{dx}(x^3) + \frac{d}{dx}(1) = 3x^2 + 0 = 3x^2 \]

Step 2: Set the Derivative Equal to 6

Next, we need to find the value of \( x \) that makes the derivative equal to 6. We set the equation:\[ 3x^2 = 6 \]

Step 3: Solve for x

To solve for \( x \), we divide both sides of the equation by 3:\[ x^2 = 2 \]Take the square root of both sides to solve for \( x \):\[ x = \pm \sqrt{2} \]

Step 4: Verify the Solution

To ensure correctness, substitute \( x = \sqrt{2} \) and \( x = -\sqrt{2} \) back into \( f'(x) \) to verify that they both result in 6:For \( x = \sqrt{2} \):\[ f'(\sqrt{2}) = 3(\sqrt{2})^2 = 3 \times 2 = 6 \]For \( x = -\sqrt{2} \):\[ f'(-\sqrt{2}) = 3(-\sqrt{2})^2 = 3 \times 2 = 6 \]Both values satisfy \( f'(x) = 6 \).

Key Concepts

Derivatives

In calculus, derivatives are a fundamental concept used to understand how functions change. A derivative represents the rate of change or the slope of the function at any given point. If you imagine a curve on a graph, the derivative at a specific point tells you how steep the curve is at that point.

• The notation for a derivative is often expressed as \( f'(x) \) or \( \frac{df}{dx} \).
• The process of finding the derivative is called differentiation.

To find the derivative of a function, various rules can be applied based on the function's components. In our exercise, we focused on differentiating a polynomial function to solve a particular problem, illustrating how useful derivatives are in practical situations beyond the abstract math.

Power Rule

The power rule is one of the basic and most commonly used rules for finding derivatives. It's an efficient method that helps differentiate expressions of the form \( x^n \). Here's the general form of the power rule:

• If \( f(x) = x^n \), then the derivative \( f'(x) = nx^{n-1} \).

In the exercise, we applied the power rule to the function \( f(x) = x^3 + 1 \) to find its derivative. Differentiating each term:

• The derivative of \( x^3 \) according to the power rule is \( 3x^2 \).
• The derivative of a constant, such as \( +1 \), is always zero.

So, the derivative of the function \( x^3 + 1 \) becomes \( f'(x) = 3x^2 \). Understanding and applying the power rule is crucial for finding derivatives quickly and correctly.

Equation Solving

After finding the derivative, we often need to solve equations to find specific values where certain conditions are met. In our exercise, we needed to find where the derivative equals 6. This required setting up the equation:

• \( 3x^2 = 6 \)

To solve this quadratic equation, we performed these steps:

• First, divide both sides by 3 to simplify: \( x^2 = 2 \).
• Then, take the square root of both sides, resulting in \( x = \pm \sqrt{2} \).

Equation solving is a vital skill in calculus, providing the means to find important points like maxima, minima, or specific slopes on a graph.

Function Analysis

Function analysis is about understanding the behavior of a function. It involves looking at the function's values, slopes, and how it increases or decreases. Analyzing functions through their derivatives offers deep insights into their characteristics.

• For instance, the sign of \( f'(x) \) (positive, negative, zero) tells us whether the function is increasing, decreasing, or has a horizontal tangent.
• Finding where the derivative equals a particular number, like 6, helps us identify specific points on the curve that feature this particular property.

In our exercise, we analyzed \( f(x) = x^3 + 1 \) through its derivative. Both solutions, \( x = \sqrt{2} \) and \( x = -\sqrt{2} \), were verified to ensure they correctly provide the condition where \( f'(x) = 6 \). Function analysis allows us to interpret and predict the behavior of functions much more effectively than evaluating numbers alone.