Problem 6 - Calculus III

Question

Find the curvature of the curve defined by the function $$ y=3 x^{2}-2 x+4 $$ at the point \(x=2\).

Step-by-Step Solution

Answer

The curvature of the curve at \( x = 2 \) is approximately 0.059560.

Step 1: Understand the Formula for Curvature

The formula for the curvature \( \kappa \) of a function \( y = f(x) \) is given by \[ \kappa = \frac{|f''(x)|}{(1 + (f'(x))^2)^{3/2}}. \] This formula requires the first and second derivatives of the function \( f(x) \).

Step 2: Find the First Derivative

Take the derivative of the function \( y = 3x^2 - 2x + 4 \) with respect to \( x \). This gives us the first derivative:\[ f'(x) = \frac{d}{dx}(3x^2 - 2x + 4) = 6x - 2. \]

Step 3: Find the Second Derivative

Differentiate the first derivative \( f'(x) = 6x - 2 \) to get the second derivative:\[ f''(x) = \frac{d}{dx}(6x - 2) = 6. \]

Step 4: Evaluate First Derivative at x=2

Substitute \( x = 2 \) into the first derivative to find its value at this point:\[ f'(2) = 6(2) - 2 = 12 - 2 = 10. \]

Step 5: Evaluate Second Derivative at x=2

Since the second derivative is a constant, \( f''(2) = 6 \).

Step 6: Calculate the Denominator of the Curvature Formula

Compute the term \( 1 + (f'(2))^2 \), which is part of the denominator in the curvature formula:\[ 1 + (10)^2 = 1 + 100 = 101. \]

Step 7: Final Calculation of the Curvature

Substitute the values of \( f''(x) \) and the result from the previous step into the curvature formula:\[ \kappa = \frac{|6|}{(101)^{3/2}} = \frac{6}{\sqrt{101^3}}. \]

Step 8: Simplify the Expression

Compute the numerical value of the expression \[ \kappa = \frac{6}{(101)^{3/2}}. \]Approximating, we get\[ \kappa \approx 0.059560. \]

Key Concepts

First Derivative

The first derivative of a function, often denoted as \( f'(x) \), represents the rate of change of the function with respect to its variable, usually \( x \). It essentially tells us how the function is changing at any particular point on its curve. Finding the first derivative involves performing differentiation, where we apply rules like the power rule to each term of the function.

• For the function \( y = 3x^2 - 2x + 4 \), we take the derivative of each term separately:
• The derivative of \( 3x^2 \) is \( 6x \), using the power rule \( nx^{n-1} \).
• The derivative of \( -2x \) is \( -2 \), and constants like \( 4 \) have a derivative of \( 0 \).

Putting this together, the first derivative is \( f'(x) = 6x - 2 \). This function \( f'(x) \) describes the slope of the tangent line to the curve at any value of \( x \). It is a linear function, indicating the slope is changing at a constant rate.
Find this step helpful? Try it with different functions to understand how different terms contribute to the overall rate of change.

Second Derivative

The second derivative of a function, denoted as \( f''(x) \), gives us information about the curvature of the function. This means it tells us how the rate of change of the function itself is changing. In a simpler sense, it can indicate whether a function is concave up or down at any given point.

• To find the second derivative, you differentiate the first derivative.
• For \( f'(x) = 6x - 2 \), the derivative is simply \( f''(x) = 6 \) because the rate of change of a linear function like \( 6x \) is constant.

The constant second derivative \( 6 \) tells us that the curvature of \( y = 3x^2 - 2x + 4 \) is consistent across its domain. This indicates a parabola where the rate at which it curves is uniform, a typical trait for quadratic functions.

Curvature Formula

Understanding the curvature formula is important for analyzing how curved a function is at a certain point. The curvature \( \kappa \) of a function \( y = f(x) \) at any given point is calculated using the formula: \[ \kappa = \frac{|f''(x)|}{(1 + (f'(x))^2)^{3/2}}. \]This allows us to quantify the sharpness or flatness of a curve. Higher curvature means a sharper turn, while lower curvature points to flatter sections.

• Use the absolute value of \( f''(x) \) to ensure curvature is always positive.
• The denominator modifications \( (1 + (f'(x))^2)^{3/2} \) account for how the slope affects the perceived curvature in a visual sense.

For the function \( y = 3x^2 - 2x + 4 \) at \( x = 2 \), we substitute into this formula:

• The previously found \( f'(2) = 10 \) and \( f''(2) = 6 \).
• Plug these into the formula for an output of approximately \( \kappa \approx 0.059560 \).

This means that at \( x = 2 \), the curve doesn't bend sharply, giving us a quantifiable measure of its smoothness.

Polynomial Function Evaluation

Evaluating polynomial functions involves replacing the variable with a given value and simplifying to find a result. This is commonly used in exercises to verify specific function values and changes along the curve of interest.

• For a polynomial like \( y = 3x^2 - 2x + 4 \), substitute \( x = 2 \) to evaluate the function at this point: \( y(2) = 3(2)^2 - 2(2) + 4 \).
• Calculating, this results in \( y = 3(4) - 4 + 4 = 12 \).

This process of substitution can be used to assess various values or changes such as derivatives or at certain key points like maxima, minima, or points of interest on a curve. Evaluating functions and derivatives at particular points allows us to specifically understand behavior at those exact positions.