Problem 29 - Calculus Single Variable

Question

In Exercises \(29-36,\) find the tangent line to the graph of \(y=f(x)\) at \(P\). \(f(x)=3 x^{2 / 3}, P=(8,12)\)

Step-by-Step Solution

Answer

The equation of the tangent line is \( y = x + 4 \).

Step 1: Find the Derivative

The first step to find the equation of the tangent line is to calculate the derivative of the function \( f(x) = 3x^{2/3} \). Using the power rule, the derivative \( f'(x) \) is computed as follows: \[ f'(x) = \frac{d}{dx}[3x^{2/3}] = 3 \times \frac{2}{3}x^{-1/3} = 2x^{-1/3}. \]

Step 2: Evaluate the Derivative at the Given Point

We need to evaluate the derivative at the given x-coordinate of point \( P \), which is \( x = 8 \). Thus, substitute \( x = 8 \) into \( f'(x) = 2x^{-1/3} \) to find the slope of the tangent line: \[ f'(8) = 2(8)^{-1/3} = 2 \times \frac{1}{2} = 1. \] Therefore, the slope of the tangent line at \( P = (8,12) \) is \( 1 \).

Step 3: Use the Point-Slope Formula

The equation of a line in point-slope form is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point \( P \). Plug in \( m = 1 \), \( x_1 = 8 \), and \( y_1 = 12 \) to get the equation: \[ y - 12 = 1(x - 8). \] Simplify this to get the equation of the tangent line: \[ y = x + 4. \]

Key Concepts

Understanding Derivatives

Derivatives are a fundamental concept in calculus, crucial for understanding how functions change. They represent the rate at which a function is changing at any given point. Essentially, the derivative gives us the slope of the tangent line to a curve at a particular point. For the function in our exercise,

• Start with the function: \( f(x) = 3x^{2/3} \).
• The derivative \( f'(x) \) tells us how \( f(x) \) changes as \( x \) changes.

To find \( f'(x) \), you differentiate \( f(x) \). Calculating this allows you to determine specific slope values where needed. By understanding derivatives, you gain insight into both the steepness and direction of the curve at each specific point.

The Power Rule Simplified

The power rule is a quick and effective tool in calculus for finding derivatives of functions that are polynomials or can be expressed as such. It states that for any real number \(n\), the derivative of \(x^n\) is \(nx^{n-1}\). This rule is instrumental for students because of its simplicity and wide applicability.For example, consider the function \(3x^{2/3}\). Applying the power rule:

• The original exponent is \(2/3\).
• Multiply by the coefficient: \(3 \times (2/3) = 2\).
• Subtract one from the exponent: \(2/3 - 1 = -1/3\).

Thus, the derivative is \(2x^{-1/3}\). The power rule reduces otherwise complex processes into simple arithmetic steps, making calculus more accessible.

Understanding Point-Slope Form

The point-slope form provides a method to express the equation of a line when you know the slope and one point on the line. This is especially helpful when dealing with lines that are tangent to curves.Point-slope form is expressed as:

• \(y - y_1 = m(x - x_1)\)
• \((x_1, y_1)\) is a point on the line.
• \(m\) is the slope of the line.

Using this form for the tangent line in our problem, with point \(P = (8, 12)\) and slope \(1\):\[ y - 12 = 1(x - 8) \]Simplifying yields the linear equation \(y = x + 4\), which describes the tangent line at the given point. This flexibility makes the point-slope form valuable for quickly formulating line equations.

Slope of the Tangent Line

The slope of the tangent line at a point on a curve provides insights into the curve's behavior at that point. For any function, the derivative at a specific point gives the slope of the tangent at that point. In our exercise, finding the derivative \(f'(x)\) gives us the necessary slope.For the function \(3x^{2/3}\), the derivative is \(2x^{-1/3}\). When all values are plugged in:

• Evaluate at \(x = 8\):
• \(f'(8) = 2 \times (8)^{-1/3} = 1\).

Hence, the slope of the tangent line at point \(P = (8, 12)\) is \(1\). A positive slope indicates the line is rising as it moves from left to right, reflecting the curve's local upward trend.