Problem 6 - Calculus: Early Transcendentals

Question

Find an equation of the tangent line to the curve at the given point. \( y = x^3 - 3x + 1 \), \( (2, 3) \)

Step-by-Step Solution

Answer

The equation of the tangent line is \( y = 9x - 15 \).

Step 1: Find the Derivative

To find the equation of the tangent line, we first need to find the derivative of the function, which gives us the slope of the tangent line at any given point. The function is given by \[ y = x^3 - 3x + 1 \]. Using the power rule, the derivative of this function is: \[ y' = 3x^2 - 3 \].

Step 2: Evaluate the Derivative at the Given Point

Next, we plug in the x-coordinate of the given point \((2, 3)\) into the derivative to find the slope of the tangent line at this point. The derivative is \( y' = 3x^2 - 3 \), so substituting \( x = 2 \), we get: \[ y'(2) = 3(2)^2 - 3 = 12 - 3 = 9 \]. Thus, the slope of the tangent line at \((2, 3)\) is 9.

Step 3: Use the Point-Slope Formula for the Equation of the Line

Now that we have the slope of the tangent line, we use the point-slope form of a line's equation, which is given by: \[ y - y_1 = m(x - x_1) \], where \( m \) is the slope and \((x_1, y_1)\) is the given point. We have \( m = 9 \) and \((x_1, y_1) = (2, 3)\). Plugging these into the formula, we get \[ y - 3 = 9(x - 2) \].

Step 4: Simplify the Equation

To get the equation of the tangent line in slope-intercept form, simplify the equation from the previous step:\[ y - 3 = 9(x - 2) \] Expanding this, we have:\[ y - 3 = 9x - 18 \] Adding 3 to both sides gives:\[ y = 9x - 15 \].This is the equation of the tangent line.

Key Concepts

Understanding Tangent Lines

A tangent line is a straight line that touches a curve at a single point without crossing it. Think of it as a line that just kisses the curve opportunistically. At the point where the tangent touches the curve, it has the same slope as the curve itself.
Understanding this concept is essential when you're trying to find the specific slope or direction of the curve at a given point. This is particularly useful for understanding how quickly or slowly a function is changing at that very point.
To find this tangent line, follow these steps:

• Differentiate the function to find its slope at any point.
• Evaluate this derivative at the point of interest to find the slope of the tangent line.
• Use this slope along with the coordinates of the given point to write the equation of the tangent line.

Grasping the concept of a tangent line allows you to better understand the geometrical and physical interpretation of many real-world problems.

Mastering the Power Rule

The power rule is one of the fundamental tools in calculus, simplifying the process of differentiating polynomial functions. You can apply the power rule when you need to take the derivative of a term in the form of \( x^n \), where \( n \) is a real number.
The rule states that you bring down the exponent as a coefficient and decrease the exponent by one. Thus, the derivative of \( x^n \) becomes \( nx^{n-1} \).
This rule makes finding derivatives quick and efficient, especially for functions like \( y = x^3 - 3x + 1 \).

• For \( x^3 \), applying the power rule gives \( 3x^2 \).
• The derivative of \( -3x \) is simply \( -3 \) since it's similar to \( x^1 \).
• Constants like \( +1 \) have a derivative of zero, as they do not change.

The power of this rule is its simplicity and the speed it introduces to calculus problems, making it crucial for finding slopes.

Using the Point-Slope Formula

The point-slope formula is a straightforward way to write the equation of a line when you know a point on the line and its slope. This formula is: \[y - y_1 = m(x - x_1)\] where \( m \) is the slope, and \( (x_1, y_1) \) are the coordinates of the known point.
It's ideal for writing tangent line equations because once you have the slope from the derivative, the rest is simple.

• Start with the slope \( m \) calculated from the derivative at the given point.
• Use the point you're given, in this case \( (2, 3) \), to substitute \( x_1 \) and \( y_1 \) into the formula.
• Plug these values into the point-slope formula: \( y - 3 = 9(x - 2) \).

Simplifying this further, you can express it in slope-intercept form, \( y = mx + b \), making it easier to analyze or graph. With this formula, you can efficiently determine the specific line that just touches the curve at your chosen point.