How do I find the equation of the line that is perpendicular to the tangent line of the curve given by y = x³ + 4x + 1 at the point (2, 1)?

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To find the equation of the line perpendicular to the tangent line of the curve defined by y = x³ + 4x + 1 at the point (2, 1), we will follow these steps:

  1. Calculate the derivative of the curve: The first step is to find the derivative of the function to determine the slope of the tangent line. The derivative, denoted as y’, gives the slope of the tangent line at any point on the curve.
  2. Find the slope of the tangent line: We will substitute the x-value of the point into the derivative to find the slope at that specific point.
  3. Determine the slope of the perpendicular line: The slope of the line perpendicular to the tangent line is the negative reciprocal of the slope of the tangent line.
  4. Use point-slope form to write the equation: Finally, we will use the point-slope form of the equation of a line to write the equation of the desired perpendicular line.

Step 1: Calculate the derivative

The derivative of the curve is: y’ = 3x² + 4

Step 2: Find the slope of the tangent line

Now, we will substitute x = 2 into the derivative:

y'(2) = 3(2)² + 4 = 3(4) + 4 = 12 + 4 = 16

So, the slope of the tangent line at point (2, 1) is 16.

Step 3: Determine the slope of the perpendicular line

The slope of the line that is perpendicular to the tangent line is the negative reciprocal of the slope we just found:

m_perpendicular = -1/16

Step 4: Use point-slope form to write the equation

We can use the point-slope formula, which is given by:

y – y_1 = m(x – x_1)

Where (x_1, y_1) is the point we are using (2, 1) and m is the slope we just found, which is -1/16.

Substituting in these values:

y – 1 = -1/16(x – 2)

By simplifying this, we arrive at:

y – 1 = -1/16x + 1/8

y = -1/16x + 1/8 + 1

y = -1/16x + 9/8

Final Answer:

The equation of the line perpendicular to the tangent line at the point (2, 1) is y = -1/16x + 9/8.

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