How do you find the equation of the tangent line to the curve y = sec(x) at the point (2, sec(2))?

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To find the equation of the tangent line to the curve y = sec(x) at the given point, we will follow these steps:

  1. Determine the point of tangency: You are given the point (2, sec(2)). First, calculate the y-coordinate at x = 2:
    • y = sec(2) = 1 / cos(2)
  2. Differentiate the function: To find the slope of the tangent line, you’ll need the derivative of y = sec(x). The derivative is:
    • dy/dx = sec(x)tan(x)
  3. Evaluate the derivative at the point: Now, substitute x = 2 into the derivative to find the slope of the tangent line:
    • m = sec(2)tan(2)
  4. Write the equation of the tangent line: The equation of a line in point-slope form is given by:
    • y - y_1 = m(x - x_1)

    Substituting the values:

    • y - sec(2) = sec(2)tan(2)(x - 2)
  5. Rearrange to standard form: You can rearrange the equation to get it into slope-intercept form (y = mx + b) or keep it in point-slope form based on preference.

Thus, the final equation of the tangent line to the curve at the given point is:

y - sec(2) = sec(2)tan(2)(x - 2)

This gives you the slope and point at which the tangent line touches the secant curve, ensuring a precise representation of the curve’s behavior at that point.

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