How do you find the equations of the tangent lines to the curve y = x^2 + 1 that are parallel to the line 2y = 5?

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To find the equations of the tangent lines to the curve y = x2 + 1 that are parallel to the line 2y = 5, we need to follow a few steps:

  1. Determine the slope of the given line.
    First, we need to rewrite the equation of the line 2y = 5 in slope-intercept form (y = mx + b). Dividing both sides by 2 gives us:
  2. y = rac{5}{2}
  3. From this, we see that the slope (m) of the line is 0, meaning it is a horizontal line.
  4. Find the derivative of the curve.
    Next, we need to find the derivative of the curve to determine the slope of the tangent lines. The curve is y = x2 + 1, so we differentiate with respect to x:
  5. dy/dx = 2x
  6. To find the points where the tangent line is horizontal, we set the derivative equal to the slope of the line:
  7. 2x = 0
  8. Solving for x, we find that:
  9. x = 0

Find the y-coordinate of the tangent point.
Now that we have the x-coordinate, we can find the corresponding y-coordinate by substituting x back into the original curve:

  1. y = 02 + 1 = 1

So the point on the curve is (0, 1).

Write the equation of the tangent line.
We now have a point (0, 1) and we know the slope is 0, which means the tangent line is horizontal. The equation of a horizontal line through the point (0, 1) is simply:

  1. y = 1

Thus, the equation of the tangent line to the curve y = x2 + 1 that is parallel to the line 2y = 5 is y = 1.

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