A tangent line is a line that touches a curve at only one point and has the same slope as the curve at that point.
The slope of a tangent line can be found by taking the derivative of the function at the point of tangency. In this case, the function is f(x) = x^2-4x+5, so the derivative is f'(x) = 2x-4. Plug in the x coordinate of the point of tangency (in this case, 0) to find the slope.
The equation of a tangent line can be written in point-slope form: y - y1 = m(x - x1), where m is the slope and (x1,y1) is the point of tangency. Plug in the slope (found in the previous question) and the coordinates of P(0,1) to get the equation y - 1 = -4x. Simplify to get the final equation of the tangent line: y = -4x + 1.
The slope of the tangent line changes as the point of tangency moves because the derivative of the function changes at different points. This means that the slope of the tangent line is not constant and will vary depending on the location of the point of tangency.
No, there can only be one tangent line to a curve at a given point. This is because a tangent line is defined as a line that touches the curve at only one point, so there cannot be multiple lines that satisfy this condition at the same point.