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The equation of the tangent and normal line of a function can be found by using calculus. For example, given the function f(x) = √x^2 - 1, at the point x = 1, we can find these lines.
To begin, we need to find the derivative of the function f(x). By using the power rule of differentiation, we obtain f'(x) = (1/2) * (x^2 - 1)^(-1/2) * 2x.
Next, we substitute x = 1 into the derivative to find the slope of the tangent line. Plugging in x = 1, we get f'(1) = (1/2) * (1^2 - 1)^(-1/2) * 2(1) = 1.
Therefore, the slope of the tangent line at x = 1 is 1.
Using the point-slope form of a line, we can write the equation of the tangent line as y - f(1) = f'(1) * (x - 1). Substituting the values, we have y - √(1^2 - 1) = 1 * (x - 1), which simplifies to y = x - 1.
Lastly, the slope of the normal line is the negative reciprocal of the slope of the tangent line. In this case, the slope of the normal line is -1.
Using the point-slope form again, we can write the equation of the normal line as y - f(1) = -1 * (x - 1). Simplifying this equation gives us y = -x + 2.
So, the equations of the tangent and normal line for the given function at x = 1 are y = x - 1 and y = -x + 2, respectively.
