Find the equation of the tangent and normal line of the equation:
f(x) = √x^2 - 1, at x = 1
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Show your complete solution.
You must have meant
f(x) = √(x^2 - 1) or else you would have written f(x) = x - 1
f ' (x) = (1/2)(x^2 - 1)^(-1/2) (2x)
= x/√(x^2 - 1)
at x = 1 , f'(1) = 1/0 , which is undefined
so we must have a vertical tangent at (1, 0)
and the equation of that tangent is x = 1
the normal would be a horizontal line, y = 0
geez @Stefan, will you stop posting this same lame problem?
To find the equation of the tangent line and normal line of the function f(x) = √(x^2 - 1) at x = 1, we need to follow these steps:
Step 1: Find the derivative of the function f(x).
Step 2: Determine the slope of the tangent line at x = 1 using the derivative.
Step 3: Calculate the equation of the tangent line using the slope and the point (1, f(1)).
Step 4: Determine the slope of the normal line by taking the negative reciprocal of the tangent line's slope.
Step 5: Calculate the equation of the normal line using the slope and the point (1, f(1)).
Now let's go through each step:
Step 1: Find the derivative of the function f(x).
To find the derivative, we need to use the chain rule. The derivative of √(x^2 - 1) with respect to x can be written as:
f'(x) = (d/dx)√(x^2 - 1)
To simplify this, we use the chain rule. Let u = x^2 - 1. Then, f(x) = √u. The derivative is given by:
f'(x) = (d/du)√u * (du/dx)
The derivative (d/du)√u can be found using the power rule for differentiation:
(d/du)√u = (1/2)u^(-1/2)
To find du/dx, we differentiate u = x^2 - 1 with respect to x:
du/dx = 2x
Substituting these values back into the equation for f'(x):
f'(x) = (1/2)(x^2 - 1)^(-1/2) * 2x
Simplifying this expression further gives:
f'(x) = x/(√(x^2 - 1))
Step 2: Determine the slope of the tangent line at x = 1 using the derivative.
To find the slope of the tangent line at x = 1, we substitute x = 1 into the derivative we found:
f'(1) = 1/(√((1)^2 - 1)) = 1/√0
The denominator is zero, which means that the derivative is undefined at x = 1. This indicates that the tangent line is vertical, and thus the slope is undefined.
Step 3: Calculate the equation of the tangent line using the slope and the point (1, f(1)).
Since the slope is undefined, the equation of the tangent line is simply x = 1.
Step 4: Determine the slope of the normal line by taking the negative reciprocal of the tangent line's slope.
Since the slope of the tangent line is undefined, the slope of the normal line is zero.
Step 5: Calculate the equation of the normal line using the slope and the point (1, f(1)).
Since the slope of the normal line is zero, the equation of the normal line is y = f(1), where f(1) is the value of the function at x = 1.
To find f(1), substitute x = 1 into the original function:
f(1) = √(1^2 - 1) = √0 = 0
Therefore, the equation of the normal line is y = 0.
To summarize:
- Equation of the tangent line: x = 1
- Equation of the normal line: y = 0
The equation of the tangent line at x = 1 is given by:
y = 2x - 1
The equation of the normal line at x = 1 is given by:
y = -1/2x + 3/2