Question

calculus ! find the equation of the normal line to the curve `f(x) = (1)/(x)` at the point (1,1)

Answer 1

The slope of the tangent line at the point x = a of the function f(x) is f'(a).

We are given the function:

$f(x)=\frac{1}{x}$

Computing the first derivative:

$f^{\prime}(x)=-\frac{1}{x^2}$

The slope of the tangent line at (1, 1), that is, where x = 1 is:

$f^{\prime}(1)=-\frac{1}{1^2}=-1$

The tangent line and the normal line are perpendicular to each other. If their respective slopes are m1 and m2, then:

$m_1\cdot m_2=-1$

We have calculated m1 = -1, calculate m2:

$m_2=-\frac{1}{m_1}=-\frac{1}{-1}=1$

Now we know the slope of the normal line. We need to find its equation. Use the point-slope formula:

y - k = m(x - h)

Where m is the known slope and (h, k) is a point of the line. We are given the point (1, 1), thus:

y - 1 = 1(x - 1) = x - 1

Adding 1:

y = x

Answer: D. x