# How do you find the equation of the tangent and normal line to the curve #y=1/x# at x=2?

By signing up, you agree to our Terms of Service and Privacy Policy

To find the equation of the tangent line to the curve (y = \frac{1}{x}) at (x = 2), we need to find the derivative of the function (y) with respect to (x), then evaluate it at (x = 2) to find the slope of the tangent line. Once we have the slope, we can use the point-slope form of a line to find the equation of the tangent line.

First, let's find the derivative of (y) with respect to (x): [y = \frac{1}{x}] [y' = -\frac{1}{x^2}]

Now, evaluate the derivative at (x = 2): [y'(2) = -\frac{1}{2^2} = -\frac{1}{4}]

So, the slope of the tangent line at (x = 2) is (-\frac{1}{4}).

Now, we need to find the (y)-coordinate corresponding to (x = 2) by plugging (x = 2) into the original function: [y(2) = \frac{1}{2} = 0.5]

Now that we have the slope and a point on the tangent line, we can use the point-slope form of a line to find the equation of the tangent line: [y - y_1 = m(x - x_1)] [y - 0.5 = -\frac{1}{4}(x - 2)]

This is the equation of the tangent line.

To find the equation of the normal line, we use the fact that the normal line is perpendicular to the tangent line. Therefore, the slope of the normal line is the negative reciprocal of the slope of the tangent line. So, the slope of the normal line is the negative reciprocal of (-\frac{1}{4}), which is (4).

Using the point-slope form with the same point ((2, 0.5)), we find the equation of the normal line: [y - 0.5 = 4(x - 2)]

This is the equation of the normal line.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- What is the equation of the line tangent to # f(x)=secx - 2cosx # at # x=pi/3#?
- If f is defined by #f(x)= x^3 +2x^2 + x#, how do you find the value of x when the average rate of change of f on the interval #x=-1# to #x =2# is equal to the instantaneous rate of change on the interval from #x=-1# to #x=2#? The answer is x=0.786?
- How do you find the instantaneous rate of change of #y=5x+9# at x=2?
- What is the equation of the line normal to #f(x)= sqrt(2x^3-x) # at #x=1#?
- What is the equation of the line normal to #f(x)=-4x^2 -5x +1 # at #x=-3#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7