# What is the equation of the line normal to #f(x)=x ^3-3x^2 # at #x=4#?

The gradient of the normal line is the reciprocal of this value:

Now, let's express the equation of the normal line in point-slope form:

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

To find the equation of the line normal to the function f(x) = x^3 - 3x^2 at x = 4, we need to determine the slope of the tangent line at that point and then find the negative reciprocal to obtain the slope of the normal line.

To find the slope of the tangent line, we take the derivative of the function f(x). The derivative of f(x) = x^3 - 3x^2 is f'(x) = 3x^2 - 6x.

Next, we substitute x = 4 into the derivative to find the slope of the tangent line at x = 4. f'(4) = 3(4)^2 - 6(4) = 48 - 24 = 24.

Since the slope of the normal line is the negative reciprocal of the slope of the tangent line, we have: Slope of normal line = -1/24.

Now, we have the slope of the normal line and the point (4, f(4)) = (4, 4^3 - 3(4)^2) = (4, 16 - 48) = (4, -32).

Using the point-slope form of a line, we can write the equation of the line normal to f(x) at x = 4: y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point. Therefore, the equation of the line normal to f(x) = x^3 - 3x^2 at x = 4 is: y - (-32) = (-1/24)(x - 4).

Simplifying the equation, we get: y + 32 = (-1/24)x + 1/6.

Rearranging the equation, we have: y = (-1/24)x - 31/6.

Thus, the equation of the line normal to f(x) = x^3 - 3x^2 at x = 4 is y = (-1/24)x - 31/6.

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

- The mass, m(t), in grams, of a tumor t weeks after it begins growing is given by #m(t) = (te^t)/(80)#. What is the average rate of change, in grams per week, during the fifth week of growth?
- What is the equation of the line tangent to #f(x)=x sin2x # at #x=pi/12#?
- Using the limit definition, how do you differentiate #f(x)=sec x#?
- How do you find the instantaneous slope of #y=4# at x=100?
- How does Leibniz notation work for second derivatives?

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