How do you find the equation of a line tangent to the function #y=x^2(x-2)^3# at x=1?

Question

William Abdalla · Accepted Answer

I have solved this way! Please, see the answer below:

Isaiah Allen · Answer

The equation is #y=9x-10#.

To find the equation of a line, you need three pieces: the slope, an #x# value of a point, and a #y# value. The first step is to find the derivative. This will give us important information about the slope of the tangent. We will use the chain rule to find the derivative. #y=x^2(x-2)^3# #y=3x^2(x-2)^2(1)# #y=3x^2(x-2)^2# The derivative tells us the points what the slope of the original function looks like. We want to know the slope at this particular point, #x=1#. Therefore, we simply plug this value into the derivative equation. #y=3(1)^2(1-2)^2# #y=9(1)# #y=9# Now, we have a slope and an #x# value. To determine the other value, we plug #x# into the original function and solve for #y#. #y=1^2(1-2)^3# #y=1(-1)# #y=-1# Therefore, our slope is #9# and our point is #(1,-1)#. We can use the formula for the equation of a line to get our answer. #y=mx+b# #m# is the slope and #b# is the vertical intercept. We can plug in the values we know and solve for the one we don't. #-1=9(1)+b# #-1=9+b# #-10=b# Finally, we can construct the equation of the tangent. #y=9x-10#

Scarlett Allison · Answer

undefined To find the equation of a line tangent to a function at a specific point, you can follow these steps:

1. Find the derivative of the function.
2. Substitute the given x-value into the derivative to find the slope of the tangent line.
3. Use the point-slope form of a line, y - y₁ = m(x - x₁), where (x₁, y₁) is the given point and m is the slope, to write the equation of the tangent line.

For the function y = x^2(x - 2)^3, the derivative is y' = 6x(x - 2)^2 + 2x^2(x - 2)^3.

Substituting x = 1 into the derivative, we get y' = 6(1)(1 - 2)^2 + 2(1)^2(1 - 2)^3 = -6.

The slope of the tangent line at x = 1 is -6.

Using the point-slope form with the given point (1, f(1)), we can write the equation of the tangent line as y - f(1) = -6(x - 1).