Determine equations for the tangents to the points on the curve y = -x^4 + 8x^2 such that the tangent lines are perpendicular to the line x = 1.
I've found the derivative, but not really sure where to go from here.
The first derivative is the slope of the tangent at a given point
hoping you had
-4x^3 + 16x for the derivative, which, when x = 1 becomes -4+16 or 12
Since the new line is perpendicular to the tangent , it's slope must be -1/12 ,the negative reciprocal.
-4x^3 + 16x = -1/12
4x^3 - 16x - 1/12 = 0
48x^3 - 192x - 1 = 0
perhaps your problem is solving this cubic.
It does not factor , so it is hard to do, and quite messy
You could use something like Wolfram to solve it, .....
It shows 3 real x values, appr. -2, +2, and 0
https://www.wolframalpha.com/input/?i=48x%5E3+-+192x+-+1+%3D+0
I was expecting some nicer numbers, since we are not done yet
pick x = 2.0026, sub this into the original to get y = 15.99989
let's go with (2,16)
so the tangent at the point (2,16) is
y - 16 = (-1/12)(x - 2)
thanks reiny! my problem was that it was indeed cubic, and I thought i may have perhaps been doing the question wrong. You've helped me on 2 questions today, appreciate it.
Of course, the maxima are at x = ±√8, so if (2,16) is one point of interest
is (√8-0.828,16) (just to the left of the peak), then the other point is at x = -√8-0.828 = -3.656
hmmm. and yet, the cubic works at -2, not -3.656
I'll have to ponder this...
To determine equations for the tangents to the points on the curve that are perpendicular to the line x = 1, we need to follow a step-by-step process:
Step 1: Find the derivative of the given curve.
Compute the derivative of y = -x^4 + 8x^2. Using the power rule, we get:
dy/dx = -4x^3 + 16x
Step 2: Find the slope of the tangent line.
Since the tangent lines are perpendicular to the line x = 1, their slopes must be negative reciprocals of the slope of the line x = 1. The line x = 1 is a vertical line, so its slope is undefined. However, we can consider the slope of the line to be 0 for our purposes. Therefore, the slope of the tangent lines will be the negative reciprocal of 0, which is undefined.
Step 3: Set the derivative equal to the undefined slope.
-4x^3 + 16x = undefined
Step 4: Solve for x by setting the derivative equal to undefined using algebraic manipulations.
To set the derivative equal to undefined, we need to find the values of x at which the derivative is undefined. Since the derivative is a polynomial, it is defined for all real numbers. Therefore, there are no values of x at which the derivative is undefined.
Step 5: Determine the points on the curve.
To find points on the curve corresponding to the undefined slope, we need to substitute the values of x obtained in the previous step into the original equation y = -x^4 + 8x^2.
Since there are no values of x at which the derivative is undefined, there are no points on the curve that satisfy the given conditions.
Therefore, there are no equations for the tangents to the points on the curve y = -x^4 + 8x^2 such that the tangent lines are perpendicular to the line x = 1.