Find the equation of a normal line to the given curve with the given slope
y=(4x-7)^3, slope -1/12, x>0
So far, I have:
y'=12(4x-7)^2
12(4x-7)^2=12
4x-7=+1,-1
x=2, x=3/2
well the slope of our line must be -1/(-1/12) = 12 agree
I have y' = 3 (4x-7)^2 (4) = 12 (4x-7)^2 agree
that is at (4x-7)^2 = 1
4x - 7 = +1
x = 2
or
x = 3/2 agree
let's use x = 2
then y = (4*2-7)^3 = 1
so through the point (2 , 1)
y = (-1/12) x + b
1 = (-1/12) (2) + b
b = 1 + 1/6 = 7/6 = 14/12
y = - x/12 + 14/12
12 y = -x + 14
so
y = 12 x -23
forget that last line, error
To find the equation of the normal line to the given curve with the given slope, we need to first find the derivative of the curve.
The given curve is y = (4x-7)^3, and we need to find the derivative y'.
To find the derivative, we can use the chain rule:
If y = f(g(x)), then y' = f'(g(x)) * g'(x).
In this case, let f(u) = u^3 and g(x) = 4x - 7.
First, let's find f'(u):
f'(u) = 3u^2.
Next, let's find g'(x):
g'(x) = 4.
Now we can apply the chain rule:
y' = f'(g(x)) * g'(x)
= 3(4x - 7)^2 * 4
Simplifying further:
y' = 12(4x - 7)^2.
Now we have the derivative of the curve, which represents the slope of the tangent line at any given point on the curve.
Next, we need to find the x-values where the derivative is equal to the given slope of -1/12.
Setting the derivative equal to -1/12:
12(4x - 7)^2 = -1/12.
To solve this equation, we can divide both sides by 12 to isolate the expression (4x - 7)^2.
(4x - 7)^2 = -1/12 / 12
(4x - 7)^2 = -1/144.
Now, taking the square root of both sides to eliminate the square:
4x - 7 = ±√(-1/144).
Since x > 0, we can discard the negative square root.
4x - 7 = √(-1/144).
Now we can solve for x by isolating it on one side:
4x = 7 + √(-1/144).
Finally, divide both sides by 4 to solve for x:
x = (7 + √(-1/144)) / 4.
Note that the value inside the square root is negative, which means there is no real solution for x in this case.
Therefore, there is no point on the curve where the slope of the tangent line is -1/12, so there is no normal line to the curve with the given slope.