Find an equation of the curve whose tangent line has a slope of f'(x) =2x^-10/11, given that the point (-1,-4) is on the curve.
Your function is not written correctly, since (-1,-4) does not satisfy the equation. Try again.
In any case, f(x) is the antiderivative, and you can use the point to determine C.
maybe f'(x) = (2 x^-10)/11 ?
y = [(2/-9) x^-9 ]/11 + c
-4/2 = [1/ (-1)^9 ] /99 - c/2
-2 = -1/99 - c/2
c/2 = 1/99
c = 2/99
y = [(2/-9) x^-9 ]/11 + 2/99
y = -2 /99 x^-9 + 2/99
99 y = -2x^-9 + 2
duh. what was I thinking? SMH
To find an equation of the curve whose tangent line has a slope of f'(x) = 2x^(-10/11), we need to integrate this expression with respect to x to find the original function f(x).
Integration is the reverse process of differentiation. When we integrate a function, we essentially find the antiderivative of the function.
In this case, since the derivative of f(x) is given as 2x^(-10/11), the antiderivative of this expression will give us the original function f(x).
The integral of x^(-10/11) can be found using the power rule of integration, which states that the integral of x^n with respect to x, where n is any real number except for -1, is (x^(n+1))/(n+1) + C, where C is the constant of integration.
Applying the power rule of integration to the given expression, we get:
∫2x^(-10/11) dx = 2 ∫x^(-10/11) dx
Using the power rule, we can rewrite the integral as:
2 * (x^(-10/11 + 1))/((-10/11 + 1)) + C
Simplifying further, we have:
2/(-10/11 + 1) * (x^1/11) + C
To find the constant of integration C, we can use the given point (-1, -4) that lies on the curve. Plugging in these values into the equation, we get:
-4 = 2/(-10/11 + 1) * (-1)^(1/11) + C
Simplifying the above expression, we can solve for C:
-4 = 2/(-10/11 + 1) * (-1)^(1/11) + C
-4 = 2/(1/11) * (-1)^(1/11) + C
-4 = 2 * 11/1 * (-1)^(1/11) + C
-4 = -22 * (-1)^(1/11) + C
C = -4 + 22 * (-1)^(1/11)
So, the equation of the curve whose tangent line has a slope of f'(x) = 2x^(-10/11), with the given point (-1, -4) on the curve, is:
f(x) = 2/(-10/11 + 1) * x^(1/11) + (-4 + 22 * (-1)^(1/11))