To find the area of the region bounded by the curve and the tangent line, we need to determine the x-values where the curve and the tangent line intersect.
First, let's find the equation of the tangent line to the curve y = (x^3)-4x+1 at the point (-1,4). The slope of the tangent line is given by the derivative of the function f(x) = (x^3)-4x+1 evaluated at x = -1.
f'(x) = 3x^2 - 4
f'(-1) = 3(-1)^2 - 4 = -1 - 4 = -5
The equation of the tangent line is therefore y = -5x + c, where c is the y-intercept. We can find c by substituting the coordinates of the point (-1,4) into the equation.
4 = -5(-1) + c
4 = 5 + c
c = -1
So the equation of the tangent line is y = -5x - 1.
Next, we need to find the x-values where the curve y = (x^3)-4x+1 intersects with the tangent line y = -5x - 1.
Setting the y-values equal to each other, we have:
(x^3)-4x+1 = -5x - 1
Rearranging and simplifying the equation:
x^3 + x - 2x = -2 - 1
x^3 - x = -3
We can see that x = -1 is a solution to this equation. By polynomial division, we can find the other solutions. Dividing x^3 - x by x - (-1) using synthetic division:
-1 | 1 0 -1
| -1 1
| 1 -1 0
Therefore, x^3 - x = (x + 1)(x^2 - x) = 0
The other two solutions are x = 0 and x = 1.
So, the curve y = (x^3)-4x+1 intersects the tangent line y = -5x - 1 at three points: (-1, 4), (0, 1), and (1, -6).
To find the area of the region bounded by the curve and the tangent line, we integrate the difference of the two functions between the x-values of 0 and 1.
Area = ∫[0,1] [(x^3)-4x+1 - (-5x - 1)] dx
Simplifying the integrand:
Area = ∫[0,1] (x^3 + x + 2x + 2) dx
= ∫[0,1] (x^3 - 9x + 2) dx
Integrating term by term:
Area = [1/4 * x^4] - [9/2 * x^2] + [2 * x] |_[0,1]
= [1/4 * (1)^4] - [9/2 * (1)^2] + [2 * (1)] - [1/4 * (0)^4] - [9/2 * (0)^2] + [2 * (0)]
Simplifying:
Area = 1/4 - 9/2 + 2 - 0 - 0 + 0
= (1/4) - (9/2) + 2
= 1/4 - 18/4 + 8/4
= -17/4 + 8/4
= -9/4
Since area cannot be negative, we take the absolute value:
Area = 9/4
Therefore, the area of the region bounded by the curve and the tangent line is 9/4.
The correct answer is A. 9/4.
Find the area of the region bounded by the curve y = f(x) = (x^3)-4x+1 and the tangent line to the curve y = f(x) at (-1,4).
A. 9/4
B. 8.25
C. 27/4
D. 31/4
E. 6.25
1 answer
1 answer