find the points of intersection of
(a) the tangents to the graph of y=2+2x-x^2 at the points where the graph meets the x-axis.
(b) the normals to the graph at these points.
(c) find the area of the triangle created from the two intersections and the point (0,0)
Find the roots of the polynomial.
Then find the slope at those points. y'=2-2x
Then use the point-slope form to get the line equations.
Do the same for the normals, with slope -1/y'
Find the intersections, and then you can measure the triangle.
Here's a start, showing the tangents:
http://www.wolframalpha.com/input/?i=plot+y%3D2%2B2x-x%5E2,y%3D2%E2%88%9A3+(x-1%2B%E2%88%9A3),+y%3D-2%E2%88%9A3+(x-1-%E2%88%9A3)
To find the points of intersection, tangents, normals, and the area of the triangle, let's go through the steps for each part:
(a) Finding the points of intersection of the tangents:
1. Start with the equation of the graph: y = 2 + 2x - x^2.
2. To find the points where the graph meets the x-axis, set y = 0:
0 = 2 + 2x - x^2.
3. Rearrange the equation to form a quadratic equation:
x^2 - 2x - 2 = 0.
4. Solve the quadratic equation by factoring, completing the square, or using the quadratic formula to find the roots x1 and x2, which represent the x-coordinates of the points of intersection.
Now let's move on to calculating the tangents at these points:
5. Find the derivatives of the equation y = 2 + 2x - x^2. Taking the derivative with respect to x gives: dy/dx = 2 - 2x.
6. Substitute the x-coordinates found in step 4 into dy/dx to get the slopes of the tangents.
7. Use the point-slope form of a line with the computed slopes and the corresponding points to obtain equations for the tangents.
(b) Finding the normals to the graph at these points:
8. Remember that the normal is a line perpendicular to the tangent. Thus, the slope of the normal is the negative reciprocal of the slope of the tangent.
9. Use the point-slope form of a line to derive equations for the normals at the points of intersection.
(c) Finding the area of the triangle created by the two intersections and the point (0,0):
10. Determine the coordinates of the two intersection points using the x-coordinates obtained in step 4 and the equation of the graph.
11. Use the formula for the area of a triangle: Area = 0.5 * base * height.
- The base of the triangle would be the distance between the x-coordinates of the two intersection points.
- The height can be found by subtracting the y-coordinate of (0,0) from the y-coordinate of one of the intersection points.
12. Calculate the area using the given values.
By following these steps, you should be able to find the points of intersection, the equations of the tangents and normals, as well as the area of the triangle.