Find the area of the region bounded by the parabola y=x^2, the tangent line to this parabola at (1,1) and the x-axis.

I don't really get what this question is asking.

It looks like the area of right triangle to me...try the graph, and shade the area under the tangent.

Find out where the tangent line crosses the x-axis. Then draw yourself a figure to see what area they are talking about. The tangent line will intersect the x axis at some x value that is >0. I would suggest doing this by calculating the area under the y = x^2 curve and the x axis from x=0 to x=1, using a simple integral from 0 to 1 of y(x), and subtracting the triangular area formed by the tangent line, x=1 and the x axis.

Please ignore the "#1 of 2" in my name. There is aome problem with the Jiskha software automatically entering my name wrong when I prepare answers

i need a refresher on how to find the tangent line. thanks.

The first derivative of a function evaluated at the point will give the slope of the line tangent to the function there. Since you know the point and the slope you can determine the equation for the line.
Briefly
m=f'(x_o) at (x_o,f(x_o)) and the equation for the tangent line is
y-f(x_o)=f'(x_o)(x-x_o)
where (x_o,f(x_o)) is any point where f(x) is defined.

ok. I found the equation to be y=2x-1. I draw the diagram but i am not sure which area i am suppose to find. The triangle under the tangent line and the x-axis??

Yes, you are correct. The area you need to find is the region bounded by the parabola y = x^2, the tangent line y = 2x - 1, and the x-axis.

To find this area, you need to first find the x-coordinates where the tangent line and the parabola intersect. Set the equations of the tangent line and the parabola equal to each other:

x^2 = 2x - 1

Now, solve this equation to find the x-coordinates of the points of intersection. Once you have the x-coordinates, you can find the corresponding y-coordinates by substituting them into the equation of the parabola.

Next, you can find the area of the region bounded by the tangent line, the parabola, and the x-axis. The area can be found using integration. Integrate the function (parabola) minus the tangent line from the smallest x-coordinate to the largest x-coordinate of the points of intersection. This will give you the area of the region.

In this case, the area will be the integral of (x^2 - (2x-1)) dx from the smallest x-coordinate to the largest x-coordinate.

Once you evaluate this integral, you will have the area of the region bounded by the parabola, the tangent line, and the x-axis.