Find the area of the region bounded by the parabola y=x^2 , the tangent line to this parabola at (10, 100), and the x-axis.
Well, it seems like you're asking for an area, but I'm more familiar with jokes and punchlines! Nevertheless, I'll give it a try.
To find the area bounded by the parabola, the tangent line, and the x-axis, we can use calculus. We need to find the x-coordinate where the tangent line intersects the parabola, and then integrate the difference between the parabola and the x-axis within that range.
But first, let me tell you a joke to lighten the mood: Why don't scientists trust atoms?
Because they make up everything!
Now, back to the math! The equation of the parabola is y = x^2. To find the x-coordinate where the tangent line intersects the parabola, we'll need to find the derivative of the parabola function.
The derivative of y = x^2 is 2x. Plugging in the x-coordinate of the tangent point (10) into the derivative, we get a slope of 20. So, the equation of the tangent line is y = 20x - 100.
Next, we need to find the two x-coordinates where the tangent line intersects the parabola. Setting the equation of the parabola equal to the equation of the tangent line, we get:
x^2 = 20x - 100
Rearranging, we have:
x^2 - 20x + 100 = 0
Solving this quadratic equation, we find that x = 10 ± √100. So, we have two intersection points: (10 - √100, 10 - √100)^2 and (10 + √100, 10 + √100)^2.
Now, we can find the area by integrating the difference between the parabola and the x-axis within this range. Since the parabola is always above the x-axis in this scenario, the integral will be simply:
∫[10 - √100, 10 + √100] (x^2) dx
Integrating the above expression will give you the area of the region bounded by the parabola, tangent line, and the x-axis. But let me remind you that my real talent lies in humor, and math might not be my strongest suit!
Keep smiling and remember to laugh through the numbers!
To find the area of the region bounded by the parabola y=x^2, the tangent line at (10, 100), and the x-axis, we can use integration.
Step 1: Determine the points of intersection.
We find the x-coordinate of the intersection points by equating the parabola equation to 0, which gives x^2 = 0. Therefore, the parabola intersects the x-axis at the origin (0, 0).
Next, we find the x-coordinate of the point of tangency on the parabola. Since the tangent line passes through (10, 100), we substitute these coordinates into the parabola equation: 100 = (10)^2. Solving for x, we get x = 10.
Step 2: Set up the integral.
Since the region is bounded by the parabola, the tangent line, and the x-axis, the limits of integration will be from 0 to 10.
To find the area, we integrate the difference between the equations of the parabola and the tangent line with respect to x. The equation of the tangent line at (10, 100) is y = mx + c, where m is the slope of the tangent line and c is the y-intercept. The slope of the tangent line can be found by taking the derivative of the parabola equation, which gives dy/dx = 2x. Thus, the slope at x = 10 is 2(10) = 20.
Therefore, the equation of the tangent line is y = 20x + c. Substituting the point (10, 100) into the equation, we get 100 = 20(10) + c. Solving for c, we find that c = 100 - 200 = -100.
So, the equation of the tangent line is y = 20x - 100.
Step 3: Calculate the integral.
The integral to find the area is given by:
∫[0,10] (x^2 - 20x + 100) dx
Integrating this expression gives:
∫[0,10] x^2 dx - ∫[0,10] 20x dx + ∫[0,10] 100 dx
Integrating each term gives:
(1/3) * x^3 - 10 * (1/2) * x^2 + 100 * x | [0,10]
Evaluating this expression at the limits of integration:
(1/3) * (10)^3 - 10 * (1/2) * (10)^2 + 100 * (10) - [(1/3) * (0)^3 - 10 * (1/2) * (0)^2 + 100 * (0)]
Simplifying the expression:
(1/3) * 1000 - 10 * (1/2) * 100 + 1000 - 0 = 1000/3 - 500 + 1000
Finally, simplifying further, we get:
(1000 - 1500 + 3000)/3 = 2500/3
Therefore, the area of the region bounded by the parabola y=x^2, the tangent line at (10, 100), and the x-axis is 2500/3 square units.
To find the area of the region bounded by the parabola y = x^2, the tangent line to this parabola at (10, 100), and the x-axis, we need to identify the points where these curves intersect.
First, let's find the equation of the tangent line at (10, 100). The slope of the tangent line is equal to the derivative of the function at x = 10. We can find this derivative by taking the derivative of the function y = x^2.
dy/dx = 2x
Since we want the slope at x = 10, we substitute x = 10 into the derivative equation:
dy/dx = 2(10) = 20
Now we have the slope of the tangent line. Using the point-slope form of a linear equation, we can write the equation of the tangent line:
y - 100 = 20(x - 10)
Simplifying:
y - 100 = 20x - 200
y = 20x - 100
Next, we need to find the x-coordinates of the intersection points between the parabola y = x^2 and the tangent line y = 20x - 100. Since we want the points where they intersect with the x-axis, we set y = 0 in both equations:
For the parabola: 0 = x^2
For the tangent line: 0 = 20x - 100
Solving the first equation, we have x = 0. This is one of the intersection points.
Solving the second equation, we have 20x = 100, which gives us x = 5. This is the other intersection point.
Now we have the x-coordinates of the intersection points, which are x = 0 and x = 5. To find the y-coordinates, we substitute these x-values into either the parabola equation or the tangent line equation.
Using the parabola equation, when x = 0, we have y = 0^2 = 0. When x = 5, we have y = 5^2 = 25.
So the points of intersection are (0, 0) and (5, 25).
Now, we can calculate the area of the region bounded by the curves. We need to find the area under the parabola between x = 0 and x = 5, and subtract the area under the tangent line between the same x-values.
The area under the parabola can be found using the definite integral:
∫[0, 5] x^2 dx
Evaluating this integral, we get:
(1/3)x^3 | [0, 5] = (1/3)(5^3) - (1/3)(0^3) = 125/3
The area under the tangent line can be found using the definite integral:
∫[0, 5] (20x - 100) dx
Simplifying and evaluating the integral, we get:
10x^2 - 100x | [0, 5] = 10(5^2) - 100(5) - 10(0^2) + 100(0) = 75
Finally, subtracting the area under the tangent line from the area under the parabola, we get:
Area = (125/3) - 75/3 = 50/3
Therefore, the area of the region bounded by the parabola y = x^2, the tangent line at (10, 100), and the x-axis is 50/3 square units.