Use a graphing calculator to graph f(x)=x^4-6x^3+11x^2-6x. Then use upper sums to approximate the area of the region in the first quadrant bounded by f and the x-axis using four subintervals.
The function is in the shape of a W with the middle of the W above the x-axis between x=1 and 2.
Subdivide the region into 4 subintervals and find the area using the technique you have learned in class, trapezoidal rule, Simpson's rule, integration, etc.
To graph the function f(x) = x^4 - 6x^3 + 11x^2 - 6x, you can use a graphing calculator like Desmos or any other graphing software. Here's how you can do it on Desmos:
1. Open a web browser and go to https://www.desmos.com/calculator.
2. In the input box, type the equation: y = x^4 - 6x^3 + 11x^2 - 6x.
3. Press Enter to see the graph of the function.
Now, to approximate the area of the region bounded by f(x) and the x-axis in the first quadrant using four subintervals, we can use upper sums. Here's how you can do it manually:
1. Divide the x-axis into four equal subintervals: [a, b], [b, c], [c, d], [d, e]. Choose your values for a, b, c, and d such that a < b < c < d < e, and all these points lie within the first quadrant.
2. Calculate the width of each subinterval: Δx = (e - a) / 4.
3. Calculate the x-values for each subinterval: x1 = a + Δx, x2 = b + Δx, x3 = c + Δx, x4 = d + Δx.
4. Evaluate the function at each x-value to get the heights: y1 = f(x1), y2 = f(x2), y3 = f(x3), y4 = f(x4).
5. Calculate the area of each rectangle: A1 = y1 * Δx, A2 = y2 * Δx, A3 = y3 * Δx, A4 = y4 * Δx.
6. Add up the areas of all the rectangles to get the approximate total area: Approximate Area = A1 + A2 + A3 + A4.
By following the steps above, you can manually approximate the area of the region in the first quadrant bounded by f(x) and the x-axis using four subintervals.