Projects

Use scientific notebook to find the smallest positive zero, a, of the function h(x) = x - cos(x/nn), where nn is the last two digits of your social security number. If your last two digits are 00, then use 99. Accuracy required: |h(a)| < 10^-8.

Plot h(x). Use a plot interval which has the smallest positive zero of h in the center of the plot.

After you perform the iteration, define A = iteration vector. Then let scientific notebook numerically evaluate h(A).

Turn in a hard copy of your project. Projects will be graded on neatness, completeness, accuracy, readability, and mathematical correctness.

Calculate the Lower and Upper sums (s₁₆ and S₁₆) for f(x) = x⁷ over the interval [10, 20]. Use 16 partitions of equal size. Draw the picture for partitions and rectangles for the upper and lower sums.

Remember, for to calculate lower sums, for each interval [x_j, x_j+1] choose c_j so that f(c_j) is less than or equal f(x) for each x in [x_j, x_j+1]. Let D x = x_j+1 - x_{j .}

Then s_n = D x {(f(c₀) + f(c₁) + ...+ f(c_n-1)}

For increasing functions, c_j is just the left endpoint. For a picture just use {maple -> calculus -> Plot Approx. Integral} and in the Plot Components Dialog box set the range to 10-20, the number of boxes to 16 and select Right Boxes.

S_n is calculated similarly. Just choose C_j in the interval [x_j, x_j+1] so that f(C_j) is equal to or greater than f(x) for each x in the interval [x_j, x_j+1].

Find formulas for s_n and S_n. Simplify s_n and S_n so that each is a sum of powers of (1/n). Then take the limit as n goes to infinity.

Finally, use scientific notebook to calculate the integral of f(x) = x⁷ on the interval from 10 to 20 and see if this answer agrees with both of the limits obtained above.

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