MATH 156 MIDTERM OCTOBER 2006In this exam you will work with the integral NiMtSSRpbnRHNiI2JCwmKiYiIiUiIiItSSVzcXJ0RzYkSSpwcm90ZWN0ZWRHRi5JKF9zeXNsaWJHRiU2IywmIiIjRioqJEkieEdGJUYyISIiRipGKkYyRjUvRjQ7IiIhRio= . It ends up that it is equal to NiMlI1BpRw==. So this exam has as purpose to find approximations of NiMlI1BpRw==.Answers all the questions. When an explanation is required, your answer should not just be: "I saw it on the graph" or "The numbers show that A<B". Explain your answers using the properties of the function involved. Do not forget the student and plots packages. If you are graphing two sums on the same graph, the order you are displaying them plays a role: you may need to interchange the order of display to see both of them e.g. display(A, B) will not produce the same result as display(B, A).1. Graph the function NiMtJSJmRzYjJSJ4Rw===4*sqrt(2-x^2)-2 in the range [0,1]. 2. Use Maple to decide whether the function NiMtJSJmRzYjJSJ4Rw== is increasing or decreasing.3. Use Maple to decide whether the function NiMtJSJmRzYjJSJ4Rw== is concave upwards or concave downwards.4. Plot on the same graph the function and the left-hand sums and right-hand sums with the same number of subintervals for NiMlIm5H=2, 4, 8, e.g. LHS(8) and RHS(8). Which are larger and why?5. Plot on the same graph the function and the left-hand sums and midpoint sums with the same number of subintervals NiMlIm5H=2, 4, 8, e.g. LHS(8) and MID(8). Which are larger and why?6. Plot on the same graph the function and the right-hand sums and midpoint sums with the same number of subintervals NiMlIm5H=2, 4, 8, e.g. RHS(8) and MID(8). Which are larger and why?7. Compute the left-hand sums, right-hand sums, midpoint sums and trapezoid rule with NiMlIm5H=2, 4, 8, 16, 32, 64, 126, 256, 512, 1024 subintervals. Which ones are overestimates and which ones are underestimates for NiMlI1BpRw==? Why? Put these numbers in increasing order, e.g. LHS(8)<NiMlI1BpRw==<TRAP(8)<MID(8)<RHS(8). This order is not correct, your work should show the correct order. Explain your ordering, i.e. explain why LHS(8)<TRAP(8) or the other way around. For this part you should use a loop. If you do not know how to use a loop, stop at NiMlIm5H =126.8. How many decimal digits of NiMlI1BpRw== can you get using left-hand sums and right-hand sums? How many digits of NiMlI1BpRw== can you get using the trapezoid rule and midpoint sums?9. Plot on the same graph the function and the trapezoid rule with NiMvJSJuRyIiIg== subintervals. Explain why it is an underestimate or an overestimate.10. Plot on the same graph the function and the trapezoid rule with NiMvJSJuRyIiIw== subintervals. Which is larger? TRAP(1) or TRAP(2)? Why?with(student): with(plots):