In this problem we deal with the Actual Errors = Actual value of integral - Approximations, and the Estimates of Errors using the Error Bounds given on the first page of this project. Consider the function f(x)= and the integral dx. (Give answers with 6 decimal places) 1 1 1 x X 1 dx. A) In this part we find the actual value of the errors when approximating X (i) Find M₁0 T10 = and S10 (ii) You can evaluate the integral dx using MATH 9 in your calculator 41 fnInt(1/X, X, 1, 4) or by hand dx = ln 4 = X 1 dx (iii) For n = 10, find the actual error EM = M10 = X the actual error ET= and the actual error Es= B) It is possible to estimate these Errors without finding the approximations M10, T10, and S10. In this part we find an estimate of the errors using the Error Bounds formulas. Error Bounds for Midpoint and Trapezoidal Rules: Suppose that f(x) ≤K, for a ≤ x ≤b. Then |EM| ≤³ K₁(b-a)³ 24n² and ET S K₁(b-a)³ 12n² Error Bounds for Simpson's Rules: Suppose that f(¹)(x) ≤ K₂ for a ≤ x ≤b. Then Es|≤ K₂(b-a)³ 180n4 (1) Find the following derivatives of f(x)=-=: f'(x) = ,J)=r_g)= AY (ii) To find K₁, sketch the graph of y=f(x) on the interval [1,4] by pressing Y MATH 12/x^3 to get Y₁ = abs(2/x³) The maximum value of f"(x) is K₁=_ 2 Or use the following inequalities: 1≤x≤4⇒1≤x³ ≤64⇒ 64 So f"(x) ≤2= K₁ (iii) With n = 10 partitions and using the above formulas for Error Bounds, find ( Show your work) LEMIS K₁(ba)³_2(4-1)³ 24n² 24(10)² = = , and ET ≤ (iv) Sketch the graph of y=f(x) on the interval [1, 4] to find K₂ an Upper Bound (or Maximum) of f(¹)(x)|, K₂ =_ and Es ≤ (v) Are the Actual Errors found in part A) compatible with the Error Bounds in part B)? x³ f(¹)(x) = C) (i) Use the Error Bound formulas to find the maximum possible error (i.e. an upper bound for the error) in approximating dx with n = 50 and using the Trapezoidal rule. |E₁|≤ (ii) Use the Error Bound formulas to find the maximum possible error in approximating dx with n = 10 using the Simpson's rule. | Es|≤ (iii) Using your answers to part (i) and (ii), the number of partitions needed to approximate ₁dx correct to 2 decimal places is approximately: X n = with the Trapezoidal rule, and n = with the Simpson's rule. D) Use the Error Bound formulas to find how large do we have to choose n so that the approximations T‚ M„, and S, to the integral dx are accurate to within 0.00001: 1 x Trapezoidal rule: |ET| ≤ K₁(b-a)³ 12n² <0.00001 2(4-1)³ < 0.00001 12n² 2(3)³ n> n = 12(0.00001) Midpoint rule: n = (show work) Simpson's rule: n = (show work)