Name ________________________ Date _________________ Please Circle Answers Show all work for credit Part 1: Complete U-substitution 1. Evaluate ∫ 8𝑥 (2𝑥 − 1)4 𝑑𝑥 {simplify answer} by complete u – substitution 2 4𝑥 2. Evaluate ∫0 𝑑𝑥 by complete u – substitution {2 decimal place 2𝑥 1 accuracy} Part 2 𝟏 3. Integral Approximation (Trapezoid Rule) 𝒉 (f(x0) 2f(x1) 𝟐 2(x2) 2f(x3) …. f(xn)) pattern 3 1 A. Use the trapezoid rule to evaluate ∫1 𝑑𝑥 with N = 5 2𝑥 1 trapezoids. {rounded to 3 decimal places) {Sketch the x –coordinates used} B. Show the work to find the actual value {3 places} Type ____ U = ___ U’ =___ C. State the error ________________ 4. A. 2 Evaluate ∫0 (2𝑥 1)2 𝑑𝑥 by Simpsons Rule 1 h (f(x0) 3 4f(x1) 2(x2) 4f(x3) …. f(xn)) pattern (rounded to 3 decimal places) with (n = 4 intervals) Sketch the x – coordinates used. Simpson’s Rule approximation ________________ B. Show work for and find the actual value with the antiderivative (Rounded to 3 decimal places) Type___ U = ____U’ =___ C. State the error of approximation in Part A. _____ Multivariable Calculus Find the partial derivatives, but Show All Chains for Credit 5. f(x,y) = 2(x2 y2)4 fx = fxy = fy fyx 6. f(x,y) = 2ln⁡(3𝑥 2𝑦 2 ) Show All Chains for Credit a. fx = b. fxy = 7. f(x,y) = fy fyx fx fxy 2 3𝑒 𝑥 3𝑦 Show All Chains for Credit 8. f(x,y) = 2𝑥 (3𝑥 𝑦 2 )4 simplify answers Show all chains for credit and Find fy fyy (requires product rule) 9. f(x,y) = 3𝑥 (𝑥 2 4𝑦)3 Find fx needs product rule Bonus. 𝑓(𝑥, 𝑦) = ⁡ 𝑥 3 − 𝑦 2 − 12𝑥 2𝑦⁡⁡𝑑𝑜𝑙𝑙𝑎𝑟𝑠 Use the D-test to determine relative extrema and /or saddle points D-Test : 𝑫 = 𝒇𝒙𝒙 𝒇𝒚𝒚⁡⁡ − (𝒇𝒙𝒚 ) 𝟐 If D0 then If 𝑓𝑥𝑥 0⁡𝑡ℎ𝑒𝑛⁡𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙⁡𝑝𝑜𝑖𝑛𝑡⁡𝑖𝑠⁡𝑎⁡𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒⁡𝑚𝑖𝑛 Step 1. Find critical point(s) Step 2: Use. D Test to determine Extrema and Saddle Points