Test 4 Version B MA 141-010 Directions: Be sure to put your name and the version of the test in your blue book. Start each question on a new page. Show your work at each step to get full credit for each question. No calculators, phones, notes, or books may be used on this exam. 1. (20 points.) Approximate the area under the curvef(x) =-sin(2x) + 3 fromx= 0tox= 2πusing the left sum of 4 rectangles.2. (13 points.) Find(Δx) iff(x) = 2x2+ 4. What does(Δx)Δxrepresent?3. (26 points.) Calculate the following limits:(a) limx→03 sin(5x)4 tan(x)(b) limx→0x3x-24. (26 points.)Calculate the following integrals using any method we have learned inclass.(a)Rxcsc2(x)dx(b)R0-3√9-x2dx5.(15 points.)Find the area between the curvesf(x) = 9xandf(x) =x3in the firstquadrant. You should simplify as much as possible, but you will receive full credit aslong as everything up to arithmetic simplification is correct. 1
6.(Bonus: 5 points) Fill in the blanks for the proof of part i of the Fundamental Theoremof Calculus.
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