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OHIO MATH 263B - Exam Guide

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MATH263B – Final Exam – Spring 2008 – Name:Each problem is worth 10 points. There are 160 points total.Do not simplify your answers. These are the questions that appeared on one exam. Thequestions on your final can be taken from any of the material covered in the class. You mustdo and understand all of the homework and review all of the material to be well-prepared.1. Find the most general anti-derivative:f(u) =u4+ 3√uu2.2. T he speed of a r unner increased steadily during the first three seconds of a race. Herspeed at half-second intervals is given in the table. Find lower and upper estimates forthe distance she traveled during these three s econds.t (s) 0 0.5 1 1.5 2 2.5 3v (ft/s) 0 6.2 10.8 14.9 18.1 19.4 20.23. Evaluate the Riemann su m for f(x) = 2 − x2, 0 ≤ x ≤ 4, with fou r equal subintervals,taking the sample points to be the right endpoints.4. Evaluate the integral.Zπ/40sec2(t) dt5. Find the average value of the function g(x) = cos x, on the interval [0, π/2].6. Evaluate the indefinite integral.Z2x(x2− 3)4dx7. Evaluate the integral:Z21ln xx2dx.8. Evaluate the integral:Zx − 9(x + 5)(x − 2)dx.9. Use the (a) Trapezoid rule and (b) Simpson’s rule to approximate the given integral withthe specified value of n . Do not simplify.Z3011 + y4dy, n = 6.10. Determine whether the integral is convergent or divergent. Find its value if it is conver-gent.Z∞−∞xe−x2dx11. Sketch the region in the plane consisting of points whose polar coordinates satisfy thegiven conditions.0 ≤ r < 4, −π/4 ≤ θ < π/6.12. Sketch the bounded region enclosed by the given curves. Decide whether to integratewith respect to x or y. Draw a typical approximating rectangle and label its height andwidth, then find the area of the region.y = 12 − x2, y = x2− 6.13. (a) Eliminate the parameter to find the Cartesian equation of the curve.(b) Sketch the plot by using the parametric equation to p lot points. Indicate with anarrow the direction in which the cur ve is traced as t increases.x = e2t, y = t + 114. Find dy/dx and d2y/dx2.x = t − et, y = t + e−t.15. Set up, but do not evaluate, the integral that represents the length of the curve.x = t − t2, y = t3/2; 1 ≤ t ≤ 2.16. Find the volume of the solid obtained by r otating the region bounded by the given curvesabout the specified line. Sketch the region, the s olid and a typical d isk or washer.y = x, y =√x; about y =

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