Math 1A Calculus Fall, 2006Prof. HaimanPractice Final Exam1. Differentiate the functiony =(x + 1)√x + 23√x + 3.2. Evaluate the limit if it exists (possibly as an infinite limit).(a) limx→1+x1 − x(b) limx→1−x1 − x(c) limx→1x1 − x3. Find all points P on the curve y = x2+ 1 with the property that the tangent line at Ppasses through the origin.4. Use a linear approximation to estimate√37.5. If sin(y − x) = y + x, express dy/dx in terms of x and y.6. Find the constant a for which f(x) = x3+ ax2has an inflection point at x = 1. For thisvalue of a, find the intervals of concavity of f(x).7. Use Newton’s method to find the root of x4+ x − 4 = 0 in the interval [1, 2], correct to 6decimal places.8. Find the points on the parabola y = x2closest to (0, 1).9. Find the limit.limx→11ln x−1x − 110. Evaluate the integral.Z21x√x − 1 dx11. Find the area enclosed by the lines x = 0, y = 1 and the curve y =3√x.12. Evaluate the integral.Zπ/20cos x −12dx.113. Differentiate the functionf(x) =Z2xxettdt.14. Find the most general function f(x) for which f00(x) = cos x.15. Find an interval [0, c] on which the average value of the function f(x) = x2+ 2 is equalto 5.16. Set up an integral for the volume of the solid obtained by rotating the region enclosedby the x axis, the line x = 2, and the curve y = ln x about the y axis, using(a) the method of slices;(b) the method of cylindrical shells.Evaluate one of these integrals to find the volume.17. Find the volume of a pyramid with a square base of length 2 on each side, and height 3.18. Evaluate the limit by expressing it as an
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