Math 1A, Calculus Final Exam Haiman, Fall 2004NameStudent ID NumberDiscussion Section Instructor(If you don’t remember your section instructor’s name, give section time instead.)Instructions:• Wait until you are told to begin before looking at the questions.• After being told to start, put your name on each page, in case they get separated.• Write answers in the space provided, and turn in only the exam paper. Show enoughwork so that we can see how you got your answers.• You may use one prepared sheet of notes. No other notes, books or calculators areallowed.• There are 15 questions, on both sides of the page. All questions have equal value.For grading use only1 92 103 114 125 136 147 158 Total11. Evaluate the limit if it exists (possibly as an infinite limit).(a) limx→11ln x(b) limx→11(ln x)22. Differentiate the function y = sin(sin(sin x)).3. Find (a) all local maxima and minima of the functionf(x) =xx2+ 1,and (b) the intervals of increase or decrease of f(x).24. Find the linear approximation to the function f(x) = ln x near x = 2.5. If y = exy, express dy/dx in terms of x and y.6. Suppose we use Newton’s method to approximate the root r of the function whose graphis shown, using x1= 1 for the first approximation.1r2-11For the next approximation x2, decide whether x2< r or x2> r. Justify your answer.37. Find the largest area of a rectangle with horizontal and vertical sides, lower-left corner atthe origin (0, 0), and upper-right corner on the curve y = e−x.8. Find the limit.limx→∞x1/(1+lnx)9. IfRxaf(t) dt = x ln x for all x > 0, find the function f(x) and the constant a.410. Evaluate the integral.Z20xe−x2dx11. Evaluate the indefinite integral.Z(x + 1)(x + 2)x2dx12. Sketch the region enclosed by the lines x = 2, y = 2 and the curve xy = 1, and find itsarea.513. Find the average value of the function f(x) = 1/x on the interval [1, 3].14. Find the volume of the circular cone obtained by rotating the triangle enclosed by thex and y axes and the line x + y = 1 about the y axis. (Solve the problem using integration.Do not just cite a formula you might already know for the volume of a cone.)15. Set up, but do not evaluate, an integral for the volume of the solid obtained by rotatingthe region enclosed by the x axis, the line x = 2, and the curve y = xe−xabout the y
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