Math 1A Calculus Haiman Fall 2004 Final Exam Name Student ID Number Discussion 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 enough work so that we can see how you got your answers You may use one prepared sheet of notes No other notes books or calculators are allowed There are 15 questions on both sides of the page All questions have equal value For grading use only 1 1 9 2 10 3 11 4 12 5 13 6 14 7 15 8 Total 1 Evaluate the limit if it exists possibly as an infinite limit a lim x 1 1 ln x b 1 x 1 ln x 2 lim 2 Differentiate the function y sin sin sin x 3 Find a all local maxima and minima of the function f x x x2 1 and b the intervals of increase or decrease of f x 2 4 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 graph is shown using x1 1 for the first approximation 1 1 r 2 1 For the next approximation x2 decide whether x2 r or x2 r Justify your answer 3 7 Find the largest area of a rectangle with horizontal and vertical sides lower left corner at the origin 0 0 and upper right corner on the curve y e x 8 Find the limit lim x1 1 ln x x 9 If Rx a f t dt x ln x for all x 0 find the function f x and the constant a 4 10 Evaluate the integral 11 Evaluate the indefinite integral Z Z 2 2 xe x dx 0 x 1 x 2 dx x2 12 Sketch the region enclosed by the lines x 2 y 2 and the curve xy 1 and find its area 5 13 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 the x 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 rotating the region enclosed by the x axis the line x 2 and the curve y xe x about the y axis 6
View Full Document
Unlocking...