Math151 Sample Problems for the Final Fall 2011 1 Given vectors a 2 b 2 3 Find a a unit vector u that has the same direction as 2 b a b angle between a and b c comp b a proj b a 2 Find the work done by by a force of 20 lb acting in the direction N50o W in moving an object 4 ft due west 3 Find the distance from the point 2 3 to the line 3x 4y 5 0 4 Find vector and parametric equations for the line passing through the points A 1 3 and B 2 1 5 Find all points of discontinuity for the function 2 x 1 if x 2 f x x 2 if x 2 6 Find the vertical and horizontal asymptotes of the curve y 7 Find dy for each function dx x2 4 3x2 3 a y sin x x 5 2x 1 x2 4 2 b y 3 1 3x c y t sin 1 t x t cos 1 t2 d 2x2 2xy y 2 x 8 Find the equation of the tangent line to the curve y x 5 x at the point 1 2 9 A particle moves on a vertical line so that its coordinate at time t is y t3 12t 3 t 0 a Find the velocity and acceleration functions b When is the particle moving upward c Find the distance that particle travels in the time interval 0 t 3 10 The vector function r t t 25t 5t2 represents the position of a particle at time t Find the velocity speed and acceleration at t 1 11 Find y if y e 5x cos 3x 12 Find d50 cos 2x dx50 1 13 A ladder 10 ft long rests against a vertical wall If the bottom of the ladder slides away from the wall at a rate of 0 9 ft s how fast is the angle between the ladder and the ground changing when the bottom of the ladder is 8 ft from the wall 14 Find the quadratic approximation of 1 x for x near 4 15 If f x x x2 ex and g x f 1 x find g 1 16 Solve the equation ln x 6 ln x 3 ln 5 ln 2 5 1 sin 17 Find cos 4 18 Evaluate each limit sin x sin 2x x 0 sin 3x b lim cot x csc x a lim x 0 c lim xsin x x 0 19 Find the absolute maximum and absolute minimum values of f x x3 2x2 x on 1 1 20 For the function y x2 ex find a All asymptotes b Intervals on which the function is increasing decreasing c All local minima local maxima d Intervals on which the function is CU CD e Inflection points 21 A cylindrical can without a top is made to contain V cm3 of liquid Find the dimensions that will minimize the cost of the metal to make the can 22 Find the derivative of the function f x Rx 0 t2 dt t2 1 23 Evaluate the integral 2 R2 1 a x dx x 1 b R2 x2 1 dx x 1 c 2 R cos t 2 sin t dt 0 24 Find the area under the curve y x above the x axis between 0 and 4 2 25 A particle moves in a straight line and has acceleration given by a t t2 t Its initial velocity is v 0 2 cm s and its initial displacement is s 0 1 cm Find the position function s t 26 Find the vector function r t that gives the position of a particle at time t having the acceleration a t 2t initial velocity v 0 and initial position 1 0 3
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