Math 151 WIR Spring 2010 c Benjamin Aurispa Math 151 Exam 2 Review 1 Use differentials to approximate the value of 4 16 2 2 Find the linear approximation to f x x 2 3 at x 4 and use it to approximate the value of 2 053 1 Math 151 WIR Spring 2010 c Benjamin Aurispa 3 Find the quadratic approximation to f x cos x at x 3 4 Calculate the following limits x 8 a lim x 6 1 2 x 6 e5x e 2x x e6x e 4x b lim 2 Math 151 WIR Spring 2010 c Benjamin Aurispa 5 Find an equation of the tangent line to the graph of f x xex 6 Consider the function f x x3 4 x3 9 a Show that f x is one to one b Find the inverse of f x 3 5 x at x 1 Math 151 WIR Spring 2010 c Benjamin Aurispa 7 Given f x e3x 3 4x3 2 find g 0 3 where g is the inverse of f 8 A 5 meter drawbridge is raised so that the angle of elevation changes at a rate of 0 1 rad s At what rate is the height of the drawbridge changing when it is 2 m off the ground 9 A dog sees a squirrel at the base of a tree 10 ft away The dog takes off running toward the tree with a speed of 3 ft s The squirrel takes off up the tree at the same time with a speed of 5 ft s At what rate is the distance between them changing 2 seconds later 4 Math 151 WIR Spring 2010 c Benjamin Aurispa 10 A 25 ft long trough has ends which are isosceles triangles with height 5 ft and a length of 4 ft across the top Water is poured in at a rate of 15 ft3 min but water is also leaking out of the trough at a rate of 3 ft3 min At what rate is the water level changing when the width of the water across the trough is 2 ft 11 Calculate the following limits cos x tan 9x 1 x 0 sin x a lim x cos 7x sin 4x x 0 3 sin2 10x b lim 5 Math 151 WIR Spring 2010 c Benjamin Aurispa 12 Find the values of x 0 x 2 where the tangent line to f x sin2 x cos x is horizontal 3 13 Find y 0 for the equation sin 3x y xy 2 ex y 14 Find the slope of the tangent line to the graph of 5x2 y y 3 12 at the point 1 3 6 Math 151 WIR Spring 2010 c Benjamin Aurispa 15 For what value s of a are the curves x2 y 2 5 and y 2 4x2 5 orthogonal at the point 1 3 2 a 9 4 16 Find a tangent vector to the curve r t cos3 3t 2 sin 5t cos 4t at the point where t 4 7 Math 151 WIR Spring 2010 c Benjamin Aurispa 17 The position of an object is given by the function r t a What is the speed of the particle at time t 1 b What is the acceleration at time t 1 18 Find f 29 x where f x e 4x cos 3x 8 t2 8t t 3t 2 Math 151 WIR Spring 2010 c Benjamin Aurispa 19 Consider the curve x t2 6t y 2t3 9t2 a Find the slope of the tangent line at the point 5 11 b Find the points on the curve where the tangent line is horizontal or vertical 20 Find an equation of the tangent line to the curve x tan 2t at the point where t 0 9 4 y t2 3t 2 4 t 1 5 2 t 1 2 Math 151 WIR Spring 2010 c Benjamin Aurispa 21 Suppose F x x f f0 g g0 0 2 5 2 4 1 1 6 1 24 g 7x 1 f 5x g cos x g f 3x Use the table of values below to find F 0 0 p 3 2 7 3 3 1 22 Show that the function f x e2x cos x satisfies the differential equation y 00 4y 0 5y 0 10 Math 151 WIR Spring 2010 c Benjamin Aurispa 23 Differentiate the following a f x tan2 5x sec 3x2 x3 b g t 3t2 5t 4t3 t2 c h x x62 4 8 x 8x6 x 5 cot 9x ee 11
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