Fall 2009 Math 151 9 Find a tangent vector of unit length to the curve r t t sin t i t cos t j at the point where t LIFE Review for Final Exam 10 Find the slope intercept equation of the line 1 tangent to the curve x tet y at the t point e 1 x 2 2x 11 Compute lim x 2 x2 2x courtesy David J Manuel covering 1 1 6 4 d 1 Compute dx 0 sin x p 4 5t4 dt 12 Find the values of a and c which make the function below continuous and differentiable at x 3 ax2 9x c if x 3 f x 2ax2 a2 x 5 if x 3 x2 4 13 Compute lim x 4x 1 1 tan 2x 14 Compute lim 1 2 n X 3 3i 4 2 Compute lim n n n i 1 3 A conical tank shown below is losing water at a rate of one cubic foot per minute How fast is the height of the water changing when the height is 6 feet x 3x 15 Given f x f 4 h f 4 lim h 0 h x2 9 find 16 Find the point s on the curve xy 2 1 where the tangent line also passes through the point 0 1 17 The function f x e2x 2x3 1 is one toone If g f 1 what is g 2 4 Suppose you have 600 grams of a radioactive substance which decays exponentially After 5 years 2 3 of the substance remains When will only 10 of the original substance remain 18 Simplify tan sin 1 x 3 x 3 19 Given f x and f 2 3 x 1 x 0 find f x 5 A rectangular beam is cut from a cylindrical log of radius 10cm The strength of the beam is proportional to the width and the square of the depth of a cross section Find the dimensions of the strongest beam 1 2 20 Given f 3 6 f 3 0 and f 3 4 State everything you know about the graph of f at x 3 21 Find the linear approximation to f x 3 x 27 at x and use it to approximate 3 3 8 6 Let a be the vector 3i 2j and b be the vector i 4j Find a unit vector in the direction of 2a 3b 7 A triangle has vertices A 3 2 B 1 2 and C 0 4 Find the angle of the triangle at vertex A 22 Approximate the area under the graph of f x 16 x2 between x 2 and x 4 using a partition P 2 0 4 and taking x i midpoint 8 Find parametric equations of the line tangent 6t 3 j at to the graph of r t t t i t 1 the point where t 1 23 Find the derivative of g x ln x2 e4x e2 x2e x 1 24 Compute lim x 0 ln 1 x x x2 2 25 Find f if f x e 1 2 x 4x2 x x2 4 x 2 d x 27 Find dx cos2 4x 26 Compute lim 28 Determine the interval s on which f x x5 15x4 6 is decreasing and concave down 2
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