test2-04 - (6%) (c) Find the critical points, and nd where...

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Department of Mathematics, University of Toronto Term Test 2 – January 12, 2005 MAT 137Y, Calculus! Time Alloted: 1 hour 50 minutes 1. For the following, simplify your answers unless otherwise stated. (8%) (i) Evaluate lim x 0 cos5 x - cos3 x x 2 . (8%) (ii) Find the derivative of f ( x ) = - x , where x < 0, using the definition of derivative . (8%) (iii) Find the equation of the tangent line to the curve 2 ( x 2 + y 2 ) 2 = 25 ( x 2 - y 2 ) at the point ( 3 , 1 ) . (8%) (iv) Using Newton’s Method with x 1 = 2, find the next approximation x 2 to the root of x 4 - 20 = 0. (12%) 2. A car leaves an intersection at noon and travels due south at a speed of 80 km/h. Another car has been heading due east at 60 km/h and reaches the same intersection at 1:00 p.m. At what time were the cars closest together? Verify that your answer is correct by showing that at that given time the distance is minimized. 3. Let f ( x ) = 3 p ( x 2 - 1 ) 2 . (3%) (a) Show that f 00 ( x ) = 4 9 ( x 2 - 3 )( x 2 - 1 ) - 4 / 3 . (3%) (b) Find the domain, and (if any) the intercepts and asymptotes.
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Unformatted text preview: (6%) (c) Find the critical points, and nd where f is increasing and decreasing. Determine whether the critical points are local maxima, local minima, cusps, vertical tangents, or none of the above. (5%) (d) Find where f is concave up and concave down and locate the points of inection. (5%) (e) Sketch the graph of f ( x ) , using the information found above. 4. (5%) (a) State the formal denition of lim x a + f ( x ) = . (5%) (b) Prove, using the denition above, that lim x 1 + 1 x-1 = . 5. (5%) (a) State the Mean Value Theorem. (9%) (b) By applying the Mean Value Theorem to f ( x ) = x , show that 1 11 &lt; 102-10 &lt; 1 10 . (10%) 6. Show that no two points on the curve y = x 4 + 2 x 2-x share a common tangent line. 1...
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