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Calculus I Exam 1, Spring 2003, Answers 1. Find the equation of the line which goes through the point (3,-2) and is parallel to the line given by the equation 2x 3y 1 Answer. Writing the equation of the given line as y 2 3 x 1 3 , we see it has slope m 2 3. Thus, the line we seek goes through (3,-2) and has slope 2/3, so has the equation y 2 x 3 or y 2 3 2 x 3 4 2. Find the derivatives of the following functions: a) f x 3x4 8x2 x Answer. f x 3 4x3 b) g x x 1 1 x 8 2x 1, or f x 12x3 16x 1. 1 Answer. First write the function in exponential notation: g x x 1 x rule: g x 1 x 1 1 x 1 x 2 1 x c) h x 1 1 and now use the product 2 1 x 1 x2 Answer. Use the quotient rule: h x x 1 2x x2 1 x 1 2 x2 2x 1 x 1 2 3. Find the derivatives of the following functions: a) f x tan 2x 1 2 Answer. f x 4 tan 2x 1 sec2 2x b) g x 2x2 1 Answer. g x 1 2x2 1 2 4x 4x 1 2x2 2 4. Find the equation of the line tangent to the curve y x 2 1 2 at (2,25). Answer. Take differentials: dy 2 x 2 1 2x dx. Now, evaluate at x 2 and replace dy dx by y 25 x 2: y 25 2 5 4 x 2 leading to y 40x 55 5. An object moves in a straight line so that its position at time t is given by x t t cost. What is the velocity of the object when t 3π 4? Answer. Differentiating the velocity is v dx dt cost t sint. Now evaluate at t 3π 4. The corresponding point is in the second quadrant, so cos 3π 4 so v 2 2 2 2 sin 3π 4 2 2 3π 2 4 2 2 3π 2 4 1