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MATH 1210-6 Spring 2003 Midterm exam I Student Name: Student ID Number: Course Abbreviation and Number: Course Title: Instructor: Math 1210 Calculus I Vladimir Vinogradov Date of Exam: Time Period: Duration of Exam: Number of Exam Pages: (including this cover sheet) Exam Type: Additional Materials Allowed: February 7, 2003 Start time: 12:55 pm 1 hours 6 Closed Book Calculator QUESTION VALUE SCORE 1 40 2 40 3 20 TOTAL 100 *) The bonus question counts for 10 points maximum. End Time: 1:55 pm 1. (40 points) Evaluate the derivative of the function y(x) = sin(π sin(π sin(πx))) at x = 0. ANSWER: 2 2. (40 points) The relation 2x2 + y 2 = 9 determines a curve in the x − y plane (an ellipse). a) Find the equation of the line which goes through the point (2,-1) on the curve and is perpendicular to the tangent line to the curve at this point. b) At what points (x0 , y0 ) does this curve have a horizontal tangent line. ANSWER: 3 3. (20 points) Differentiate s x−1 x+1 ANSWER: 4 Bonus question (10 points). As you know (sin x)0 = cos x and (cos x)0 = − sin x. Find the hundredth derivative of y(x) = sin(2x) y (100) (x) =? ANSWER: 5 Useful formulae Constant multiple rule: (k f (x))0 = k f 0 (x) Sum rule: (f (x) + g(x))0 = f 0 (x) + g 0 (x) Product rule: (f (x)g(x))0 = f 0 (x)g(x) + f (x)g 0 (x) Quotient rule: Chain rule: Ã f (x) g(x) !0 = f 0 (x)g(x) − f (x)g 0 (x) g 2 (x) d df dg f (g(x)) = · dx dg dx d df dg dh f (g(h(x))) = · · dx dg dh dx d df dg dh du f (g(h(u(x)))) = · · · dx dg dh du dx Power rule (xα )0 = αxα−1 Trigonometric functions: (cos x)0 = − sin x (sin x)0 = cos x cos(0) = 1, sin(0) = 0 π π cos( ) = 0, sin( ) = 1 2 2 cos(π) = −1, sin(π) = 0 6