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1Afinal2004

# 1Afinal2004 - Math 1A Final 12:30-3:30 You are allowed 1...

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Math 1A Final 2004-12-15 12:30-3:30 You are allowed 1 sheet of notes. Calculators are not allowed. Each question is worth 1 mark, which will be given only for a clear correct answer and correct working. There is no partial credit for wrong answers. 1. Find a formula for the inverse of the function y = 3 x 3 + 2. 2. Evaluate the limit lim h 0 (2+ h ) 3 - 8 h . 3. Prove that e x = 2 + x has at least one real root. 4. Differentiate f ( x ) = ( ax + b ) / ( cx + d ). 5. Find the derivative of the function y = tan 3 (2 x ). 6. Find dy/dx if 4 cos( x ) sin( y ) = 1. 7. Find the critical numbers of the function f ( x ) = xe 2 x . 8. Show that the equation x 4 + 4 x + c = 0 has at most two real roots. 9. Find lim x 0 cos( x ) - 1 x 2 . 10. Find the points on the ellipse 4 x 2 + y 2 = 4 that are farthest away from the point (1 , 0). 11. Explain why Newton’s method does not work for finding the root of the equation x 3 - 3 x + 6 = 0 if the initial approximation is chosen to be x 1 = 1. 12. Use one iteration of Newton’s method applied to the initial approximation x 1 = 2 to find 36 1 / 5 correct to two decimal places.
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