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Unformatted text preview: University of Toronto at Scarborough Physical Sciences Division, Mathematics FINAL EXAMINATION MATA26Y Calculus Examiners: X. Jiang R. Sharpe S. Tryphonas Date: April 24, 2001 Duration: 3 hours 1. (a) [3 pts] State the Intermediate Value Theorem (IVT) (b) [2 pts] Does IVT apply to the function f ( x ) = x 2- 3 x 2 + x- 2 on the interval [0 , 2]? (c) [3 pts] Show that the equation x 5 + 2 x- 2 = 0 has exactly one root on the interval [0 , 1]. (d) [3 pts] Show that x 5- x 3 + 4 x 2- 4 = 0 has all its roots in the interval [- 9 , 9]. 2. Consider the curve given by x 2 3 + y 2 3 = 5. (a) [1 pt] Verify that the point P = (8 , 1) lies on the curve. (b) [3 pts] Calculate the tangent line to the curve at P . (Put your answer in the form y = mx + b .) 3. [15 pts] Sketch the graph of f ( x ) = x 2 x 2 + e x , being sure to show: (a) The regions where the function is positive and negative, increasing and decreasing....
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This note was uploaded on 10/04/2011 for the course MATH 16121 taught by Professor Rachelbelinsky during the Spring '11 term at Georgia State University, Atlanta.
- Spring '11