M408C - final review - x-axis Use the shell method Problem...

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M408C Calculus 55815/20/25 Fall 2002 Final Exam Review Problems The Final Exam will be comprehensive, covering the material from sections 2.1, 2.3–2.5, 3.1–3.3, 3.5–3.7, 4.1–4.8, 5.1–5.7, 6.1–6.3, 7.1–7.5, 7.7, 8.2 of Salas and Hille’s Calculus , 8th edition, with more emphasis on the last sections. Problem 1. Evaluate the limits that exist a. lim x 0 tan 3 x 2 x 2 + 5 x ; b. lim h 2 h 3 - 4 h h 3 - 2 h 2 . Problem 2. Differentiate a. f ( x ) = e x arctan x ; b. f ( x ) = 4 - x 2 + 2 arcsin( x/ 2); c. f ( x ) = arctan(ln x ) . Problem 3. Find the critical numbers and classify the extreme values for f ( x ) = x 3 - 3 x + 6 x [0 , 3 / 2] . Problem 4. Determine the concavity and find the points of inflection for the function f ( x ) = e - x 2 . Problem 5. Sketch the region bounded by the curves x + y = 2 y 2 and y
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Unformatted text preview: x-axis. Use the shell method. Problem 6. Sketch the region bounded by y = 3 x-x 2 and y = 3-x and find the volume of the solid generated by revolving the region about the x-axis. Use the washer method. Problem 7. Integrate a. Z x 10 x dx ; b. Z π/ 2 π/ 6 cos x 1 + sin x ; c. Z 2 1 x 3 √ 3 x 4 + 1 dx Problem 8. Find the derivative of f ( x ) = ( x + 1) x . Problem 9. Calculate g (1) by using logarithmic differentiation g ( x ) = (1 + x )(2 + x ) x (4 + x )(2-x ) Problem 10. Use integration by parts to calculate a. Z 1 / 2 x 2 cos πxdx ; b. Z x ln( x + 1) dx...
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