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MAT286-2004Fallns - MAT286 Final Exam Fall 2004 1 Suppose...

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MAT286 Final Exam Fall 2004 1. Suppose that the slope of the tangent line to a function f ( x ) at any x is given by f 0 ( x ) = 2 x 2 - 4 x + 3 . If the graph y = f ( x ) goes through the point (1 , 10), find the function f ( x ). 1
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2 2. Consider the following function f ( x ) on the interval [0 , 2]: f ( x ) = x 2 + cos x + 4 . a) Find the exact area bounded by the curve and the x -axis on the given interval. ( Show all steps including the appropriate antiderivative and the evaluations. No credit will be given for providing only the final numerical value. ) b) Find the average value of the function f ( x ) on the given interval.
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3 c) Set up, but do not evaluate the integral that gives the volume of the solid formed by revolving the region bounded by the curve y = x 2 + cos x + 4 and the x -axis on the given interval about the x -axis. d) Use your TI-83 to evaluate the integral in part c). Round to two decimal places.
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4 3. Find the following antiderivatives. Circle your answers. a) Z 2 x x + 7 x 3 - e - 3 x dx b) Z 1 x ln x dx c) Z x 2 sin( - 2 x ) dx
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5 4. Determine whether the following improper integrals converge or diverge. If the integral
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