Practice Midterm II - 1

Practice Midterm II - 1 - 2 continued on next page Name:...

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Name: Student number: 1. () Each part of the following question is worth 2 points. NO PARTIAL CREDIT will be given. (i) Z 1 2 x dx = (ii) Write out the Midpoint Rule with n = 4 to approximate Z 7 3 e x 2 dx (you don’t need to simplify). (iii) Find the average value of f ( x ) = x 1 / 3 on the interval [ - π/ 2 , π/ 2]. (iv) Z π/ 4 0 sec( t ) tan( t ) dt = (v) State the Fundamental Theorem of Calculus (as on p. 387). 1 continued on next page
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Name: Student number: 2. () Define f ( x ) = Z x - 1 sin( t 3 ) dt . (i) Compute f ( - 1) and f (1). Hint: Don’t try to compute an antiderivative for sin( t 3 ), instead use some other properties of the integral and sin( t 3 ). (ii) Find the critical point of f on the interval [ - 1 , 1]. (iii) Argue that f must have an absolute min on the interval [ - 1 , 1] at your critical point (without actually computing the absolute min).
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Unformatted text preview: 2 continued on next page Name: Student number: 3. () Use linear approximation to estimate 26. With reference to a graph, explain why your answer is an overestimate. 3 continued on next page Name: Student number: 4. () Find the volume of the solid obtained by rotating the region bounded by y = x , y = 1, y = 3, and x = 1 about the y-axis. 4 continued on next page Name: Student number: 5. () (i) Find the point on the line y = 5-3 x closest to the origin. (ii) Sketch a graph which includes the given line and a line segment from the origin to your solution point. What should the angle between the two lines be? 5 continued on next page...
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