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Math 126-103 (CRN 12418) Fall 2014 Carter This file contains an accumulation of quizzes from the past several semesters of Ma 126. The quizzes are those that were asked before the first test. I also am including the first tests from several recent semesters. This document should serve as an adequate study guide. Look where these problems come from in the book and study problems in a neighborhood of their union. 1. Estimate the area under the curve y = x2 for 0 ≤ x ≤ 1 by using an upper sum and four rectangles of equal width. 2. Estimate the area under the curve y = x2 between x = 0 and x = 1 by a lower sum with four equal subintervals. 3. Prove by induction: N X k= k=1 4. Use the formula Pn k=1 N (N + 1) 2 k 2 = [n(n + 1)(2n + 1)] /6 to compute 17 X k2. k=3 5. Use one of the formulas n X k=1 n X n(n + 1) n(n + 1)(2n + 1) k= k2 = or to compute 2 6 k=1 17 X k2. k=3 n n X X n(n + 1) n(n + 1)(2n + 1) 6. Use the formulas k= and k2 = to compute 2 6 k=1 k=1 71 X k(k − 1). k=18 7. Graph the region expressed by the integrand over the interval, and use elementary geometry to compute Z 3√ 9 − x2 dx −3 8. Find dy/dx. x Z y= 0 1 √ 1 + t2 dt. 9. Compute the following integrals: (a) Z cos (3z + 4) dz (b) √ Z 3 − 2s ds (c) 1 Z 0 x3 √ dx x4 + 9 (d) Z 1 x2 + √ x dx 0 (e) Z x1/2 sin (x3/2 + 1) dx 10. Compute the areas between the curves: (a) y = x2 and y = −x2 + 4x. (b) y = 2 sin (x) and y = sin (2x), for 0 ≤ x ≤ π. 11. Compute the volume of the tetrahedron that is indicated. It has edge lengths 3, 4, and 5. 3 4 y z 5 x 2 12. Find the volume of the solid formed by rotating region that bounded by y = x3 , the horizontal line y = 0, and the vertical line x = 2 about the x-axis. 13. Find the volume of the solid generated by revolving the region bounded by the lines and curves indicated below about the x-axis. y = x3 ; y = 0; x = 2. 14. Use cylindrical shells to find the volume of the solid formed by rotating region that is 2 bounded by y = 2 − x4 and the vertical lines x = 0 and x = 2 about the y-axis. 15. Use the shell method to compute the volume of the solid generated by revolving the region bounded by the curves around the y-axis: √ x = 0. y = 2x − 1; y = x; Z bq 16. Use the formula L = 1 + (dy/dx)2 dx to compute the length of the curve y = x3/2 from x = 0 to x = 4. a 17. Recall, that the surface area for the result of rotating a curve y = f (x) for x ∈ [a, b] is given by the formula Z b p 2πf (x) 1 + [f 0 (x)]2 dx SA = a Use this formula to compute the lateral surface of the cone that is obtained by rotating the line segment y = x/2 for 0 ≤ x ≤ 4 about the x-axis. 18. Compute the following definite and indefinite integrals. (a) Z 2y dy y 2 − 25 (b) 4 Z 1 (ln (x))3 dx 2x (c) er dr er + 1 Z (d) Z ln (π/2) 2ev cos (ev ) dv ln (π/6) (e) Z 0 3 1 2−θ dθ (f) Z xex dx (g) Z arctan y dy 19. It took 1800 Joules ( kilogram square meters per square second) of work to stretch a spring from its natural length of 2 meters to a length of 5 meters. Find the spring’s force constant. Math 126 Carter Test 1 Spring 2014 General Instructions. Write your name on only the outside of your blue books. Do not write on this test sheet, do all of your work inside your blue books. Write neat complete solutions to each of the problems in the blue book. Please put your test sheet inside the blue book as you leave. There are 111 points. An easy burrito is made from a heaping spoon full of canned beans, some leftover rice, grated cheese, salsa, and a largish tortilla. Combine the stuff on top of the tortilla with the cheese toping everything. Zap in the microwave for 1 minute or less, and then zap again. If you have some guacamole and/or sour cream, add these before rolling it up. 1. (5 points) State the fundamental theorem of calculus. 2. Compute the following definite and indefinite integrals ( 8 points each) (a) 18 Z (144 − 4h) dh 0 (b) Z (4x3 + 3x2 − 2x + 5) dx (c) π/2 Z cos (x) dx 0 (d) Z 2 √ (1 − x) 2x − x2 dx 0 (e) Z 0 4 5 x dx √ 25 + x2 (f) Z e2x cos (e2x ) dx (g) Z sec2 (x) dx 3. (10 points) Determine the volume that is obtained by revolving the region that lies above the curve x = y 2 , below the line y = 3, and to the right of the y-axis, around the x-axis. 4. (10 points) Determine the volume of the cone that is obtained by revolving the triangular region that is bounded by the lines y = 3x/2, y = 0, and x = 4 about the x-axis. 5. 10 points) Use the formula Z b 2πf (x) SA = p 1 + (f 0 (x))2 dx a that is used to compute the surface area of the result of rotating a curve y = f (x) about the x-axis to determine the surface area of the cone from the previous problem. Do not include the area of the circular base that lies to the right of the cone. 6. 10 points) A bag of sugar, initially weighing 12 lbs, was lifted from the floor lifted at a constant rate. As it rose to a height of 6 feet half of the sugar leaked out at a constant rate. How much work was done in lifting the the sugar? 7. 10 points) If a force of 90 Newtons stretches a spring 1 meter beyond its natural length, how much work is done in stretching the spring 5 meters beyond its natural length? Math 126 Carter Test 1 Summer 2014 General Instructions. Write your name on only the outside of your blue books. Do not write on this test sheet, do all of your work inside your blue books. Write neat complete solutions to each of the problems in the blue book. Please put your test sheet inside the blue book as you leave. There are 116 points. Try boiling your rice or couscous in meat broth. 1. (5 points) State the fundamental theorem of calculus. 2. (5 points) Prove that N X j= j=0 5 n(n + 1) 2 3. Compute the following definite and indefinite integrals (8 points each) (a) Z 5 (6x2 + 4x + 5) dx 1 (b) 9 Z (72 − 4x) dx 0 (c) π/3 Z sin (x) dx 0 (d) Z 4 √ 16 − x2 dx 0 (e) Z √ x dx x2 + 1 (f) Z ln (x) dx (g) Z x+3 dx (x + 4)(x − 1) 4. (10 points) Determine the volume that is obtained by revolving the region that is bounded by the curves y = x2 , y = 0 (the x-axis), x = 1, and x = 3 about the y-axis. 5. (10 points) Determine the volume of the cone that is obtained by revolving the triangular region that is bounded by the lines y = 5/3x, y = 0, and x = 9 about the x-axis. 6. 10 points) A bag of sand, initially weighing 72 lbs, was lifted at a constant rate. As it rose to a height of 9 feet, half of the sand leaked out at a constant rate. How much work was done in lifting the the sand? 7. 10 points) If a force of 100 Newtons stretches a spring 1 meter beyond its natural length, how much work is done in stretching the spring 8 meters beyond its natural length? Rbp 8. (10 points) Use the arc-length formula L = a 1 + (y 0 )2 dx to compute the arc length of the curve y = x3/2 from x = 0 to x = 4. 6