21b-practice1-solutions.pdf - Solutions to practice problem...

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Solutions to practice problem set for Midterm Exam 1 MAT 21B (UC Davis, Winter 2018) Dan Romik January 27, 2018 1 Write the sum 1 + 3 + 5 + 7 + . . . + 97 + 99 in notation, and evaluate it. You may use the formula N X m =1 m = N ( N + 1) / 2. 2 (a) Evaluate the definite integral A = dx . (An answer expressed in terms of
1 x values of standard mathematical functions is acceptable.) Z 10 1 10 9 . Without using a calculator, determine which of A and B is larger, and explain how you can tell.
(c) Define C = B - 1 + 1 10 . Without using a calculator, determine which of A and C is larger, and explain how you can tell.
0 2 4 6 8 10 0.2 0.4 0.6 0.8 1.0 (a) A is the area bounded between the graph of y = 1 x and the x -axis above the interval [1 , 10]. 0 2 4 6 8 10 0.2 0.4 0.6 0.8 1.0 (b) B is the upper sum associated with an approximation to the integral defining A , corresponding to the sum of the areas of the shaded rectangles. 0 2 4 6 8 10 0.2 0.4 0.6 0.8 1.0 (c) C is the lower sum associated with an approximation to the integral defining A , corresponding to the sum of the areas of the shaded rectangles.

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