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MATH 101 Quiz #1 (v.T1) Last Name: Thursday, January 14 First Name: Grade: Student-No: Section: Very short answer question 1. 1 mark If Z 2 f (x) dx = 4 and 0 Z 2 g(x) dx = 3, calculate 0 your answer in the box. Simplify your answer completely. Z 2 3f (x) 2g(x) dx. Please write 0 Answer: Short answer questions—you must show your work 2. 2 marks Use elementary geometry to calculate f (x) = ( Z 3 f (x) dx, where 0 x, if x 1, 1, if x > 1. 3. 2 marks The value of the following limit is equal to the area below a graph of y = f (x), integrated over the interval [0, b]: ✓ ◆ n X 4 4i lim cos 2 + n!1 n n i=1 2 . Find f (x) and b. (Do not evaluate the limit.) Please write your answers in the boxes. Answer: Answer: Long answer question—you must show your work 4. 5 marks Consider the integral: Z 5 (5 + x2 ) dx. (⇤) 0 (a) (1 mark) Approximate this integral using the left Riemann sum with n = 5 intervals. (b) (4 marks) Write down the expression for the right Riemann sum with n intervals and calculate the sum. Now take the limit n ! 1 in your expression for the Riemann sum, to evaluate the integral (⇤) exactly. You can leave your answer in “calculator-ready” form. Hint: you may use the identity n X i=1 i2 = 2n3 + 3n2 + n . 6