advertisement

MATH 101 Quiz #1 (v.T2) Last Name: Thursday, January 14 First Name: Grade: Student-No: Section: Very short answer question 1. 1 mark If Z 2 f (x) dx = 3 and 0 Z 2 g(x) dx = 4, calculate 0 Z 2 2f (x) + 3g(x) dx. Please 0 write your answer in the box. Simplify your answer completely. 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 2, 2, if x > 2. 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 log 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 3 (7 + x3 ) dx. (⇤) 0 (a) (1 mark) Approximate this integral using the left Riemann sum with n = 3 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 i3 = n4 + 2n3 + n2 . 4