C₁: closed circle, x² + y² = 9 C₂: closed circle, (x - 2)² + y² = 1 C3: closed circle, x² + y² = 4 C4: boundary of the closed box, -1 ≤ x ≤ 3, -1 ≤ y ≤ 5 S₁: boundary of the closed box, -2 ≤x≤ 2,-2 ≤ y ≤ 2,-2 ≤z≤2 S₂: boundary of the closed box, 1 ≤ x ≤ 2, 3 ≤ y ≤ 4,5 ≤z ≤6 S3: the sphere, x² + y² + z² = 4 S4: the sphere, (x - 5)2 + y² + z² = 1 S5: the sphere, x² + y² + z² = 9 Assume that all the curves above are oriented counterclockwise in R2, and all the surfaces are oriented outward in R³. (A) ₁ <-y, x>. dr (B) ₂ <-y, x> ||²||² dr (C) √₂ < x,y > dr Value Brief Justification

You have the class period to complete your exam during. You MUST SHOW WORK to receive credit! All work must be your own original work. No notes or books are allowed. Only an approved calculator may be used. No sharing of calculators allowed. Sprece, 1) Describe in words the region of R³ represented by 1 ≤ x² + y² + z² ≤5 the inequalities: X

II. Sketch the graph of the inequality. 2.) 5x + 3y 2 - 15 y 4 61 4- 3- 21 192 ms ari 2i t92 noituloz erit 16rj st6ibni noisulce.oh -6-5-4 -3 -2 -1 1 2 3 4 5 6 -11 -21 -3 -4- 2. -5- -6-

A doctor sees between 7 and 12 patients each day. On Mondays and Tuesdays, the appointment times are 15 minutes. On Wednesdays and Thursdays, they are 30 minutes. On Fridays, they are one hour long. The doctor works for no more than 8 hours a day. Here are some inequalities that represent this situation. - .25 is less than or equal to y is less than or equal to 1 - 7 is less than or equal to x is less than or equal to 12 -  What does each variable represent? What does the expression in the last inequality mean in this situation?

Sketch the region that corresponds to the given inequalities. 2x  +  4y  ≥  4 2x  −  4y  ≤  4 x      ≥  0