Calculus AB Assignment Continuity Problems 4. Suppose fÝxÞ = x?1 x 4 ?1 if x ² 2 |x?3| x?3 if 2 < x . Identify any points of discontinuity, and determine (giving reasons) if they are removable, infinite (essential), or jump discontinuities. (6 Points) 5. Find all the constants a that make the function fÝxÞ = 3Ýa?xÞ if x < 3 a 2 ?x 2 if x ³ 3 continuous for all x. (2 Points) 6.G i v e ngÝxÞ = x 3 ?2x+2 if x ²? 2 x 2 +ax+bi f ?2 < x < 1 4x 4 ?3x 3 ?5x+2 if 1 ² x , find values for a and b so that the function is continuous for all x. (4 Points)

4. The function f(x) is defined by two different expressions, one for x ≤ 2 and one for 2 < x. The point of discontinuity is at x = 2. To determine the type of discontinuity, we need to check the limits as x approaches 2 from the left and right.

The left-hand limit as x approaches 2 is (2)^2 - 1 = 3. The right-hand limit as x approaches 2 is |2-3|/(2-3) = -1. Since the left-hand limit does not equal the right-hand limit, there is a jump discontinuity at x = 2.

5. For the function f(x) to be continuous for all x, the limit as x approaches 3 from the left must equal the limit as x approaches 3 from the right.

The left-hand limit as x approaches 3 is 3*(a-3). The right-hand limit as x approaches 3 is a^2 - 3^2 = a^2 - 9. Setting these equal gives 3a - 9 = a^2 - 9. Solving for a gives a = 3.

6. For the function g(x) to be continuous for all x, the limits as x approaches -2 and 1 from both sides must be equal.

The limit as x approaches -2 from the right is (-2)^2 + a*(-2) + b = 4 - 2a + b. The limit as x approaches -2 from the left is (-2)^3 - 2*(-2) + 2 = -8 + 4 + 2 = -2. Setting these equal gives 4 - 2a + b = -2.

The limit as x approaches 1 from the left is (1)^2 + a*(1) + b = 1 + a + b. The limit as x approaches 1 from the right is 4*(1)^4 - 3*(1)^3 - 5*(1) + 2 = 4 - 3 - 5 + 2 = -2. Setting these equal gives 1 + a + b = -2.

Solving these two equations simultaneously gives a = -3 and b = 1.