Understanding Continuity and Quadratic Inequalities

The video tutorial explains how to determine the interval of continuity for a given function, f(x) = cos(x) * sqrt((25 - x^2) / (x^2 - 4)). It covers two main conditions for continuity: division by zero and the domain of the square root function. The tutorial solves the quadratic inequality 25 - x^2 >= 0 and combines it with the condition x^2 - 4 != 0 to find the interval of continuity. The final interval is expressed using interval notation, excluding points where the function is undefined.

What is the function given in the problem for determining the interval of continuity?

• F(x) = cos(x) * (x^2 - 4) / (25 - x^2)

• F(x) = sin(x) * (25 - x^2) / (x^2 - 4)

• F(x) = tan(x) * (25 - x^2) / (x^2 - 4)

• F(x) = cos(x) * (25 - x^2) / (x^2 - 4)

What is the first condition for continuity mentioned in the video?

• x^2 - 4 cannot equal zero

• x^2 - 4 must equal zero

• x^2 - 4 must be less than zero

• x^2 - 4 must be greater than zero

What are the solutions to the equation x^2 - 4 = 0?

• x = ±4

• x = 0

• x = 4

• x = ±2

What is the second condition for continuity related to the square root function?

• 25 - x^2 must be greater than zero

• 25 - x^2 must be equal to zero

• 25 - x^2 must be greater than or equal to zero

• 25 - x^2 must be less than zero

What is the solution to the inequality 25 - x^2 >= 0?

• x is between -2 and 2, inclusive

• x is between -2 and 2, exclusive

• x is between -5 and 5, exclusive

• x is between -5 and 5, inclusive

Which intervals are tested on the number line to determine the solution to the inequality?

• x = -3, x = 0, x = 3

• x = -5, x = 0, x = 5

• x = -6, x = 0, x = 6

• x = -4, x = 0, x = 4

What is the final interval of continuity for the function?

• [-5, -2) U (-2, 2) U (2, 5]

• (-5, -2) U (-2, 2) U (2, 5)

• (-5, -2] U [-2, 2] U [2, 5]

• [-5, -2] U (-2, 2) U [2, 5]

Why are -2 and 2 excluded from the final interval of continuity?

• Because they make the function positive

• Because they make the function negative

• Because they make the function zero

• Because they make the function undefined

What is the significance of closed and open points on the number line?

• Both indicate exclusion

• Both indicate inclusion

• Closed points indicate exclusion, open points indicate inclusion

• Closed points indicate inclusion, open points indicate exclusion

How is the interval of continuity expressed in interval notation?

• Using only parentheses

• Using only brackets

• Using parentheses for inclusion and brackets for exclusion

• Using brackets for inclusion and parentheses for exclusion