Analyzing Function Domains and Derivatives

The video tutorial explores a family of functions defined as f(x) = a√(b - x²), where a and b are positive constants. It covers determining the domain, critical values, and increasing-decreasing behavior using derivatives and sign charts. The tutorial concludes with a brief mention of concavity analysis, to be covered in a subsequent video.

What is the basic form of the family of functions discussed in the video?

• f(x) = a + b - x^2

• f(x) = a * sqrt(b - x^2)

• f(x) = a * (b - x^2)^2

• f(x) = a / (b - x^2)

What are the parameters 'a' and 'b' in the family of functions?

• Positive real numbers

• Variables

• Complex numbers

• Negative integers

What condition must be satisfied for the domain of the function?

• b - x^2 > 0

• b - x^2 < 0

• b - x^2 >= 0

• b - x^2 <= 0

How is the domain of the function expressed in interval notation?

• (-∞, ∞)

• [0, b]

• (-b, b)

• [-sqrt(b), sqrt(b)]

What is the first step in finding the first derivative of the function?

• Find the second derivative

• Set the derivative equal to zero

• Apply the chain rule to the radical expression

• Differentiate 'a' with respect to 'x'

What is the simplified form of the first derivative?

• -ax / sqrt(b - x^2)

• ax / sqrt(b - x^2)

• -a / sqrt(b - x^2)

• a / sqrt(b - x^2)

Where is the first derivative of the function equal to zero?

• x = -b

• x = 0

• x = b

• x = sqrt(b)

What are the critical values of the function?

• ±b

• ±sqrt(b)

• 0 and ±sqrt(b)

• 0 and ±b

What does the sign of the first derivative indicate?

• The function's concavity

• The function's domain

• The function's increasing or decreasing behavior

• The function's range

On which interval is the function increasing?

• From 0 to sqrt(b)

• From -sqrt(b) to 0

• From -b to b

• From -∞ to ∞