Critical Numbers and Derivatives

This video tutorial explains how to find critical numbers of a function. It covers the definition of critical numbers, including points where the derivative is zero or does not exist. The video provides multiple examples, including quadratic, cubic, and trigonometric functions, and demonstrates techniques like factorization and the product rule. It also discusses non-differentiable points in absolute value functions.

What is a critical number in the context of a function?

• A point where the function is undefined

• A point where the derivative is zero or undefined

• A point where the function has a horizontal asymptote

• A point where the function has a vertical tangent

In the function f(x) = 4x^2 + 8x, what is the critical number?

• x = 0

• x = -1

• x = 1

• x = 2

When finding critical numbers in a fractional function, what should you set to zero?

• Only the numerator

• Only the denominator

• Both numerator and denominator

• Neither numerator nor denominator

For the function f(x) = |5x + 8|, what is the critical number?

• x = 0

• x = 5/8

• x = -8/5

• x = 8/5

What rule is used to find the derivative of a product of two functions?

• Power Rule

• Product Rule

• Chain Rule

• Quotient Rule

In the function f(x) = x√(9-x), what is the first critical number?

• x = 3

• x = 6

• x = 0

• x = 9

What is the critical number for the function f(x) = x√(9-x) when the denominator is set to zero?

• x = 9

• x = 0

• x = 3

• x = 6

What is the derivative of sin^2(x) with respect to x?

• 2sin(x)cos(x)

• sin(x)cos(x)

• 2cos(x)

• cos^2(x)

Which of the following is a critical number for the function f(x) = sin^2(x) + cos(x) in the interval [0, 2π]?

• π/2

• 2π

• π

• π/3

In which quadrants is cosine positive?

• Quadrants II and III

• Quadrants III and IV

• Quadrants I and IV

• Quadrants I and II