Differentiability and Derivatives in Calculus

Mr. Bean introduces calculus students to key derivative rules, including the derivative of a constant, constant multiple rule, and sum and difference rule. The lesson covers horizontal tangent lines, normal lines, and differentiability in piecewise functions, emphasizing the importance of continuity and smooth curves. Students learn to apply these concepts to solve calculus problems effectively.

What is the main focus of today's calculus lesson?

• Integration techniques

• Several derivative rules

• Graphing functions

• Limits and continuity

What is the derivative of a constant, such as y = 2?

• Undefined

• 0

• 1

• 2

If y = pi, what is the derivative of y?

• pi

• 1

• Undefined

• 0

What does the constant multiple rule state?

• The derivative of a sum is the sum of the derivatives.

• The derivative of a constant times a function is the constant times the derivative of the function.

• The derivative of a product is the product of the derivatives.

• The derivative of a constant is zero.

How do you find the derivative of y = 3x^2 using the constant multiple rule?

• 9x^2

• 3x

• 6x

• x^2

What is the derivative of y = x^5/3?

• x^4/3

• 15x^4

• 5x^4/3

• 5/3 x^4

What does the sum and difference rule for derivatives state?

• The derivative of a constant is zero.

• The derivative of a quotient is the quotient of the derivatives.

• The derivative of a product is the product of the derivatives.

• The derivative of a sum is the sum of the derivatives.

How do you find the derivative of y = 4x - 5/x + 6?

• 4x^2 - 5x + 6

• 4x - 5x^2 + 6

• 4 - 5/x^2

• 4x + 5/x^2

What is a horizontal tangent line?

• A line with an undefined slope

• A line with a slope of 0

• A line with a slope of 1

• A line with a slope of -1

When does a function have a horizontal tangent line?

• When the derivative is undefined

• When the derivative is positive

• When the derivative is negative

• When the derivative is zero