Differentiability and Continuity Concepts

The video tutorial explains the differentiability of a function by examining its graph. It highlights conditions where a function is not differentiable, such as vertical tangents, discontinuities, and sharp turns. The tutorial also discusses the role of horizontal tangents and asymptotes in graph analysis, providing examples to illustrate these concepts.

What type of tangent is present at the point (3, 0) on the graph of function f?

• Oblique tangent

• No tangent

• Vertical tangent

• Horizontal tangent

Why is a function not differentiable at a point with a vertical tangent?

• The slope is zero

• The slope is undefined

• The function is continuous

• The function is smooth

At which x-value is the function f not differentiable due to discontinuity?

• x = -3

• x = 3

• x = 6

• x = 0

What is the derivative at a point with a horizontal tangent?

• Negative

• Undefined

• Positive

• Zero

What is a sharp turn in the context of differentiability?

• A point where the slope is zero

• A point where the slope is undefined

• A smooth curve

• A point with a vertical tangent

Why is a function not differentiable at a sharp turn?

• The slopes from both sides are equal

• The function is continuous

• The slopes from both sides are different

• The function is smooth

What type of asymptote is present at x = -3 in the new graph?

• Oblique asymptote

• Vertical asymptote

• No asymptote

• Horizontal asymptote

At which x-value is the function not continuous in the new graph?

• x = 1

• x = 4

• x = 0

• x = -3

What happens to the slope at a sharp point on a graph?

• It remains constant

• It becomes zero

• It becomes infinite

• It changes direction

What is the behavior of the graph as x approaches infinity in the new example?

• It approaches y = 0

• It approaches y = 4

• It approaches y = -3

• It approaches y = 6