All Flashcards
What are the differences between a corner and a cusp?
Corner: Sharp change in direction with different left and right derivatives. Cusp: A point where the curve comes to a point, often with a vertical tangent.
What are the differences between a jump discontinuity and a removable discontinuity?
Jump Discontinuity: Left and right limits exist but are unequal. Removable Discontinuity: The limit exists, but the function is either undefined or has a different value at that point.
Compare and contrast continuity and differentiability.
Continuity: Function is 'connected' at a point. Differentiability: Function has a derivative at a point (smoothness). Differentiability implies continuity, but not vice versa.
What is the difference between a vertical tangent and a horizontal tangent?
Vertical Tangent: The derivative approaches infinity. Horizontal Tangent: The derivative is equal to zero.
Compare and contrast the left-hand derivative and the right-hand derivative.
Left-hand Derivative: The derivative as x approaches a from the left. Right-hand Derivative: The derivative as x approaches a from the right. For differentiability, they must be equal.
Compare and contrast differentiability and integrability.
Differentiability: A function has a derivative at a point. Integrability: A function has an integral over an interval. A differentiable function is integrable, but an integrable function is not necessarily differentiable.
Compare and contrast continuity and integrability.
Continuity: A function is 'connected' at a point. Integrability: A function has an integral over an interval. A continuous function is integrable, but an integrable function is not necessarily continuous.
Compare and contrast a corner and a discontinuity.
Corner: A sharp change in direction with different left and right derivatives. Discontinuity: A point where the function is not continuous.
Compare and contrast a cusp and a discontinuity.
Cusp: A point where the curve comes to a point, often with a vertical tangent. Discontinuity: A point where the function is not continuous.
Compare and contrast a vertical tangent and a discontinuity.
Vertical Tangent: The derivative approaches infinity. Discontinuity: A point where the function is not continuous.
What does a sharp corner on the graph of indicate about ?
A sharp corner indicates that is undefined at that point, as the left-hand and right-hand derivatives are not equal.
What does a vertical tangent on the graph of indicate about ?
A vertical tangent indicates that approaches infinity or negative infinity, meaning the derivative is undefined at that point.
How can you identify a non-differentiable point from a graph?
Look for discontinuities (jumps, holes), sharp corners, cusps, or vertical tangents. These are all points where the function is not differentiable.
If a graph has a jump discontinuity, what does this imply about the derivative?
The derivative does not exist at the point of jump discontinuity.
If a graph has a removable discontinuity, what does this imply about the derivative?
The derivative does not exist at the point of removable discontinuity.
How does the slope of a tangent line relate to differentiability?
If a tangent line exists and has a finite slope, the function is differentiable at that point. If the tangent line is vertical or doesn't exist, the function is not differentiable.
What does a cusp in a graph indicate about the derivative?
A cusp indicates that the derivative is undefined at that point.
What does a corner in a graph indicate about the derivative?
A corner indicates that the derivative is undefined at that point.
What does a vertical tangent in a graph indicate about the derivative?
A vertical tangent indicates that the derivative is undefined at that point.
What does a discontinuity in a graph indicate about the derivative?
A discontinuity indicates that the derivative is undefined at that point.
Define continuity at a point.
A function is continuous at a point if the limit exists, the function is defined, and the limit equals the function value at that point.
Define differentiability at a point.
A function is differentiable at a point if its derivative exists at that point, meaning the left-hand and right-hand derivatives are equal.
What is a cusp?
A cusp is a point on a curve where the tangent line changes direction abruptly, and the derivative is undefined.
What is a corner?
A corner is a point on a graph where the left and right derivatives are different, resulting in a sharp turn.
What is a vertical tangent?
A vertical tangent is a tangent line to a curve that is vertical, indicating that the derivative approaches infinity at that point.
Define a jump discontinuity.
A jump discontinuity occurs when the left and right limits of a function at a point exist but are not equal.
Define a removable discontinuity.
A removable discontinuity is a point on a graph that is not continuous, but can be made continuous by redefining the function at that point.
What is the relationship between differentiability and continuity?
Differentiability implies continuity. If a function is differentiable at a point, it must also be continuous at that point.
Define left-hand derivative.
The limit of the difference quotient as x approaches a from the left.
Define right-hand derivative.
The limit of the difference quotient as x approaches a from the right.