All Flashcards
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.
Explain why differentiability implies continuity.
If a function is differentiable at a point, it has a well-defined tangent line. The existence of a tangent line implies that the function must be 'connected' at that point, hence continuous.
Explain why continuity does not imply differentiability.
A function can be continuous but have sharp corners, cusps, or vertical tangents where the derivative is undefined, thus not differentiable.
What conditions must be met for a function to be differentiable at x=a?
The function must be continuous at x=a, and the left-hand derivative must equal the right-hand derivative at x=a.
Describe how to graphically determine if a function is differentiable.
Visually inspect the graph for discontinuities, sharp corners, cusps, and vertical tangents. If none of these exist at a point, the function is likely differentiable there.
Explain the significance of a vertical tangent in terms of differentiability.
A vertical tangent indicates that the derivative approaches infinity, meaning the function is not differentiable at that point.
How do discontinuities affect differentiability?
Discontinuities always make a function non-differentiable at the point of discontinuity because differentiability requires continuity.
What is the role of limits in determining differentiability?
Limits are used to evaluate the left-hand and right-hand derivatives. If these limits are equal, the function is differentiable, assuming it is also continuous.
Explain differentiability in terms of local linearity.
A function is differentiable at a point if, when you zoom in close enough, the graph looks like a straight line (local linearity).
How does differentiability relate to the smoothness of a curve?
Differentiability implies that the curve is smooth, meaning there are no abrupt changes in direction (corners or cusps).
Explain the concept of non-differentiable points.
Non-differentiable points are locations on a function's graph where the derivative does not exist due to discontinuities, corners, cusps, or vertical tangents.
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.