Continuity

• A function is continuous at a point if its limit at that point is equal to the function’s value at that point.

• A function is continuous on an interval if it is continuous at every point in the interval.

• The derivative of a function is a measure of how fast the function is changing at a given point.

• The derivative of a function can be used to find the slope of the tangent line to the function at a given point.

• The derivative of a function can be used to determine the intervals on which the function is increasing or decreasing.

• The derivative of a function can be used to find the local extrema of a function.

• The derivative of a function can be used to solve optimization problems.

Differentiability

• A function is differentiable at a point if its derivative exists at that point.

• A function is differentiable on an interval if it is differentiable at every point in the interval.

• The derivative of a function is a linear approximation of the function near a given point.

• The derivative of a function can be used to find the Taylor series expansion of a function.

• The Taylor series expansion of a function can be used to approximate the function near a given point.

Applications of Derivatives

• Derivatives can be used to find the slope of a curve at a given point.

• Derivatives can be used to determine the intervals on which a curve is increasing or decreasing.

• Derivatives can be used to find the local extrema of a curve.

• Derivatives can be used to solve optimization problems.

• Derivatives can be used to find the velocity and acceleration of a particle in motion.

• Derivatives can be used to find the rate of change of a quantity over time.