The Physics
Hypertextbook
Opus in profectus

# Kinematics & Calculus

## Summary

• The kinematic quantities of displacement, velocity, and acceleration can be described by all sorts of functions.
• The function describing one quantity can be transformed into functions describing the other two quantities.
• The procedure for doing so is either…
• differentiation (finding the derivative) or…
• The derivative of displacement with time is velocity.
• The derivative of velocity with time is acceleration.
• integration (finding the integral).
• The integral of acceleration over time is velocity.
• The integral of velocity over time is displacement.

• The techniques of calculus can also be used to analyze functions — including those that describe motion.
• The first derivative of a function…
• is the instantaneous rate of change of the function.
• determines the slope of a line tangent to a graph of the function.
• equals zero at a local extrema (maximum or minimum) or a saddle point of the function.
• The second derivative of a function…
• is used to determine the direction of concavity of the graph of a function.
• The graph of a function is concave up if its second derivative is positive.
• The graph of a function is concave down if its second derivative is negative.
• An inflection point (the transition between two different concavities) occurs where the second derivative is zero.
• is used to distinguish extrema.
• An extremum is a local maximum if the second derivative of the function at that point is negative.
• An extremum is a local minimum if the second derivative of the function at that point is positive.
• A saddle point occurs at a location where both the first and second derivatives of a function are zero.