SUMMARY
The function f(x) = |x| is differentiable for all real numbers except at x = 0. The piecewise representation of the function is f(x) = -x for x < 0, f(x) = 0 for x = 0, and f(x) = x for x > 0. The presence of a cusp at the origin results in a discontinuity in the derivative, which is defined as the limit of the difference quotient as h approaches 0. The slopes of the lines on either side of the cusp are symmetric but differ in value, confirming the non-differentiability at that point.
PREREQUISITES
- Understanding of piecewise functions
- Knowledge of limits and the definition of derivatives
- Familiarity with the concept of cusps in calculus
- Basic graph interpretation skills
NEXT STEPS
- Study the properties of piecewise functions in calculus
- Learn about the concept of cusps and their implications on differentiability
- Explore the limit definition of derivatives in more depth
- Investigate the graphical interpretation of derivatives and their applications
USEFUL FOR
Students studying calculus, particularly those focusing on differentiation and the behavior of absolute value functions. This discussion is also beneficial for educators seeking to clarify concepts related to differentiability and piecewise functions.