Exploring non-differentiable functions in calculus reveals their unique challenges and applications. These functions, which may have cusps, corners, vertical tangents, or discontinuities, are crucial in fields like economics, physics, and computer graphics. The Weierstrass function exemplifies a continuous yet non-differentiable function, highlighting the nuanced relationship between continuity and differentiability. Understanding these functions is vital for applying calculus to theoretical and practical problems.
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Non-differentiable functions pose unique challenges in calculus due to their undefined slopes and lack of a unique tangent line
Cusps, Corners, and Vertical Tangents
Points of non-differentiability include cusps, corners, and vertical tangents, as well as any point of discontinuity
Discontinuities
Discontinuities, such as jumps or holes, also indicate non-differentiability in a function
Determining non-differentiability is crucial for analyzing mathematical models and real-world phenomena, as it can affect the function's rate of change and behavior
Non-differentiable functions have applications in economics, where they can represent cost functions with abrupt changes
In physics, non-differentiable functions, such as the Heaviside step function, can model signals with instantaneous transitions
Non-differentiable functions are also used in computer graphics to simulate natural textures and enhance the realism of rendered scenes
The Weierstrass function, defined by an infinite series, is continuous everywhere but non-differentiable at any point due to its highly oscillatory nature
The Weierstrass function serves as a counterexample to the assumption that continuity implies differentiability, showing that a function can lack a well-defined tangent at every point despite being continuous
The Weierstrass function has applications in various fields, including physics and engineering, where its non-differentiability adds complexity to the analysis of functions' behavior
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Non-differentiable points: sharp corners
Function non-differentiable at points where graph changes direction abruptly without a smooth transition.
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Non-differentiable points: vertical tangents
Function non-differentiable at points where slope of tangent becomes infinitely steep.
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The ______ function disproves the notion that a function without breaks in its graph is always differentiable.
Weierstrass
