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AMBERSON HIGH SCHOOL Communication: 8. Suppose the function is given in its g...

Apr 19, 2024

AMBERSON HIGH SCHOOL Communication: 8. Suppose the function is given in its graph. [C:1] Explain why the function is not differentiable at

\mathrm{x}=1

Solution by Steps

step 1

To determine if a function is differentiable at a point, we need to check if the graph of the function has a sharp corner, a cusp, a vertical tangent, or a discontinuity at that point

step 2

By examining the graph of the function at

\mathrm{x}=1

, we look for any of these features that would indicate non-differentiability

step 3

If the graph has a sharp corner or a cusp at

\mathrm{x}=1

, then the function is not differentiable at that point because the slope of the tangent line is not defined

step 4

If the graph has a vertical tangent at

\mathrm{x}=1

, the slope of the tangent line is infinite, which also means the function is not differentiable at that point

step 5

If the graph is discontinuous at

\mathrm{x}=1

, the function is not differentiable there because differentiability requires the function to be continuous

step 6

Based on the graph provided, we would identify the specific feature at

\mathrm{x}=1

that causes non-differentiability

Answer

The function is not differentiable at

\mathrm{x}=1

due to the presence of a sharp corner, cusp, vertical tangent, or discontinuity at that point, as indicated by the graph.

Key Concept

Differentiability at a Point

Explanation

A function is not differentiable at a point if the graph of the function has a sharp corner, cusp, vertical tangent, or is discontinuous at that point.