MAT 3749Introduction to Analysis

Section 2.3 Part I

Derivatives

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Differentiable implies continuous Something else

References

Section 2.3

HW

Do not use ε-δ.

Derivatives

Example 1

Theorem

Analysis: Step 1

Analysis

Analysis

Analysis

Analysis

Analysis

Analysis

Analysis

Analysis

Analysis

Analysis

Proof

Proof

Proof

Proof

Remark

The converse of the last theorem is false

Example 2

Give a counterexample of the converse of the last theorem.

The Derivative as a Function

Change of Variables in Limits

“Change of variables” is a technique you have used frequently to evaluate limits in a completely natural way.

To formulate this technique formally and justify it rigorously, however, is a task requiring some subtlety.

Example 3

2

sin2

lim

2x

x

x

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