Calculus MTH 250 Lecture 7 Previous Lectures Summary

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Calculus (MTH 250) Lecture 7

Calculus (MTH 250) Lecture 7

Previous Lecture’s Summary • Limits by definition • Formal definition of one-sided limits •

Previous Lecture’s Summary • Limits by definition • Formal definition of one-sided limits • Comparison theorem • Sandwich theorem • Golden rules for limits • Introduction to continuity • Discontinuity • Piecewise continuous functions • Continuous extension • Intermediate value theorem

Today’s Lecture • Formal definition of continuity • Important results of continuity • Hyperbolic

Today’s Lecture • Formal definition of continuity • Important results of continuity • Hyperbolic trigonometric functions • Inverse hyperbolic trigonometric functions • Boundedness of continuous functions • Uniform continuity • Tangent lines

Formal definition of continuity

Formal definition of continuity

Important continuity results

Important continuity results

Important continuity results Exponential functions are continuous everywhere since

Important continuity results Exponential functions are continuous everywhere since

Important continuity results Sine and Cosine functions are continuous everywhere since

Important continuity results Sine and Cosine functions are continuous everywhere since

Important continuity results

Important continuity results

Important continuity results

Important continuity results

Important continuity results

Important continuity results

Important continuity results

Important continuity results

Important continuity results Example: |x| is continuous everywhere.

Important continuity results Example: |x| is continuous everywhere.

Important continuity results

Important continuity results

Hyperbolic functions

Hyperbolic functions

Hyperbolic functions •

Hyperbolic functions •

Inverse hyperbolic functions

Inverse hyperbolic functions

Inverse hyperbolic functions

Inverse hyperbolic functions

Inverse hyperbolic functions

Inverse hyperbolic functions

Inverse hyperbolic functions

Inverse hyperbolic functions

Boundedness of continuous functions

Boundedness of continuous functions

Boundedness of continuous functions

Boundedness of continuous functions

Boundedness of continuous functions

Boundedness of continuous functions

Boundedness of continuous functions

Boundedness of continuous functions

Uniform continuity

Uniform continuity

Tangent lines

Tangent lines

Tangent lines

Tangent lines

Tangent lines

Tangent lines

Tangent lines

Tangent lines

Tangent lines Example: Find the tangent line to

Tangent lines Example: Find the tangent line to

Tangent lines Example

Tangent lines Example

Tangent lines

Tangent lines

Lecture Summary • Formal definition of continuity • Important results of continuity • Hyperbolic

Lecture Summary • Formal definition of continuity • Important results of continuity • Hyperbolic trigonometric functions • Inverse hyperbolic trigonometric functions • Boundedness of continuous functions • Uniform continuity • Tangent lines