REAL ANALYSIS EXAM I THEOREMS

REAL ANALYSIS EXAM I THEOREMS

1. RULES OF DIFFERENTIATION

   • *PICTURE OF RULES*

2. MEAN VALUE THEOREM IN ONE DIMENSION

3. RATIO MEAN VALUE THEOREM

4. L'HOPITAL'S RULE

5. INTERMEDIATE VALUE THEOREM FOR DERIVATIVES

6. EQUIVALENT CONDITIONS FOR LINEAR TRANSFORMATION TO BE CONTINUOUS

   • PROOFS

7. CONTINUITY OF ALL LINEAR MAPS

   • IF A LINEAR MAP IS AN ISOMORPHISM, IT IS BIJECTIVE. ALL THAT IS LEFT TO PROVE IS THE INVERSE MAPPING IS CONTINUOUS

8. UNIQUENESS OF THE DERIVATIVE

9. DIFFERENTIABILITY IMPLIES CONTINUITY

10. C1 IMPLIES DIFFERENTIABILITY

11. CHAIN RULE IN MATRIX FORM

12. MEAN VALUE THEOREMS

13. IF DF = 0, F IS CONSTANT AND DT = T

    • PROOF

14. INVERSE FUNCTION THEOREMS

15. IMPLICIT FUNCTION THEOREM

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