Download Calculus mathematics formula sheet and more Cheat Sheet Calculus in PDF only on Docsity! Calculus I Formula Sheet Chapter 4 Section 4.1 1. Definition of the Extrema of a function: Let f be defined on interval I : • ( )f c is abs min when ( ) ( )f c f x≤ on I • ( )f c is abs max when ( ) ( )f c f x≥ on I 2. Exterme Value Theorem: If f is cts on [ , ]a b Then f has both max/min on [ , ]a b 3. Definition of Relative Extrema: • If ( )f c is max on ( , )a b (open interval) Then ( )f c is rel max • If ( )f c is min on ( , )a b (open interval) Then ( )f c is rel min 4. Definition of a Critical Number: Let f be defined at c Then c is a critical number if o ( ) 0f c′ = or o ( )f c′ DNE 5. Relative extrema occur only at c.n. 6. Find extrema on [ , ] :a b • f cts on [ , ]a b o Find c.n. on ( , )a b o Eval f at: a , all c.n., b o Smallest = abs max o Largest = abs min Section 4.2 7. Rolle’s Theorem • f cts on [ , ]a b • f diff on ( , )a b • ( ) ( )f a f b= ⇒ there is at least one c in ( , )a b such that ( ) 0f c′ = 8. Mean Value Theorem • f cts on [ , ]a b • f diff on ( , )a b ⇒ there exists a c in ( , )a b such that ( ) ( ) ( )f b f af c b a −′ = − Section 4.3 9. Definition of Increasing and Decreasing • Increasing: ( ) ( )1 2 1 2x x f x f x< ⇒ < • Decreasing: ( ) ( )1 2 1 2x x f x f x< ⇒ > 10. Test for Increasing and Decreasing • f cts on [ , ]a b • f diff on ( , )a b o ( ) 0f x′ > on ( , )a b ⇒ increasing o ( ) 0f x′ < on ( , )a b ⇒ decreasing o ( ) 0f x′ = on ( , )a b ⇒ constant 11. Find interval of increasing and decreasing • f cts on ( , )a b • Find c.n. on ( , )a b • Create intervals • Find the sign of ( )f x′ on each interval o + ⇒ increasing o −⇒ decreasing 12. The First Derivative Test • c is a c.n. in ( , )a b • f cts on ( , )a b • f diff on ( , )a b except possibly at c o ( )f c′ change – to + ⇒ ( )f c is rel min o ( )f c′ change + to – ⇒ ( )f c is rel max o + to + or – to – ⇒neither max nor min
