Calculus Techniques and Theorems for Optimization Problems

1. 12 10 18 16

3. Limit and L’H number 1 Evaluate Number/0 1) 2) 3) 1)common denominator, or 2)rationalization (conjugate) 3)common factor 4) 2 squeeze

4. Limit and L’H 121 F143

5. Continuity&Diff F171 F151

6. Five theorems 1)Newton’s method 2)Roll’s thm 3)MVT 4)EVT 5)IVT

7. THE EXTREME VALUE THEOREM THE EXTREME VALUE THEOREM attains an absolute maximum f(c) and minimum f(d) value 1 f(x) is continuous on How to find absolute Max and Min 2 Closed interval [a, b] • Find all critical points ( ƒ’= 0 , or ƒ’ undefined ) • Evaluate ƒ at all criticals and endpoints a, b • Take the largest value and the smallest

8. Sec 4.3:HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH First Derivative Test: is a critical number of 1 is cont in an open interval containing 2 _ + 3 First Derivative Test: is a critical number of 1 is cont in an open interval containing 2 _ + has local max at 3

9. 1) Newton’s method

10. Critical Points Inflection Points 1)Critical points 1)Inflection points domain domain _ + +

11. Increasing Concave up or down 1) 2)Study the sign : 1)Critical points 2) Study the sign domain domain ‘

12. Optimization Problems WE1 (WE are ONE) I)What quantity (Q) to Max or Min Look for the word: Largest Smallest Minimum maximum Shortest III)1express Q in one variable only Eliminate all variables from the equation just express this quantity Q in one variable only. II)Equation is needed to express Q (may be in several variables) A =area V=volume P=product S=distance I)What to Max or Min III)1 one variable II)Equation Area of rectangle Find the global max A Critical: -2, 2

13. Sec 4.2: The Mean Value Theorem MEAN VALUE THEOREM there is at least one number c in (a, b) 1 f(x) is continuous on [a, b] 2 f(x) is differentiable on (a, b) ROLLE’S THEOREM 1 there is at least one number c in (a, b) f(x) is continuous on [a, b] 2 f(x) is differentiable on (a, b) 3

14. Sec 4.9: Antiderivatives Find the most general antiderivative

15. Linear approximation Differentials The tangent line is considered as an approximation of the curve y=f(x) • 0,014 • 0.001 • 0.01 • 0.021 • 0.045 Change in x and y Relative Change in x and y Percentage Change in x and y

16. Differentiation 3.4 Chain Rule 3.11 Hyp 3.5 Implicit Diff Implicit Diff 3.5 Inv. Trig 3.6 Log Diff variables appear in the base and exponent

17. Multiple choice Exam 1) Skim all 28 problems to select these you know then mark them Look for the keyword: local , inc, concave, inflection, Newton,….. 2) Start by solving these problems (keep bubbling at the end) you have 180min/28Prob = 6.4min each 3) Eliminate answers that are wrong 4) Try working backwards. Plug the answers into the question to see if they make sense

18. Multiple choice Exam

19. Multiple choice Exam 4) If you have absolutely no idea what the answer is, can't use any of the above techniques, choose one letter 1) Pattern: Look for similarities

20. 1 DONOT GIVE UP (RELAX use 2Hours) 7 SKIM and Classify (Easy, M, Hard) 2 Start by Easy 3 No bubble , just circle 8 4 Next M then H Guess (be smart) 1)Eliminate 2)Look for pattern 3)Approximate 4)Subsitute by values 5)Count and be consistent (b) 5 Use answers to Question 6 9 Last 15min bubble At end, revisit and check (use different method) 10 Say and pray