Maxima and Minima of Functions of Two Independent Variables.
As the name suggests, this topic is devoted to the method of finding the maximum and the minimum values of a function in a given domain.
Maxima and Minima of Functions of Two Independent Variables.
As the name suggests, this topic is devoted to the method of finding the maximum and the minimum values of a function in a given domain.
3. Calculus Rules
1. Chain Rule
dx
u nu n1 u
d n
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
4. Calculus Rules
1. Chain Rule
dx
u nu n1 u
d n
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
e.g. i y 2 x 1
2
5. Calculus Rules
1. Chain Rule
dx
u nu n1 u
d n
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
e.g. i y 2 x 1
2
dy
2 2 x 1 2
1
dx
6. Calculus Rules
1. Chain Rule
dx
u nu n1 u
d n
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
e.g. i y 2 x 1
2
dy
2 2 x 1 2
1
dx
4 2 x 1
7. Calculus Rules
1. Chain Rule
dx
u nu n1 u
d n
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
e.g. i y 2 x 1
2
dy
2 2 x 1 2
1
dx
4 2 x 1
ii y 3x 4
7
8. Calculus Rules
1. Chain Rule
dx
u nu n1 u
d n
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
e.g. i y 2 x 1
2
dy
2 2 x 1 2
1
dx
4 2 x 1
ii y 3x 4
7
dy
7 3 x 4 3
6
dx
9. Calculus Rules
1. Chain Rule
dx
u nu n1 u
d n
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
e.g. i y 2 x 1
2
dy
2 2 x 1 2
1
dx
4 2 x 1
ii y 3x 4
7
dy
7 3 x 4 3
6
dx
21 3 x 4
6
10. Calculus Rules
1. Chain Rule
dx
u nu n1 u
d n
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
e.g. i y 2 x 1 iii y x 10
2 2 3
dy
2 2 x 1 2
1
dx
4 2 x 1
ii y 3x 4
7
dy
7 3 x 4 3
6
dx
21 3 x 4
6
11. Calculus Rules
1. Chain Rule
dx
u nu n1 u
d n
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
e.g. i y 2 x 1 iii y x 10
2 2 3
3 x 10 2 x
dy dy
2 2 x 1 2
1 2 2
dx dx
4 2 x 1
ii y 3x 4
7
dy
7 3 x 4 3
6
dx
21 3 x 4
6
12. Calculus Rules
1. Chain Rule
dx
u nu n1 u
d n
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
e.g. i y 2 x 1 iii y x 10
2 2 3
3 x 10 2 x
dy dy
2 2 x 1 2
1 2 2
dx dx
4 2 x 1 6 x x 10
2 2
ii y 3x 4
7
dy
7 3 x 4 3
6
dx
21 3 x 4
6
13. Calculus Rules
1. Chain Rule
dx
u nu n1 u
d n
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
e.g. i y 2 x 1 iii y x 10
2 2 3
3 x 10 2 x
dy dy
2 2 x 1 2
1 2 2
dx dx
4 2 x 1 6 x x 10
2 2
ii y 3x 4 iv y 5 4 2 x
7 6
dy
7 3 x 4 3
6
dx
21 3 x 4
6
14. Calculus Rules
1. Chain Rule
dx
u nu n1 u
d n
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
e.g. i y 2 x 1 iii y x 10
2 2 3
3 x 10 2 x
dy dy
2 2 x 1 2
1 2 2
dx dx
4 2 x 1 6 x x 10
2 2
ii y 3x 4 iv y 5 4 2 x
7 6
dy dy
7 3 x 4 3 30 4 2 x 2
6 5
dx dx
21 3 x 4
6
15. Calculus Rules
1. Chain Rule
dx
u nu n1 u
d n
“BRING DOWN the POWER, LOWER the POWER, DIFF the
INSIDE”
e.g. i y 2 x 1 iii y x 10
2 2 3
3 x 10 2 x
dy dy
2 2 x 1 2
1 2 2
dx dx
4 2 x 1 6 x x 10
2 2
ii y 3x 4 iv y 5 4 2 x
7 6
dy dy
7 3 x 4 3 30 4 2 x 2
6 5
dx dx
21 3 x 4 60 4 2x
6 5
18. v y x2 3
1
x 3
2 2
1
x 3 2 2 x
dy 1 2
dx 2
19. v y x2 3
1
x 3
2 2
1
x 3 2 2 x
dy 1 2
dx 2
1
x x 3
2
2
20. v y x2 3
1
x 3
2 2
1
x 3 2 2 x
dy 1 2
dx 2
1
x x 3
2
2
x
x2 3
21. v y x2 3 dy dy dt
1
x 3
2 2 dx dt dx
1
x 3 2 2 x
dy 1 2
dx 2
1
x x 3
2
2
x
x2 3
22. v y x2 3 dy dy dt
1
x 3
2 2 dx dt dx
1
x 3 2 2 x
dy 1 2 vi x 4t y 2t 2
dx 2
1
x x 3
2
2
x
x2 3
23. v y x2 3 dy dy dt
1
x 3
2 2 dx dt dx
1
x 3 2 2 x
dy 1 2 vi x 4t y 2t 2
dx 2 dx
1 4
x x 3
2 dt
2
x
x2 3
24. v y x2 3 dy dy dt
1
x 3
2 2 dx dt dx
1
x 3 2 2 x
dy 1 2 vi x 4t y 2t 2
dx dy
dx 2
1 4 4t
x x 3
2 dt dt
2
x
x2 3
25. v y x2 3 dy dy dt
1
x 3
2 2 dx dt dx
1
x 3 2 2 x
dy 1 2 vi x 4t y 2t 2
dx dy
dx 2
1 4 4t
x x 3
2 dt dt
2
dy dy dt
x
dx dt dx
x2 3
26. v y x2 3 dy dy dt
1
x 3
2 2 dx dt dx
1
x 3 2 2 x
dy 1 2 vi x 4t y 2t 2
dx dy
dx 2
1 4 4t
x x 3
2 dt dt
2
dy dy dt
x
dx dt dx
x2 3 1
4t
4
27. v y x2 3 dy dy dt
1
x 3
2 2 dx dt dx
1
x 3 2 2 x
dy 1 2 vi x 4t y 2t 2
dx dy
dx 2
1 4 4t
x x 3
2 dt dt
2
dy dy dt
x
dx dt dx
x2 3 1
4t
4
t
28. v y x2 3 dy dy dt
1
x 3
2 2 dx dt dx
1
x 3 2 2 x
dy 1 2 vi x 4t y 2t 2
dx dy
dx 2
1 4 4t
x x 3
2 dt dt
2
dy dy dt
x
dx dt dx
x2 3 1
4t
4
t
Exercise 7E; 1adgj, 2behk, 3bd, 4adgj, 5ad, 6b, 7b, 8b,
10bdg, 11a, 13