11 x1 t10 06 maxima & minima (2012)
TRANSCRIPT
Maxima & Minima
Problems
Maxima & Minima
Problems When solving word problems we need to;
Maxima & Minima
Problems When solving word problems we need to;
(1) Reduce the problem to a set of equations
(there will usually be two)
Maxima & Minima
Problems When solving word problems we need to;
(1) Reduce the problem to a set of equations
(there will usually be two)
a) one will be what you are maximising/minimising
Maxima & Minima
Problems When solving word problems we need to;
(1) Reduce the problem to a set of equations
(there will usually be two)
a) one will be what you are maximising/minimising
b) one will be some information given in the problem
Maxima & Minima
Problems When solving word problems we need to;
(1) Reduce the problem to a set of equations
(there will usually be two)
a) one will be what you are maximising/minimising
b) one will be some information given in the problem
(2) Rewrite the equation we are trying to minimise/maximise with
only one variable
Maxima & Minima
Problems When solving word problems we need to;
(1) Reduce the problem to a set of equations
(there will usually be two)
a) one will be what you are maximising/minimising
b) one will be some information given in the problem
(2) Rewrite the equation we are trying to minimise/maximise with
only one variable
(3) Use calculus to solve the problem
e.g. A rope 36 metres in length is cut into two pieces. The first is bent
to form a square, the other forms a rectangle with one side the
same length as the square.
Find the length of the square if the sum of the areas is to be a
maximum.
e.g. A rope 36 metres in length is cut into two pieces. The first is bent
to form a square, the other forms a rectangle with one side the
same length as the square.
Find the length of the square if the sum of the areas is to be a
maximum.
e.g. A rope 36 metres in length is cut into two pieces. The first is bent
to form a square, the other forms a rectangle with one side the
same length as the square.
Find the length of the square if the sum of the areas is to be a
maximum. s
s
e.g. A rope 36 metres in length is cut into two pieces. The first is bent
to form a square, the other forms a rectangle with one side the
same length as the square.
Find the length of the square if the sum of the areas is to be a
maximum. s
s s
r
e.g. A rope 36 metres in length is cut into two pieces. The first is bent
to form a square, the other forms a rectangle with one side the
same length as the square.
Find the length of the square if the sum of the areas is to be a
maximum. s
s s
r
12 rssA
e.g. A rope 36 metres in length is cut into two pieces. The first is bent
to form a square, the other forms a rectangle with one side the
same length as the square.
Find the length of the square if the sum of the areas is to be a
maximum. s
s s
r
12 rssA 23626 rs
e.g. A rope 36 metres in length is cut into two pieces. The first is bent
to form a square, the other forms a rectangle with one side the
same length as the square.
Find the length of the square if the sum of the areas is to be a
maximum. s
s s
r
12 rssA 23626 rs
2in subject the make r
e.g. A rope 36 metres in length is cut into two pieces. The first is bent
to form a square, the other forms a rectangle with one side the
same length as the square.
Find the length of the square if the sum of the areas is to be a
maximum. s
s s
r
12 rssA 23626 rs
2in subject the make r
sr
sr
318
6362
e.g. A rope 36 metres in length is cut into two pieces. The first is bent
to form a square, the other forms a rectangle with one side the
same length as the square.
Find the length of the square if the sum of the areas is to be a
maximum. s
s s
r
12 rssA 23626 rs
2in subject the make r
sr
sr
318
6362
1 into substitute
e.g. A rope 36 metres in length is cut into two pieces. The first is bent
to form a square, the other forms a rectangle with one side the
same length as the square.
Find the length of the square if the sum of the areas is to be a
maximum. s
s s
r
12 rssA 23626 rs
2in subject the make r
sr
sr
318
6362
1 into substitute
sssA 3182
e.g. A rope 36 metres in length is cut into two pieces. The first is bent
to form a square, the other forms a rectangle with one side the
same length as the square.
Find the length of the square if the sum of the areas is to be a
maximum. s
s s
r
12 rssA 23626 rs
2in subject the make r
sr
sr
318
6362
1 into substitute
sssA 3182
2
22
218
318
ssA
sssA
2218 ssA
2218 ssA
sds
dA418
2218 ssA
sds
dA418
42
2
ds
Ad
2218 ssA
sds
dA418
42
2
ds
Ad
0 occur when points Stationary ds
dA
2218 ssA
sds
dA418
42
2
ds
Ad
0 occur when points Stationary ds
dA
2
9
0418 i.e.
s
s
2218 ssA
sds
dA418
42
2
ds
Ad
0 occur when points Stationary ds
dA
2
9
0418 i.e.
s
s
04,2
9when
2
2
ds
Ads
2218 ssA
sds
dA418
42
2
ds
Ad
0 occur when points Stationary ds
dA
2
9
0418 i.e.
s
s
04,2
9when
2
2
ds
Ads
maximum a is area themetres, 2
14 is square theoflength side when the
(ii) An open cylindrical bucket is to made so as to hold the largest
possible volume of liquid.
If the total surface area of the bucket is , find the height,
radius and volume.
2units 12
(ii) An open cylindrical bucket is to made so as to hold the largest
possible volume of liquid.
If the total surface area of the bucket is , find the height,
radius and volume.
2units 12
h
r
(ii) An open cylindrical bucket is to made so as to hold the largest
possible volume of liquid.
If the total surface area of the bucket is , find the height,
radius and volume.
12 hrV
2units 12
h
r
(ii) An open cylindrical bucket is to made so as to hold the largest
possible volume of liquid.
If the total surface area of the bucket is , find the height,
radius and volume.
12 hrV
2212 2 rhr
2units 12
h
r
(ii) An open cylindrical bucket is to made so as to hold the largest
possible volume of liquid.
If the total surface area of the bucket is , find the height,
radius and volume.
12 hrV
2212 2 rhr
2in subject the make h
2units 12
h
r
(ii) An open cylindrical bucket is to made so as to hold the largest
possible volume of liquid.
If the total surface area of the bucket is , find the height,
radius and volume.
12 hrV
2212 2 rhr
2in subject the make h
r
rh
rrh
2
12
122
2
2
2units 12
h
r
(ii) An open cylindrical bucket is to made so as to hold the largest
possible volume of liquid.
If the total surface area of the bucket is , find the height,
radius and volume.
12 hrV
2212 2 rhr
2in subject the make h
r
rh
rrh
2
12
122
2
2
1 into substitute
2units 12
h
r
(ii) An open cylindrical bucket is to made so as to hold the largest
possible volume of liquid.
If the total surface area of the bucket is , find the height,
radius and volume.
12 hrV
2212 2 rhr
2in subject the make h
r
rh
rrh
2
12
122
2
2
1 into substitute
r
rrV
2
12 22
2units 12
h
r
(ii) An open cylindrical bucket is to made so as to hold the largest
possible volume of liquid.
If the total surface area of the bucket is , find the height,
radius and volume.
12 hrV
2212 2 rhr
2in subject the make h
r
rh
rrh
2
12
122
2
2
1 into substitute
r
rrV
2
12 22
2units 12
h
r
3
2
16 rrV
3
2
16 rrV
2
2
36 r
dr
dV
3
2
16 rrV
2
2
36 r
dr
dV
rdr
Vd3
2
2
3
2
16 rrV
2
2
36 r
dr
dV
rdr
Vd3
2
2
0 occur when points Stationary dr
dV3
2
16 rrV
2
2
36 r
dr
dV
rdr
Vd3
2
2
0 occur when points Stationary dr
dV
2
4
62
3
02
36 i.e.
2
2
2
r
r
r
r
3
2
16 rrV
2
2
36 r
dr
dV
rdr
Vd3
2
2
0 occur when points Stationary dr
dV
2
4
62
3
02
36 i.e.
2
2
2
r
r
r
r
06,2when 2
2
dr
Vdr
3
2
16 rrV
2
2
36 r
dr
dV
rdr
Vd3
2
2
0 occur when points Stationary dr
dV
2
4
62
3
02
36 i.e.
2
2
2
r
r
r
r
06,2when 2
2
dr
Vdr
maximum a is ,2when V r
3
2
16 rrV
2
2
36 r
dr
dV
rdr
Vd3
2
2
0 occur when points Stationary dr
dV
2
4
62
3
02
36 i.e.
2
2
2
r
r
r
r
06,2when 2
2
dr
Vdr
maximum a is ,2when V r
3
2
16 rrV
2
22
212 ,2when
2
h r
2
2
36 r
dr
dV
rdr
Vd3
2
2
0 occur when points Stationary dr
dV
2
4
62
3
02
36 i.e.
2
2
2
r
r
r
r
06,2when 2
2
dr
Vdr
maximum a is ,2when V r
3
2
16 rrV
2
22
212 ,2when
2
h r
8
222
V
2
2
36 r
dr
dV
rdr
Vd3
2
2
0 occur when points Stationary dr
dV
2
4
62
3
02
36 i.e.
2
2
2
r
r
r
r
06,2when 2
2
dr
Vdr
maximum a is ,2when V r
3
2
16 rrV
2
22
212 ,2when
2
h r
8
222
V
units 2both areheight and radius when theunits 8 is volumemaximum 3
Exercise 10H; evens (not 30)
Exercise 10I; evens