This document discusses optimization problems and their solutions using calculus. It provides examples of finding maximum and minimum values for functions by taking the derivative and setting it equal to zero. The key steps are to identify what is being optimized (highest, longest, etc.), write an equation relating the variables, take the derivative of that equation, set the derivative equal to zero to find critical points, and then determine if those points are maxima or minima using the second derivative test or nature of the function.

1. Block 2
Optimisation

2. What is to be learned?
• What is meant by optimisation
• How to use differentiation to solve
practical problems
• How to identify these types of questions

3. • Highest
• Tallest
• Longest
• Smelliest
• Most economical
• Most difficult
→All types of maximum
Find SP →→ dydy
//dxdx = 0= 0

4. • Lowest
• Shortest
• Weeest (spelling!)
• Lightest
• Least economical
• Least difficult
→All types of minimum
Find SP →→ dydy
//dxdx = 0= 0

5. has 64m of fencing
wants the largest
possible area to
enclose sheep
wall

6. has 64m of fencing
wants the largest
possible area to
enclose sheep
wall
find max TP
→→ derivative = 0derivative = 0
x
y
x
Area = xy
must have only one letter
y + 2x = 64
y = 64 – 2x
Area = x y(64 – 2x)
a) Prove

7. A = x(64 – 2x)
A = 64x – 2x2
dA
/dx = 64 – 4x
for SPs dA
/dx = 0
64 – 4x = 0
4x = 64
x = 16

8. has 64m of fencing
wants the largest
possible area to
enclose sheep
wall
Find A!

9. A = x(64 – 2x)
A = 64x – 2x2
dA
/dx = 64 – 4x
for SPs dA
/dx = 0
64 – 4x = 0
4x = 64
x = 16
A = 16(64 – 2(16))
= 512m2
Should prove it is max TP
→Nature Table

10. Optimisation
Asked for
• highest, tallest etc
or
• lowest, smallest etc
Find Max/Min TP using differentiation
Often asked to prove stupid formula first

11. Easy Example
The sum of two numbers (x and y) is 18
a) Prove that their product P = x(18 – x)
b) Find maximum value of P.
a)Sum
x + y = 18
Product
P = xy
from * y = 18 – x
P = x
**
y(18 – x) as required!

12. b) For Max derivative = 0
P = x(18 – x)
= 18x – x2
dP
/dx = 18 – 2x
18 – 2x = 0
18 = 2x
x = 9
Find p
P = 9(18 – 9)
= 9(9)
= 81

13. The height (h km) reached by a cannonball fired
from Sandra’s cannon after time (t secs) is given
by the equation, h = 4t – 3t2
Find the maximum height the cannonball reaches
and how long this takes.
For SPs dh
/dt = 0
dh
/dt = 4 – 6t
4 – 6t = 0
t = 2
/3 second
H = 4(2
/3)– 3(2
/3)2
= 4
/3km
Key Question

14. Ex. Percy’s Taxis Fuel costs (C£) depend on
his average speed (v mph): C = 10v + 625000
He wants to drive at a speed which will give
him the lowest costs.
What will his costs be?
rearrange C = 10v + 625000v-2
dC
/dv = 10 – 1250000v-3
for SPs dC
/dv = 0
10 – 1250000v-3
= 0
10 =1250000v-3
v2
Find Derivative

15. 10 =1250000v-3
10 = 1250000
cross multiply
10v3
= 1250000
v3
= 1250000
10
v3
= 125000
v = 3
√125000
= 50
Prove Min TP →
v3
Nature table

16. Should drive at 50 mph
Costs C = 10(50) + 625000
= £750
502