Applications of differentiation
Finding greatest / least values
Example
A rectangular beam is to be cut out of a cylinder.
The diameter of the cylinder is 40cm.
The breadth of the beam is b cm.
The depth of the beam is d cm.
The strength of the beam is given by the formula
S = 1.7b(400 – b2)
What dimensions of the beam are required for the beam to
have maximum strength ?
Check that this is a maximum
Maximum occurs when b = 20÷√3 cm
Find d
Higher Derivatives
|
Newton |
Liebniz |
|
f(x) |
y=f(x) |
1st Derivative |
f’(x) |
dy/dx |
2nd Derivative |
f’’(x) |
d2y/dx2 |
3rd Derivative |
f’’’(x) |
d3y/dx3 |
nth Derivative |
fn(x) |
dny/dxn |
Example
Rectilinear motion (straight line motion)
If an object moves in a straight line along the x-axis,
then after t seconds it has moved a distance s units
from the origin.
Displacement is a function of time, so s = f(t)
Newton’s equations of Motion:-
Example
A ball is thrown vertically upwards .
The height reached by the ball after t seconds is
h = 6t- t2.
a) How long does the ball take to reach its maximum height ?
b) What is the maximum height ?
c) What is the velocity of the ball 5 seconds after being thrown ?
d) What is the velocity of the ball when it hits the ground ?
e) Is the ball accelerating or decelerating at this point ?
Integrating methods
Example
A car starts from rest and its acceleration
t seconds after the start is 1/10(20-t) ms-2.
What is its speed after 20 seconds ?
The car ceases to accelerate and continues
at this uniform velocity. What is the total distance
covered 30 seconds after it started ?
Approximating roots of an equation : Newton’s Method
Example
| Xn | f(x) | f'(x) | f(x)/ f'(x) | Xn+1 |
|---|---|---|---|---|
| 1.5 | 23 | 78 | 0.294872 | 1.205128 |
| 1.205128 | 4.735397 | 46.92505 | 0.100914 | 1.104214 |
| 1.104214 | 0.470216 | 37.72811 | 0.012463 | 1.091751 |
| 1.091751 | 0.006773 | 36.64313 | 0.000185 | 1.091566 |
| 1.091566 | 1.48E-06 | 36.62712 | 4.04E-08 | 1.091566 |
| 1.091566 | 7.11E-14 | 36.62712 | 1.94E-15 | 1.091566 |
© Alexander Forrest