Further Pure Mathematics

1. Further Pure Mathematics Notes
1.1 Logarithmic Functions and Indices>1.2 The Quadratic Function>1.3 Identities and Inequalities>1.4 Graphs>1.5 Series>1.6 The Binomial Series>1.7 Scalar and Vector Quantities>1.8 Rectangular Cartesian Coordinates>1.9 Calculus>1.10 Trigonometry>

Type 1: Area between a curve and x-axis (y > 0)

Type 2: Area between a curve and x-axis (y < 0)

Type 3: Area between a curve and x-axis (-∞ < y <∞)

Type 4: Area between two curves

Coordinates of a Stationary Point

Step 1: Differentiate the equation and equate to 0 (f'(x) = 0)

Step 2: Substitute the value of x into the equation to find y

Finding rate of change of a part of a usually 3D shape (e.g. radius):

Eg: Area of volume of cylinder info
- 50cm³/s (rate of sand poured)
- V of cone increases in a way that r of base is always 3 times the h of the cone
- Find rate of change of radius of cone, when radius is 10cm

Working:

dV/dt = 50 (given)

dr/dt = need to find

dr/dt = dV/dt x dr/dV

r = 3h (given)

h = r/3

V = 1/3πr2h

V = 1/3πr²(r/3)

V = 1/9πr³

dV/dr = 1/3πr²

dr/dt = 50 x 1/(1/3π(10)²) = 0.477 cm/s