Year 12 Maths

Logarithms

100

log₂8 = ?

3

200

log₅(1/25) = ?

-2

300

Solving for x in the logarithmic equation: log₂(x+1)=3

x = 7

400

Rewrite 6² = 36 in logarithmic form with the index as the subject of the equation.

2 = log₆36

Differentiation

100

Differentiate f(x) = 4x²+3x+2

8x + 3

200

Differentiate f(x) = 6x^-3 - 2x + sqrt(x)

-18x^-4 - 2 + (1/2)x^-(1/2)

300

Differentiate f(x) = (2x² + 3)³

12x(2x²+3)²

400

Differentiate f(x) = x⁶(x⁴+4)

6x⁵(x⁴+4) + 4x⁹ = 10x⁹ + 24x⁵

Integration

100

Integrate x⁶ + 2x + 3

(x⁷/7) + x² + 3x + C

200

Integrate (1/x³) - sqrt(x)

-(x^-2/2) - 2x^(3/2)/3 + C

300

Given that f′(x)=6x²−4x+1 and f(x) passes through the point (2,7), find the function f(x)

f(x) = 2x³-2x²+x-3

400

Integrate 2cos(3x) + 5

(2/3)sin(3x) + 5x + C