12 Math Methods Unit 3

Integrals

100

y=∫ 8x³ dx

y= 2x⁴

200

y=∫ Cos x + x dx.

y= sin x + x²/2 + C 

300

Determine the Area between the curves (TA)

y = sin (x) and y = x²+2, over the domain of -1<x<2

8.04355

400

Determine the area bound by the x-axis and the curve y = (x-2)² - 4

32/3

500

The acceleration of a particle a(t) = 5 + e^2t.

Given the particle has an initial velocity of 0m/s and an initial displacement of 3m.

Determine the expression for position for this object.

x(t) = 5t²/2 + e^2t/4 - t/2 + 11/4

Derivatives

100

Derive y=(2x +4) ³

dy/dx = 6(2x+4)²

200

Derive f(x)= (3x − 5)/(4x +7)

f'(x) = f '(x )= 41/(4x +7)² 

300

Determine the equation of the tangent to the function f(x) = x³+2 at the point x=-1

y=3x + 3

400

Derive y= ln(3x+5)

dy/dx=3/(3x+5)

500

The position of a ball is given by the function x(t) = -4.9t²+40t+3. 

Determine the acceleration of the ball after 5 seconds

9.8

Log Laws

100

Simplify 
log⁡₂16-log₂32

-1

200

Simplify log_e(1/e^3x)

-3

300

Solve 

log₅125 = y

3

400

log₃ (x + 25) − log₃ (x − 1) = 3

x=2 

500

log₉ (x − 5) + log₉ (x + 3) = 1

x=6

Functions

100

y= Log(x), x must be

Greater than 0

200

Exponential functions have a _________ asymptote 

horizontal

300

An initial population of 850 increases by 6% every half year, where t is measured in years. Determine the expression that gives the population at anytime.

N(t)=850 xx 1.06^(2t)

400

Determine the Asymptote, Range and Domain of the following function

y=log_e(-x+2)-2

Range = R

Domain = x>-2

Asymptote = x=-2

500

Sketch and determine the key features of the following function

y=x²-4x-12

y int = -12

x int = 6, -2

tp (2, -16)

Trigonometric Functions

100

Determine sin (pi/6)

1/2

200

Determine tan (5pi/3)

-Sqrt(3)

300

2cos(x)=-1, 0<x<2pi

x=2pi/3 & x = 4pi/3

400

d/dx tan (x)

1/cos²(x)

500

sqrt(2) * sin (2x) = -1, where -pi<x<pi

x1 = -3pi/8

x2 = -pi/8

x3 = 5pi/8

x4 = 7pi/8