C4

Question 1

Differentiating Trig Functions

Answer 1

y = sinx ^dy/_dx = cosx

y = cosx ^dy/_dx = -sinx

y = tanx ^dy/_dx= sec²x

Question 2

Graph Transformations

Answer 2

Reflection in x axis = -f(x)

Reflection in y axis = f(-x)

Stretch with scale factor a, parallel to x axis = f(¹/_ax)

Stretch with scale factor a, parallel to y axis = af(x)

Translation (^a₀) = f(x-a)

Translation (⁰_a) = f(x)+a

Question 3

Definitions of Even and Odd Functions

Answer 3

Even: There is a line of symmetry on the y axis.

Odd: The graph has a rotational symmetry of order 2.

Question 4

Differentiating Natural Logs

Answer 4

y = aln(x) ^dy/_dx = ^a/_x

y = ln(ax) ^dy/_dx = ¹/_x

y = ln(f(x)) ^dy/_dx = ^f’(x)/_f(x)

Question 5

Differentiating Exponentials

Answer 5

y = ae^xdy/_dx = ae^x

y = e^axdy/_dx = ae^ax

y = e^f(x)dy/_dx = f’(x)e^f(x)

Question 6

Reciprocal Trig Functions

Answer 6

sin²θ + cos²θ = 1

1 + cot²θ = cosec²θ (÷sinθ)

1 + tan²θ = sec²θ (÷cosθ)

Question 7

Double Angle Formula

Answer 7

sin2θ = 2sinθcosθ

cos2θ = cos²θ - sin²θ

2cos²θ - 1

1 - 2sin²θ

tan2θ = 2tanθ

1-tan²θ

Question 8

Finding the angle between two vectors

Answer 8

cosθ = a·b

|a||b|

Question 9

Equation of planes

Answer 9

2D planes:

r = λa + μb

3D planes:

r = a + λ(b-a) + μ(c-a)

(r-a) · n = 0

r · n = a · n

n₁x + n₂y + n₃z + d = 0

Question 10

Finding the angle between a line and a plane

Answer 10

The angle α between a line and plane is given by:

α = 90 - θ

(where) cosθ = n · p

|n||p|

Question 11

Finding the angle between two planes

Answer 11

cosθ = n₁ · n₂

|n₁||n₂|

Question 12

Exponential relationship

Answer 12

e^x+y = e^x e^y

Question 13

Natural logarithm relationship

Answer 13

ln(a) + ln(b) = ln(ab)

ln(a) - ln(b) = ln(^a/_b)