Flashcards in C4 Deck (42):

1

## sin2A = ?

### 2sinAcosA

2

## cos2A = ?

###
cos²A - sin²A

2cos²A - 1

1 - 2sin²A

3

## tan2A = ?

### 2tanA/(1 - tan²A)

4

## sin4x = ?

### 2sin2xcos2x

5

## d/dx sinax = ?

### acosax

6

## d/dx cosax = ?

### -asinax

7

## d/dx sinx = ?

### cosx

8

## d/dx cosx = ?

### -sinx

9

## d/dx sin(ax + b) = ?

### acos(ax+b)

10

## d/dx cos(ax + b) = ?

### -asin(ax +b)

11

## How would you differentiate sin²x?

###
Chain rule of (sinx)²

to obtain 2⋅(d/dx sinx)⋅(sinx)²⁻¹

= 2⋅cosx⋅sinx

= sin2x

12

## How would you differentiate cos²x?

###
Chain rule of (cosx)²

to obtain 2⋅(d/dx cosx)⋅(cosx)²⁻¹

= 2⋅-sinx⋅cosx

= -2sinxcosx

= - sin2x

13

## How do you make parametric equations cartesian?

### Eliminate the parameter

14

## dy/dx of a parametric equation?

### dy/dx = dy/dt ⋅ dt/dx = g'(t)/f'(t)

15

## Implicit: d/dx y = ?

### dy/dx

16

## Implicit: d/dx y² = ?

### 2y⋅dy/dx

17

## Implicit: d/dx xy = ?

###
u = x, v = y

u' = 1, v' =dy/dx

∴ d/dx xy = vu' + uv'

∴ d/dx xy = x⋅dy.dx + y

18

## ∫ sinax dx = ?

### -1/a cosax + c

19

## ∫ cosax dx = ?

### 1/a sinax + c

20

## ∫ sec²ax dx = ?

### 1/a tanax + c

21

## ∫ kf'(x)/f(x) dx = ?

### kln|f(x)| + c

22

## Formula for Integration by parts?

###
∫ uv' dx = uv - ∫ vu' dx

where u is a value that will differentiate to a constant, or is lnx

23

## Steps for Integration by substitution?

###
|> u = f(x)

|> Make obvious substitution (change f(x) for u)

|> Differentiate u = f(x) and rearrange for dx

|> Rearrange u = f(x) for x=

|> Integrate with respect to u

|> Change limits

|> Replace u with f(x)

24

## a/b ⋅ c/d = ?

### ac/bd

25

## a/b ÷ c/d = ?

### ad/bc

26

## Binomial Expansion formula?

### (1 + x)ⁿ = 1 + nx + n(n-1)/2! ⋅ x² + n(n-1)(n-2)/3! ⋅ x³

27

## What form must binomial expansions be in?

### MUST be in the form (1 + x)ⁿ

28

## When do binomial expansions converge?

### Converge as |x| < 1

29

## How would you write (3x² - 1)/(x-1)(2x-1)² as partial fractions?

###
= A/x-1 + B/(2x-1) + C(2x-1)²

let x = 1 [cancels (x-1)],

A = 2

let x = 1/2 [cancels (2x-1)],

-1/2⋅C = -1/4,

C = 1/2

let x = 0 [takes note of repeated factor],

-1 = A + B - C.

-1 = 2 + B - 1/2

B = -5/2

30

## Magnitude of unit vectors?

### 1

31

## Length of unit vectors?

### 1

32

## magnitude/length of unit vectors?

### 1

33

## Where do position vectors start from?

### The origin

34

## -AB-> = ?

### -OB-> − -OA->

35

## for x = ai + bj + ck, |x| = ?

###
√(a² + b² + c²)

magnitude/length

36

## Features of a vector line?

###
have a position vector, a direction vector and a parameter

in the form: position vector + parameter ⋅ direction vector

37

## When are vector lines parallel?

### Direction vectors are multiples of one another

38

## When are vector lines skew?

### If you can prove they are not parallel, and do not intersect

39

##
where do the vector lines

r = (ai + tpi) + (bj + tqj) + (ck + trk)

&

r = (di + smi) + (ej + snj) + (ek + sok)

intersect?

###
a + tp = d + sm

b + tq = e + sn

c + tr = f + so

40

## Equation for scalar product?

###
(-a->) ⋅ (-b->) = |a||b|cosθ

where θ is the angle between the vectors

41

## If lines are perpendicular, (-a->) ⋅ (-b->) = ?

###
0

because cos90 = 0

42