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Maths Manual > Yr12A HSC > Flashcards

Yr12A HSC Flashcards

1
Q

1st Derivative

A

f’(x) > 0 increasing gradient
f’(x) < 0 decreasing gradient
f’(x) = 0 stationary (horizontal tangent)

you can determine nature with table of slopes or second derivative
f’‘(x) > 0 concave up (min TP)
f’‘(x) < 0 concave down (max TP)

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2
Q

2nd Derivative

A

f’‘(x) > 0 concave up
f’‘(x) < 0 concave down
f’‘(x) = 0 concavity changes or horizontal point of inflexion

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3
Q

Differentiation by First Principle

A
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4
Q

Curve Sketching Menu; 7 steps

A
  1. Domain: Find the domain of f(x)
  2. Symmetry: Find whether the function is even, odd, or neither
  3. A: Intercepts: Find the y-ints and the zeroes
    B: Sign: Use a table of test values of f(x),
    that is, a table of signs, to find where the
    function is positive and negative
  4. A: Vertical Asymptotes: Examine any discontinuities to see whether there are vertical asymptotes
    B: Horizontal Asymptotes: Examine the behaviour of f(x) as x –> ∞, and as x –> -∞
  5. The First Derivative
    A: Find the zeroes and discontinuities of
    f’(x)
    B: Use a table of test values of f’(x), that is a table of slopes, to determine the nature of stationary points and the slope of the function throughout its domain
  6. The Second Derivative
    A: Find the zeroes and discontinuities of
    f’‘(x)
    B: Use a table of test values of f’‘(x), that is, a table of concavities, to find any points of inflexion, and the concavity of the function throughout its domain
  7. ANY OTHER FEATURES
    A ROUTINE WARNING OF INCOMPLETENESS
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5
Q

why isn’t f’‘(x) = 0 a sufficient condition for a point of inflexion?

A

the sign of f’‘(x) must also change around the point (the concavity must change)

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6
Q

define e

A

the number such that the exponential function y=eˣ has exactly gradient 1 at its y-int

e = 2.7183…

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7
Q

oddness & evenness of the trigonometric functions

A

ODD
sin(-x) = -sinx
tan(-x) = -tanx

cosecx & cotx

EVEN
cos(-x) = cosx

secx

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8
Q

Measures of Location

A

mode
median
mean

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9
Q

Measures of Spread

A

range, IQR, variance, standard deviation

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10
Q

The 5 Number Summary

A

min (Q0)
Q1
Q2 (median)
Q3
max (Q4)

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11
Q

Expected Value (weighted mean)

A

measure of central tendency
if the experiment is carried out experimentally a large number of times we would expect that the average of the outcomes would approach the expected value

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12
Q

Standard deviation

A

square root of the variance
A distribution with a larger standard deviation is more spread out than a distribution with a smaller standard deviation. Both are zero if the distribution only takes one value - that is, if it is not spread out at all. If the distribution is stretched (multiplied) by a constant k, the standard deviation also increases by a factor k

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13
Q

One to one functions

A

passes VLT
passes HLT

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14
Q

many to one functions

A

passes VLT
fails HLT

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15
Q

one to many relations

A

fails VLT
passes HLT

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16
Q

many to many relations

A

fails VLT
fails HLT

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17
Q

Type of Relation & which line test they pass/fail

A

VLT pass: to one
VLT fail: to many

HLT pass: one to
HLT fail: many to

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18
Q

definitions of the 3 trig functions (in terms of x, y, r)

A

sinx = y/r
cosx = x/r
tanx = y/x

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19
Q

(trig) 3 Pythagorean identities

A

sin²x + cos²x = 1

tan²x + 1 = sec²x

cot²x + 1 = cosec²x

20
Q

in a combination of transformations, (vertical translations)

A

do them last

21
Q

horizontal translations and phase of trig functions

A

y = sin(x + α)
y = cos(x + α)
y = tan(x + α)

all have phase α (shifting original functions left by α)

22
Q

the periods of the trig functions

A

y = sinx (&cosecx)
y = cosx (&secx)
a full revolution (2pi)

y = tanx (&cotx)
half a revolution (pi)

23
Q

the complementary angle identities

A

cos (90-α) = sinα
cot(90-α) = tanα
cosec(90-α) = secα

24
Q

vertical dilations

A

to stretch vertically by a

factor of a
replace y by y/a

new function rule is y=af(x)

25
Q

horizontal dilations

A

to stretch horizontally by a

factor of a
replace x by x/a

new function rule is y=f(x/a)

26
Q

axis of dilation for horizontal and vertical dilations

A

vertical dilations: x axis
horizontal dilations: y axis

27
Q

enlargements with centre the origin

A

the composition of 2 dilations with same factor, one horizontal and one vertical

to apply an enlargement with factor a
replace x by x/a
replace y by y/a

new function rule is y=af(x/a)

28
Q

dilations with fractional or negative factor

A

let a = dilation factor

if 0 < a < 1 graph is compressed

if a < 0 dilation with positive factor and a reflection (order does not matter)

29
Q

other names for
rotation of 180° about the origin

A

an enlargement with factor -1
reflection in the origin

30
Q

commuting transformations

A

any 2 translations commute

any 2 dilations commute (including reflections)

a translation and dilation commute if one is vertical and the other is horizontal

31
Q

horizontal dilations and period (trig functions)

A

y = sin nx
y = cos nx
have period 2π/n

y = tan nx
has period π/n

(all a result of stretching horizontally by factor 1/n)

32
Q

vertical dilations and amplitude (trig functions)

A

y = a sin x
y = a cos x
have amplitude a

(result of stretching vertically with factor a)

33
Q

skewed data

A

data is skewed in direction of the tail, NOT the peak

skewed to the right (positively skewed): bigger tail on RHS

skewed to the left (negatively skewed): bigger tail on LHS

34
Q

A Universal Formula Involving All Four Transformations

A
  1. stretch horizontally with factor 1/a
  2. shift left b
  3. stretch vertically with factor k
  4. shift up c

(step 3 then 4 can be done before step 1 then 2)

35
Q

Horizontal Asymptotes eg1

highest degree of x in numerator and denominator is the same

A

highest degree of x in numerator and denominator is the same
consider coefficient of x

36
Q

Horizontal Asymptotes eg2

degree of x in numerator is less than denominator

A

degree of x in numerator is less than denominator
horizontal asymptote is at y=0

37
Q

Horizontal Asymptotes eg3

degree of x in numerator is MORE than denominator

A

degree of x in numerator is MORE than denominator
no horizontal asymptote

38
Q

Radian <==> Degree Conversion

A
39
Q

Exact Trig Values

A
40
Q

Variance <3

A

V(X) = E(X²) - E(X)²

V(X) = Σx²p(x) - μ²

41
Q

Denominator of Trapezoidal Rule

A

if there are 5 function values, there are 4 subintervals

–> denom is 2n, where n is subintervals

42
Q

y top semicircles

A

y = √9-x²

43
Q

y bottom semicircles

A

y = -√9-x²

44
Q

x right semicircles

A

x = √9-y²

45
Q

x left semicircles

A

x = -√9-y²

46
Q

Calculating SD

A
  1. MODE
  2. STAT (2)
  3. 1-VAR (1)
  4. Enter Data (=)
  5. ON
  6. SHIFT STAT (1)
  7. VAR (4)
  8. σx (3)
  9. =

Maths Manual (12 decks)

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