Yr12A HSC Flashcards
1st Derivative
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)
2nd Derivative
f’‘(x) > 0 concave up
f’‘(x) < 0 concave down
f’‘(x) = 0 concavity changes or horizontal point of inflexion
Differentiation by First Principle
Curve Sketching Menu; 7 steps
- Domain: Find the domain of f(x)
- Symmetry: Find whether the function is even, odd, or neither
- 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 - 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 –> -∞ - 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 - 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 - ANY OTHER FEATURES
A ROUTINE WARNING OF INCOMPLETENESS
why isn’t f’‘(x) = 0 a sufficient condition for a point of inflexion?
the sign of f’‘(x) must also change around the point (the concavity must change)
define e
the number such that the exponential function y=eˣ has exactly gradient 1 at its y-int
e = 2.7183…
oddness & evenness of the trigonometric functions
ODD
sin(-x) = -sinx
tan(-x) = -tanx
cosecx & cotx
EVEN
cos(-x) = cosx
secx
Measures of Location
mode
median
mean
Measures of Spread
range, IQR, variance, standard deviation
The 5 Number Summary
min (Q0)
Q1
Q2 (median)
Q3
max (Q4)
Expected Value (weighted mean)
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
Standard deviation
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
One to one functions
passes VLT
passes HLT
many to one functions
passes VLT
fails HLT
one to many relations
fails VLT
passes HLT
many to many relations
fails VLT
fails HLT
Type of Relation & which line test they pass/fail
VLT pass: to one
VLT fail: to many
HLT pass: one to
HLT fail: many to
definitions of the 3 trig functions (in terms of x, y, r)
sinx = y/r
cosx = x/r
tanx = y/x
(trig) 3 Pythagorean identities
sin²x + cos²x = 1
tan²x + 1 = sec²x
cot²x + 1 = cosec²x
in a combination of transformations, (vertical translations)
do them last
horizontal translations and phase of trig functions
y = sin(x + α)
y = cos(x + α)
y = tan(x + α)
all have phase α (shifting original functions left by α)
the periods of the trig functions
y = sinx (&cosecx)
y = cosx (&secx)
a full revolution (2pi)
y = tanx (&cotx)
half a revolution (pi)
the complementary angle identities
cos (90-α) = sinα
cot(90-α) = tanα
cosec(90-α) = secα
vertical dilations
to stretch vertically by a
factor of a
replace y by y/a
new function rule is y=af(x)
horizontal dilations
to stretch horizontally by a
factor of a
replace x by x/a
new function rule is y=f(x/a)
axis of dilation for horizontal and vertical dilations
vertical dilations: x axis
horizontal dilations: y axis
enlargements with centre the origin
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)
dilations with fractional or negative factor
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)
other names for
rotation of 180° about the origin
an enlargement with factor -1
reflection in the origin
commuting transformations
any 2 translations commute
any 2 dilations commute (including reflections)
a translation and dilation commute if one is vertical and the other is horizontal
horizontal dilations and period (trig functions)
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)
vertical dilations and amplitude (trig functions)
y = a sin x
y = a cos x
have amplitude a
(result of stretching vertically with factor a)
skewed data
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
A Universal Formula Involving All Four Transformations
- stretch horizontally with factor 1/a
- shift left b
- stretch vertically with factor k
- shift up c
(step 3 then 4 can be done before step 1 then 2)
Horizontal Asymptotes eg1
highest degree of x in numerator and denominator is the same
highest degree of x in numerator and denominator is the same
consider coefficient of x
Horizontal Asymptotes eg2
degree of x in numerator is less than denominator
degree of x in numerator is less than denominator
horizontal asymptote is at y=0
Horizontal Asymptotes eg3
degree of x in numerator is MORE than denominator
degree of x in numerator is MORE than denominator
no horizontal asymptote
Radian <==> Degree Conversion
Exact Trig Values
Variance <3
V(X) = E(X²) - E(X)²
V(X) = Σx²p(x) - μ²
Denominator of Trapezoidal Rule
if there are 5 function values, there are 4 subintervals
–> denom is 2n, where n is subintervals
y top semicircles
y = √9-x²
y bottom semicircles
y = -√9-x²
x right semicircles
x = √9-y²
x left semicircles
x = -√9-y²
Calculating SD
- MODE
- STAT (2)
- 1-VAR (1)
- Enter Data (=)
- ON
- SHIFT STAT (1)
- VAR (4)
- σx (3)
- =
