C3 Flashcards
if y=eˣ, then x = ?
ln(y)
ln(a) + ln(b) = ?
ln(ab)
ln(a) - ln(b) = ?
ln(a/b)
kln(a) = ?
ln(aᵏ)
ln(e) = ?
1
ln(1) = ?
0
ln(eˣ) = ?
x
eˡⁿˣ = ?
x
lnx is the inverse of ?
eˣ
eˣ is the inverse of ?
lnx
How do you display the root of two lines?
Intersection
How do you show that a root lies between 4 & 5?
input x=4 into f(x)
input x=5 into f(x)
if there is a sign change then root lies between the two values of x
What form must equations be in for iteration?
x =
What value should you first input into iteration?
lowest value of restriction range
How should you treat reflections?
As stretches
If stretches & translations are in different axes, then the order ____ ?
Doesn’t matter
If stretches & translations are both in the y axes, then the order ____ ?
Stretch, then translate
If stretches & translations are both in the x axes, then the order ____ ?
Translate, then stretch
when does |x|= x?
when x≥0
when does |x| = -x?
when x≤0
If |x| = k, then x = ?
x = ±k
If |x| = |y| then x² =, x = ?
x² = y², x = ±y
Restrictions when |x| ≤ k
-k ≤ x ≤ k
Restrictions when |x| ≥ k
x ≥ k or x ≤ -k
If |𝓍| ≤ |y|, then 𝓍² ____ ?
𝓍² ≤ y²
y = eˣ, dy/d𝓍 = ?
dy/d𝓍 = eˣ
y = eᶠ⁽ˣ⁾, dy/d𝓍 = ?
dy/d𝓍 = f’(𝓍)eᶠ⁽ˣ⁾
y = ln𝓍, dy/d𝓍 = ?
dy/d𝓍 = 1/x
y = ln(f(𝓍)), dy/d𝓍 = ?
dy/d𝓍 = f’(x)/f(x)
y = [f(𝓍)]ⁿ, dy/d𝓍 = ?
dy/d𝓍 = nf’(x)[f(x)]ⁿ⁻¹
If y = u⋅v, then dy/d𝓍 = ?
dy/d𝓍 = vu’ + uv’
If y = u/v, then dy/d𝓍 = ?
dy/d𝓍 = (vu’ - uv’)/v²
dy/d𝓍 x d𝓍/dy = ?
1
d𝓍/dy = ?
1/(dy/d𝓍)
cosec𝓍 = ?
1/sin𝓍
sec𝓍 = ?
1/cos𝓍
cot𝓍 = ?
1/tan𝓍 = cos𝓍/sin𝓍
CAST = ?
C - only cosine is positive (270-360) (repeats at 360-α)
A - All trig are positive (0-90)
S - only sine is positive (90-180) (repeats at 180-α)
T - only tan is positive (180-270) (repeats at 180+α)
sin(A+B) = ?
sinAcosB + cosAsinB
sin(A-B) = ?
sinAcosB - cosAsinB
cos(A+B) = ?
cosAcosB - sinAsinB
cos(A-B) = ?
cosAcosB + sinAsinB
tan(A+B) = ?
(tanA + tanB)/(1 - tanAtanB)
tan(A-B) = ?
(tanA - tanB)/(1 + tanAtanB)
sin2A = ?
2sinAcosA
cos2A = ?
= cos²A - sin²A
= 1 - 2sin²A
= 2cos²A - 1
tan2A = ?
2tanA/(1 - tan²A)
sin4A = ?
2sin2Acos2A
asinθ + bcosθ can also be written as?
Rsin(θ±α) or Rcos(θ±α)
How do you write in the form Rcos(θ±α) from
Acos𝓍 - Bsin𝓍?
Equate cos𝓍 & sin𝓍 terms so Rcosα = A & Rsinα = B ∴ tanα = B/A R² = A² + B² R = √(A² + B²)
What does R give?
max/min value
tanA = ?
sinA/cosA
sin²A + cos²A = ?
1
secA = ?
1/cosA
cosecA = ?
1/sinA
cotA = ?
1/tanA = cosA/sinA
cosec²A = ?
1 + cot²A
sec²A = ?
tan²A + 1
∫eᶠ⁽ˣ⁾ d𝓍 = ?
eᶠ⁽ˣ⁾/f’(𝓍) + c
∫1/𝓍 d𝓍 = ?
ln|𝓍| + c
Reverse chain rule only works for?
Linear functions inside bracket
∫(a𝓍 + b)ⁿ d𝓍 = ?
1/(a(n+1) ⋅ (a𝓍 + b)ⁿ⁺¹ + c
∫f’(𝓍)/f(𝓍) d𝓍 = ?
ln|f(𝓍)| + c
∫n⋅f’(𝓍)/f(𝓍) d𝓍 = ?
n⋅ln|f(𝓍)| + c
Volume of revolution around the 𝓍 - axis?
Vₓ = π ᵇ∫ₐ [f(𝓍)]²
Volume of revolution around the y - axis?
Vᵧ = π ᵇ∫ₐ [g(y)]²
Volume of revolution between two curves?
Vₓ = π ᵇ∫ₐ [f(𝓍)]² - π ᵇ∫ₐ [g(𝓍)]²
Define: 𝓍∈R
𝓍 belongs to the set of real numbers
When are functions one-to-one?
When each member of 𝓍 maps to only one member of y
How to find the vertex of a graph?
Complete the square
min/max point @ (-p,q)
Properties of Inverse functions
reflections in y=𝓍
only one-to-one functions have an inverse
