Quiz 3 Flashcards
(27 cards)
Derivative of trig functions
sin(x): cos(x)
cos(x): -sin(x)
tan(x): sec²(x)
cot(x): -cosec²(x)
sec(x): sec(x)tan(x)
cosec(x): -cosec(x)cot(x)
Solving trig quotient limits
lim (sin3x/3x) = 1
x -> 0
lim (sin3x/8x)
x -> 0
=
lim (sin3x/3x) * (3x/8x) = 1 * 3/8 = 3/8
x -> 0
Velocity as a function (or a derivative)
v = ∆ in position / ∆ in time
= ∆s / ∆t
= ds/dt
= instantaneous velocity
Composite functions derivatives
d/dx = f(g(x) = f’( g(x) ) * g’(x)
or
dy/dx = dy/du * du/dx
Implicit differentiation (the formulas)
x² + y² = 1
y = ± √1-x²
y’ = -y/x
e differenciation
y = eᵘ⁽ˣ⁾
y’ = eᵘ⁽ˣ⁾ * u’(x)
Log/ln precalc rules
Power rule:
lnbˣ = xln b
Product rule:
ln(a.b) = ln(a) + ln(b)
Quotient rule:
ln(a/b) = ln(a) - ln(b)
Log/ln transformation rule
y = logₑx <=> eʸ = x
y’ = 1/eʸ = 1/x
Average and marginal costs given a cost function
Cost function: C(x)
Average cost: C(x)/(x)
Marginal cost: C’(x)
Implicit differentiation steps
Rule of thumb:
any term with only x:
differentiate as normal
any term with only y:
differentiate y normally then multiply by dy/dx
eg. y³ -> 3y² dy/dx
any term with x and y:
1. remove coefficient and y term and only differentiate x term
eg. differentiate x³
2. multiply the differentiate of x by the coefficient and the y term
- differentiate y term only without coefficient and multiply by dy/dx
- multiply the differentiate of y by the coefficient and the x term
- combine terms
Simple interest
after n months:
p (1 + r/n)
after a year:
p (1 + r/n)ⁿ
p - principle amount
r - rate of interest as a decimal
n - number of months
Log/Ln differentiation rules
d/dx ln(x) = 1/x
d/dx ln(u(x)) = u’(x)/u(x)
Log domain (0, ∞)
h(x) = logₐu(x)
h’(x) = ( 1/ln(a) ) * ( u’(x)/u(x) )
h(x) = logₐx
h’(x) = 1 / (xlna )
h(x) = eˣ ˡⁿ ᵇ
h’(x) = bˣ * ln(b)
Inverse trig functions
sin⁻¹ (x) -> 1 / √1 - x²
cos⁻¹ (x) -> -1 / √1 - x²
tan⁻¹ (x) -> 1 / 1 + x²
csc⁻¹ (x) -> -1 / |x| √x² -1
sec⁻¹ (x) -> 1 / |x| √x² -1
cot⁻¹ (x) -> -1 / 1 + x²
multiply by derivative of x
Important note about inverse trig functions
Is:
y = sin⁻¹ (x) <=> sin(y) = x
Is not:
sin⁻¹ (x) ≠ 1 / sin(x) or csc(x)
General rule for finding inverse functions
y = f⁻¹(x)
f(y) = x
y’ = 1/f’(y)
Domain and range of inverse trig functions
sin⁻¹(x):
D [-1, 1]
R [-π/2, π/2]
cos⁻¹(x):
D [-1, 1]
R [0, π]
tan⁻¹(x):
D [-∞, ∞]
R [-π/2, π/2]
Rates of change in terms of area of circle
How fast is the radius changing when a) r =__ :
dA/dt = 2πr dr/dt
dr/dt = __ / 2πr cm/s
b) circumference = __ :
dr/dt
= __ / 2πr
= __ / circumference cm/s
Velocity v speed
If you know the velocity, the speed is the modulus of the velocity
Highest point of an object thrown
when velocity equals 0
Speed increasing on interval..
From highest point (not inclusive) to time where it hits the ground [inclusive]
d/dx 8e^x =
8e^x
x^√x
e^(√xlnx)
e
power - exponent, ln, base
ln values
ln(1) = 0
Derivative of log(basea)x
1/(xlna) *always ln
