Formulas and Theorems 437
- Displacement fromt 1 tot 2 =
∫t 2
t 1
v(t)
=s
(
t 2
)
−s
(
t 1
)
.
- Total Distance Traveled fromt 1 to
t 2 =
∫t 2
t 1
∣∣
v(t)
∣∣
dt.
- Business Formulas:
Profit=Revenue−Cost P(x)=R(x)−C(x)
Revenue=(price)(items sold) R(x)=px
Marginal Profit P′(x)
Marginal Revenue R′(x)
Marginal Cost C′(x)
P′(x),R′(x),C′(x)are the instantaneous
rates of change of profit, revenue, and cost
respectively.
- Exponential Growth/Decay Formulas:
dy
dt
=ky,y>0 andy(t)=y 0 ekt.
- Logistic Growth Models:
dP
dt
=kP
(
1 −
P
M
)
or
dP
dt
=
(
k
M
)
(P)(M−P).
P=
M
1 +Ae−kt
- Integration by Parts:
∫
udv=uv−
∫
vdualso written as
∫
f(x)g′(x)dx=f(x)g(x)−
∫
f′(x)g(x)dx
Note: When matchinguanddv,begin with
uand follow the order of the acronym
LIPET (Logarithmic, Inverse Trigonometric,
Polynomial, Exponential, and Trigonometric
functions).
- Derivatives of Parametric Functions:
dy
dx
=
dy
dt
dx
dt
,
dx
dt
/=0,
and
d^2 y
dx^2
=
dy′
dt
dx
dt
,
dx
dt
/= 0.
- Vector Functions:
Givenr(t)= f(t)i+g(t)j:
(a)
dr
dt
=
df
dt
i+
dg
dt
j
(b)
∫b
a
r(t)dt=
(∫b
a
f(t)dt
)
i
+
(∫b
a
g(t)dt
)
j
- Arc Length of a Curve:
(a) L=
∫b
a
√
1 +
(
dy
dx
) 2
dx,y=f(x)
(b) Parametric Equations:
L=
∫b
a
√(
dx
dt
) 2
+
(
dy
dt
) 2
dt,
x=f(t)andy=g(t)
(c) Polar Equations:
L=
∫b
a
√
r^2 +
(
dr
dθ
) 2
dθ, r=f(θ)
- Polar Curves:
(a) Slope ofr=f(θ)at(r,θ)
dy
dx
=
dy
dθ
dx
dθ
=
f′(θ)sinθ+ f(θ)cosθ
f′(θ)cosθ−f(θ)sinθ
,
dx
dθ
/=0,
or written asm=
r+tanθ
dr
dθ
−rtanθ+
dr
dθ
.
