Test 3 Handout
Math 1B
Work (Sec. 6.4)
Calculus of Parametric Equations (10.2)
General Formula:
dy
dt
dy
dx = dx
dt
Work = Force x Distance
Hooke’s Law (springs):
d dy
dt dx
d2y
dx
dx2 =
dt
f(x) = kx,
where k is the spring constant and
x is the number of feet the spring is stretched
beyond its natural length
Area, A =
Arc Length (8.1 & 10.2)
Arc Length, L =
Where ds =
!
or
or
ds =
ds =
"
b
ds
"
b
y
a
dx
dt
dt
Polar Equations (10.3 & 10.4)
a
!
Rectangular relationships:
 dy2
1 +  dx dx
x = r cosθ
 dx2
1 +  dy dy
y = r sinθ
y
tanθ = x
x2 + y2 = r2
 dx2
 dy2
 dt  +  dt  dt
Slope of Tangent Line
dr
dθ · sinθ + r · cosθ
dy
dx = dr
dθ · cosθ – r · sinθ
Surface Area (Sec. 8.2)
Surface Area, A =
#
b
2"r ds
a
Area of a Polar Region
Where r is the radius of revolution.
ds for surface area is the same as for arc length.
!
y
!
2
Area, A =
1 2
r d"
2
a
y
y
y = tan -1x
y = ln x
!
y = ex
1
x
-!
2
#
b
x
1
x
Trigonometry
Some Trigonometric Identities:
1.
cos2x + sin2x = 1
2.
tan2x + 1 = sec2x
3.
sin(2x) = 2sinx cosx
4.
cos(2x) = cos2x – sin2x
5.
cos2x = 2 [1 + cos(2x)]
6.
sin2x = 2 [1 – cos(2x)]
1
1
Some Trigonometric Integrals:
1.
⌠
⌡ tanu du = ln | secu | + C
2.
⌠
⌡ cotu du = ln | sinu | + C
3.
⌠
⌡ secu du = ln | secu + tanu | + C
4.
⌠
⌡ cscu du = ln | cscu – cotu | + C
5.
u
1
1
⌠
⌡ u2 + a2 du = a tan-1 a + C
6.
⌠

⌡
u2 + a2
du =
u
7.
⌠
⌡
u
a2
u2 + a2 du = 2 u2 + a2 – 2 ln (a + u2 + a2 ) + C
a + u2 + a2 


u2 + a2 – a ln
 +C
u
Quadrant I Trig Values
Degrees
0°
30°
45°
60°
90°
Radians
0
π
6
π
4
π
3
π
2
sinθ
0
1
2
2
2
3
2
1
cosθ
1
3
2
2
2
1
2
0
tan θ
0
3
3
1
3
undefined