Student Solutions Manual to Accompany Calculus: Single and Multivariable 7th Edition | Complete Step-by-Step Solutions

,
,Lines Distance and Midpoint Formulas
Slope of line through (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ): Distance 𝐷 between (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ):
𝑦2 − 𝑦1 √
𝑚= 𝐷 = (𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
𝑥2 − 𝑥1

Point-slope equation of line through (𝑥1 , 𝑦1 ) Midpoint of (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ):
with slope 𝑚: ( )
𝑥1 + 𝑥2 𝑦1 + 𝑦2
,
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) 2 2

Slope-intercept equation of line with slope 𝑚 Quadratic Formula
and 𝑦-intercept 𝑏: If 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, then
√
𝑦 = 𝑏 + 𝑚𝑥 −𝑏 ± 𝑏2 − 4𝑎𝑐
𝑥=
2𝑎
Factoring Special Polynomials

Rules of Exponents 𝑥2 − 𝑦2 = (𝑥 + 𝑦)(𝑥 − 𝑦)
𝑥3 + 𝑦3 = (𝑥 + 𝑦)(𝑥2 − 𝑥𝑦 + 𝑦2 )
𝑎𝑥 𝑎𝑡 = 𝑎𝑥+𝑡 𝑥3 − 𝑦3 = (𝑥 − 𝑦)(𝑥2 + 𝑥𝑦 + 𝑦2 )
𝑎𝑥
= 𝑎𝑥−𝑡
𝑎𝑡 Circles
(𝑎𝑥 )𝑡 = 𝑎𝑥𝑡
Center (ℎ, 𝑘) and radius 𝑟:
Definition of Natural Log
(𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟2
𝑦
𝑦 = ln 𝑥 means 𝑒 = 𝑥
ex: ln 1 = 0 since 𝑒0 = 1 Ellipse
𝑦 𝑥2 𝑦2
𝑦=𝑒 𝑥 + =1
𝑎2 𝑏2
1 𝑦 = ln 𝑥
𝑥 𝑦
1 𝑏


𝑥
−𝑎 𝑎

Identities
−𝑏

ln 𝑒𝑥 = 𝑥 Hyperbola
𝑒ln 𝑥 = 𝑥
𝑥2 𝑦2
Rules of Natural Logarithms − =1
𝑎2 𝑏2

𝑦
ln(𝐴𝐵) = ln 𝐴 + ln 𝐵 𝑦 = 𝑏𝑥∕𝑎
𝐴
( )
ln = ln 𝐴 − ln 𝐵
𝐵 𝑥
𝑝
ln 𝐴 = 𝑝 ln 𝐴 𝑎

𝑦 = −𝑏𝑥∕𝑎

, Geometric Formulas
Conversion Between Radians and Degrees: 𝜋 radians = 180◦
Triangle Circle Sector of Circle
𝐴 = 12 𝑏ℎ 𝐴 = 𝜋𝑟2 𝐴 = 12 𝑟2 𝜃 (𝜃 in radians)
1
= 𝑎𝑏 sin 𝜃 𝐶 = 2𝜋𝑟 𝑠 = 𝑟𝜃 (𝜃 in radians)
2




𝑎 𝑟 𝑟 𝑠
ℎ

𝜃 𝜃
✛ 𝑏 ✲ 𝑟

Sphere Cylinder Cone
𝑉 = 43 𝜋𝑟3 𝐴 = 4𝜋𝑟2 𝑉 = 𝜋𝑟2 ℎ 𝑉 = 13 𝜋𝑟2 ℎ




Trigonometric Functions
𝑦 (𝑥, 𝑦)
sin 𝜃 =
✛




𝑟 sin(𝐴±𝐵) = sin 𝐴 cos 𝐵±cos 𝐴 sin 𝐵
✻
𝑥
cos 𝜃 =
𝑟 cos(𝐴±𝐵) = cos 𝐴 cos 𝐵∓sin 𝐴 sin 𝐵
𝑟




𝑦
𝑦
tan 𝜃 = sin(2𝐴) = 2 sin 𝐴 cos 𝐴
𝑥
✛




sin 𝜃 𝜃 ❄ cos(2𝐴) = 2 cos2 𝐴−1 = 1−2 sin2 𝐴
tan 𝜃 = ✛ 𝑥 ✲
cos 𝜃
cos2 𝜃 + sin2 𝜃 = 1
𝑦 𝑦 𝑦
𝑦 = tan 𝑥

1 𝑦 = sin 𝑥 1 𝑦 = cos 𝑥

𝑥 𝑥 𝑥
𝜋 2𝜋 𝜋 2𝜋 −𝜋 𝜋
−1 −1



The Binomial Theorem
𝑛(𝑛 − 1) 𝑛−2 2 𝑛(𝑛 − 1)(𝑛 − 2) 𝑛−3 3
(𝑥 + 𝑦)𝑛 = 𝑥𝑛 + 𝑛𝑥𝑛−1 𝑦 + 𝑥 𝑦 + 𝑥 𝑦 + ⋯ + 𝑛𝑥𝑦𝑛−1 + 𝑦𝑛
1⋅2 1⋅2⋅3
𝑛(𝑛 − 1) 𝑛−2 2 𝑛(𝑛 − 1)(𝑛 − 2) 𝑛−3 3
(𝑥 − 𝑦)𝑛 = 𝑥𝑛 − 𝑛𝑥𝑛−1 𝑦 + 𝑥 𝑦 − 𝑥 𝑦 + ⋯ ± 𝑛𝑥𝑦𝑛−1 ∓ 𝑦𝑛
1⋅2 1⋅2⋅3