Download Useful Formulae Exam 1 - Iowa State University

Leader: Wheaton
Useful Formulae
Course: Math 165
Supplemental Instruction
Instructor: Dr. Tokorcheck
Iowa State University
Date: 02/11/15
Below is a list of formulae that it will be helpful to know moving forward in calculus that we
have learned or needed to know to this point. While this list is by no means exhaustive, it should
be a useful starting point.
Trigonometric Formulae
Limit Laws
lim 𝑘 = 𝑘
Angle Sum Identities
𝑥→𝑐
sin(𝛼 + 𝛽) = sin(𝛼) cos(𝛽) + cos(𝛼) sin⁡(𝛽)
lim 𝑥 = 𝑐
𝑥→𝑐
cos(𝛼 + 𝛽) = cos(𝛼) cos(𝛽)
− sin(𝛼) sin⁡(𝛽)
𝑐𝑜𝑠 2 (𝑥) + 𝑠𝑖𝑛2 (𝑥) = 1
1 + 𝑡𝑎𝑛
𝑥→𝑐
lim[𝑓(𝑥) + 𝑔(𝑥)] = lim 𝑓(𝑥) + lim 𝑔(𝑥)
Pythagorian Identities
2 (𝑥)
lim 𝑘𝑓(𝑥) = 𝑘 ∗ lim 𝑓(𝑥)
𝑥→𝑐
2
= sec (𝑥)
1 + 𝑐𝑜𝑡 2 (𝑥) = csc 2 (𝑥)
Odd-Even Identities
sin(−𝑥) = − sin(𝑥)
cos(−𝑥) = cos(𝑥)
tan(−𝑥) = − tan(𝑥)
𝑥→𝑐
𝑥→𝑐
𝑥→𝑐
lim[𝑓(𝑥) − 𝑔(𝑥)] = lim 𝑓(𝑥) − lim 𝑔(𝑥)
𝑥→𝑐
𝑥→𝑐
𝑥→𝑐
lim[𝑓(𝑥) ∗ 𝑔(𝑥)] = lim 𝑓(𝑥) ∗ lim 𝑔(𝑥)
𝑥→𝑐
𝑥→𝑐
lim [
𝑥→𝑐
𝑥→𝑐
lim 𝑓(𝑥)
𝑓(𝑥)
] = 𝑥→𝑐
𝑔(𝑥)
lim 𝑔(𝑥)
𝑥→𝑐
lim[𝑓(𝑥)]𝑛 = [lim 𝑓(𝑥)]
𝑥→𝑐
𝑛
𝑥→𝑐
𝑛
lim √𝑓(𝑥) = 𝑛√lim 𝑓(𝑥)
𝑥→𝑐
𝑥→𝑐
Continuity
Differentiable
For a function to be continuous it must satisfy
three conditions:
A function 𝑓(𝑥) is differentiable if 𝑓′(𝑐)
exists for every 𝑐 in a given interval.
1) 𝑐 is in the domain of 𝑓
2) lim 𝑓(𝑥) ⁡𝑒𝑥𝑖𝑠𝑡𝑠
𝑥→𝑐
3) lim 𝑓(𝑥) = 𝑓(𝑐)
An infinitely differentiable function is called
smooth (examples include 𝑠𝑖𝑛(𝑥) and
𝑐𝑜𝑠(𝑥))
𝑥→𝑐
Derivative Definitions
Derivative Rules
𝑓(𝑐 + ℎ) − 𝑓(𝑐)
ℎ→0
ℎ
This definition is useful to evaluate the
derivative at a given point 𝑐.
Power Rule
𝑓′(𝑐) = lim
𝑑 𝑛
(𝑥 ) = 𝑛𝑥 𝑛−1
𝑑𝑥
𝑓(𝑥 + ℎ) − 𝑓(𝑥)
ℎ→0
ℎ
𝑓 ′ (𝑥) = lim
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This definition gives a function that
represents derivative of the function 𝑓(𝑥)
𝑓(𝑥) − 𝑓(𝑐)
𝑥→𝑐
𝑥−𝑐
This definition is useful to calculate inverse
functions.
𝑓 ′ (𝑥) = lim
Trig Derivatives
𝑑
(cos(𝑥)) = − sin(𝑥)
𝑑𝑥
𝑑
(sin(𝑥)) = cos(𝑥)
𝑑𝑥
𝑑
(tan(𝑥)) = sec 2 (𝑥)
𝑑𝑥
𝑑
(cot(𝑥)) = − csc 2 (𝑥)
𝑑𝑥
𝑑
(sec⁡(x)) = tan(𝑥) sec(𝑥)
𝑑𝑥
𝑑
(csc⁡(x)) = cot(𝑥) csc(𝑥)
𝑑𝑥
Squeeze Theorem Limits
lim (
𝑥→0
1 − 𝑐𝑜𝑠(𝑥)
)=0
𝑥
𝑠𝑖𝑛(𝑥)
lim (
)=1
𝑥→0
𝑥
Common Exponential Derivatives
𝑑 𝑥
(𝑒 ) = 𝑒 𝑥
𝑑𝑥
𝑑 𝑥
(𝑎 ) = ln(𝑎) ∗ 𝑎 𝑥
𝑑𝑥
Linearity of Derivatives
𝑑
𝑑
(𝑐 ∗ 𝑓(𝑥)) = 𝑐
(𝑓(𝑥))
𝑑𝑥
𝑑𝑥
𝑑
𝑑
𝑑
(𝑓(𝑥) + 𝑔(𝑥)) =
𝑓(𝑥) +
𝑔(𝑥)
𝑑𝑥
𝑑𝑥
𝑑𝑥
Product Rule
𝑑
𝑑
(𝑓 ∗ 𝑔)(𝑥) = 𝑓(𝑥) 𝑔(𝑥)
𝑑𝑥
𝑑𝑥
𝑑
+ 𝑔(𝑥) 𝑔(𝑥)
𝑑𝑥
Quotient Rule
𝑑 𝑓
𝑓 ′ (𝑥) ∗ 𝑔(𝑥) − 𝑔′ (𝑥) ∗ 𝑓(𝑥)
( ) (𝑥) =
𝑑𝑥 𝑔
𝑔2 (𝑥)
Composition Rule
𝑑
(𝑓°𝑔)(𝑥) = 𝑓 ′ (𝑔(𝑥)) ∗ 𝑔′(𝑥)
𝑑𝑥
𝑑
(𝑓°𝑔°ℎ)(𝑥) = 𝑓 ′ (𝑔(ℎ(𝑥)))
𝑑𝑥
∗ 𝑔′ (ℎ(𝑥)) ∗ ℎ′(𝑥)
U substitution composition
𝑑
𝑑𝑓(𝑢) 𝑑𝑢
𝑓(𝑥) =
∗
𝑑𝑥
𝑑𝑢
𝑑𝑥
𝑑
𝑑𝑓(𝑣) 𝑑𝑣(𝑢) 𝑑𝑢
𝑓(𝑥) =
∗
∗
𝑑𝑥
𝑑𝑣
𝑑𝑢
𝑑𝑥
Common Log Derivatives
𝑑
1
(ln(𝑥)) =
𝑑𝑥
𝑥
𝑑
1
(log 𝑎 (𝑥)) =
𝑑𝑥
𝑥𝑙𝑛(𝑎)
Inverse Functions
A function 𝑓(𝑥) is one to one if for every pair
of 𝑥1 , 𝑥2 then 𝑓(𝑥1 ) ≠ 𝑓(𝑥2 ) (vertical line
test)
must pass vertical line test to have inverse
(may have to restrict domain)
Inverse function theorem:
𝑑𝑥
1
Simply stated: 𝑑𝑦 = 𝑑𝑦⁄𝑑𝑥
Easy way to solve for an inverse is switch and
solve (exchange all x’s and y’s, solve for y)