MATH 211 Calculus I
Formula Sheet
Module 00: Functions
Composition
Transformation of Functions and Graphs
In y = a(x-h)^2 + k:
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The sign of a reflects the graph
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If 0 < a < 1, the graph compresses, and stretches when a > 1
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b shift horizontally, right if b>0 and left if b<0
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k shifts vertically
Inverse functions
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Module 01: Limits
Average velocity (i.e slope of a secant line)
𝑣𝑎𝑣 =
𝑠(𝑡1)−𝑠(𝑡0)
𝑡1 − 𝑡0
Instantaneous velocity (i.e slope of a tangent line)
𝑣𝑖𝑛𝑠𝑡 = lim
𝑡 → 𝑡0
𝑠(𝑡)−𝑠(𝑡0)
𝑡 − 𝑡0
One sided limits
Right-hand limit: lim 𝑓(𝑥) = 𝐿
+
𝑥→𝑏
Left-hand limit: lim 𝑓(𝑥) = 𝐿
−
𝑥→𝑏
For a limit to exist, the right-hand and left-hand limit must be equal.
Limit Laws
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The Squeeze Theorem
Infinite Limits
If f(x) grows arbitrarily large for all x sufficiently close, but not a: lim 𝑓(𝑥) =± ∞
𝑥→𝑎
Vertical Asymptotes
The line x=a is a vertical asymptote for f if any of the following hold:
lim 𝑓(𝑥) =± ∞
𝑥→𝑎
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lim 𝑓(𝑥) =± ∞
+
𝑥→𝑎
lim 𝑓(𝑥) =± ∞
−
𝑥→𝑎
To find the vertical asympote you need to set denominator equal to zero and solve for x.
Infinite Limits Analytically
In lim
𝑥→
𝑝(𝑥)
𝑞(𝑥)
where p(a) is approaching a non-zero number, and q(x) is zero, the limit is
infinite. To analyse the sign of the function, you factor the polynomial.
Limits at Infinity
If f(x) grows arbitrarily close to a finite number L for all sufficiently large and positive x,
then: lim 𝑓(𝑥) = 𝐿
𝑥→∞
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End behavious for transcendental functions
Continuity
A function f is continuous at a if f(a) is defined, lim 𝑓(𝑥) or lim 𝑓(𝑥) = 𝑓(𝑎) .
𝑥→𝑎
𝑥→𝑎
Discontinuity
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Jump discontinuity where the left and right hand side values for a limit
approaching a value do not match.
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Removable discontinuity occurs at a point where the graph of a function has a
hole in it.
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Infinite discontinuity if theres a vertical asymptote at a.
Continuity on an Interval
Continuity Theorems
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Module 02: Derivatives Part 1
3.1 Introducing Derivatives
As h -> 0, the slope of the tangent line is now given as: lim (𝑓(𝑎 + ℎ) − 𝑓(𝑎)) / ℎ
ℎ→0
The derivative of f at a, denoted f’(a), is given by either of the two following limits,
provided the limits exist and a is in the domain of f:
𝑓'(𝑎) = lim
𝑥→𝑎
𝑓(𝑥)−𝑓(𝑎)
𝑥−𝑎
𝑂𝑅 lim
ℎ→0
𝑓(𝑎+ℎ)−𝑓(𝑎)
ℎ
where if f’(a) exists, we say that f is differentiable at a.
The Derivative as a Function
Sketching graphs of the derivative
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Where there are minimums and minimums there is a derivative of zero.
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Where the function is increasing the derivative is negative.
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Where the function is decreasing the derivative is positive.
Continuity
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If f is differentiable at a, then f is continuous at a
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If f is not continuous at a, then f is not differentiable at a
Rules of Differentiation
1. Constant Rule
a. f(x) = c
b. 𝑓'(𝑥) = lim
ℎ→0
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
= lim
ℎ→0
𝑐−𝑐
ℎ
= lim
ℎ→0
0
ℎ
=0
c. If c is a real number, then d/dx(c) = 0
2. Power Rule
a. f(x) = x^n , where n is a non-negative integer
b. For f(x), 𝑓'(𝑥) = lim
ℎ→0
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
= lim
ℎ→0
𝑥+ℎ−𝑥
ℎ
= lim
ℎ→0
ℎ
ℎ
=1
c. So d/dx x = 1
d. If f(x) = x^2, 𝑓'(𝑥) = lim
ℎ→0
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
2
= lim
ℎ→0
2
(𝑥+ℎ) −𝑥
ℎ
= lim 2𝑥 + ℎ = 2𝑥
ℎ→0
e. So d/dx x^2 = 2x
f.
If n is a nonnegative integer, then d/dx(x^n) = nx^n-1
3. Constant Multiple Rule
a. If f is differentiable at x and c is a constant, then d/dx (cf(x))=cf’(x)
4. Sum Rule
a. If f and g are differentiable at x, then d/dx(f(x) + g(x)) = f’(x) + g’(x)
5. Difference Rule
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a. If f and g are differentiable at x, then d/dx(f(x) - g(x)) = f’(x) - g’(x)
6. The number e and the exponential function satisfies lim
ℎ→0
ℎ
𝑒 −1
ℎ
=1
a. The function f(x) = e^x is differntiable for all real numbers x and d/dx(e^x) =
e^x
The Product Rule
If f and g are differentiable at x, then
𝑑
𝑑𝑥
(𝑓(𝑥)𝑔(𝑥)) = 𝑓'(𝑥)𝑔(𝑥) + (𝑓𝑥)𝑔'(𝑥)
The Quotient Rule
If f and g are differentiable at x and g(x) does not equal 0, then the derivative of f/g at x
exists and, then
𝑑
𝑑𝑥
𝑓(𝑥)
( 𝑔(𝑥) ) =
𝑔(𝑥)𝑓'(𝑥) − 𝑓(𝑥)𝑔'(𝑥)
2
(𝑔(𝑥))
The Power Rule for negative integers
𝑑
𝑑𝑥
𝑛
(𝑥 ) =
𝑑
𝑑𝑥
(
1
𝑚
𝑥
𝑚
)=
𝑚−1
𝑥 * 0 − 1*(𝑚𝑥
𝑚 2
(𝑥 )
m=-n, so -m=n
So d/dx x^n = -mx^-m-1 = nx^n-1
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)
𝑚−1
=
−𝑚𝑥
2𝑚
𝑥
(𝑚−1)−2𝑚
=− 𝑚𝑥
Derivatives of Trigonometric Functions
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Derivatives as Rates of Change
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Module 03: Derivatives Part 2
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