Cheat Sheet Flashcards

1
Q

Definition of Derivative with h

A

lim h-> 0 f(x+h) - f(x) / h = f’(x)

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2
Q

Definition of Derivative with a

A

lim x-> a f(x) - f(a) / x-a = f’(x)

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3
Q

Horizontal Tangent

A

f’(x) = 0
numerator of dy/dx = 0

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4
Q

Vertical Tangent

A

f’(x) = undefined
denominator of dy/dx = 0

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5
Q

Horizontal asymptote

A

divide by denominator’s highest degree

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6
Q

lim x-> ∞ e^x

A

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7
Q

lim x-> -∞ e^x

A

0

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8
Q

Average rate of change

A

f(b) - f(a) / b - a

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9
Q

Average value

A

1 / b - a ∫ (a to b) f(x)

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10
Q

Tan line

A

y - y1 = m (x - x1)

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11
Q

slope (m) of tan line for perpendicular (normal line)

A
  • 1 / m
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12
Q

Critical Points

A

f’(x) = 0
f’(x) = undefined

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13
Q

Absolute Min and Max FRQ
(3)

A
  • Endpoints
  • Critical values and evaluate
  • Compare all evaluations with a TABLE
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14
Q

Inflection Points

A

f” (x) = 0
changing from f”(x) > 0 to f”(x)< 0 and vice versa

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15
Q

F’ and F”

A

f’(x)> 0 = f(x) increases
f’(x)< 0 = f(x) decreases

f”(x)> 0 = f(x) concave up
f”(x)< 0 = f(x) concave down

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16
Q

Quoetient Rule

A

g(x)f ‘ (x) - f(x)g ‘ (x) / (g(x))^2

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17
Q

d/du e^u

A

e^u du

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18
Q

d/dx sin

A

cos

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19
Q

d/dx cos

A

-sin

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20
Q

d/dx tan

A

sec^2

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21
Q

d/dx cotx

22
Q

d/dx sec

23
Q

d/dx csc

24
Q

1 / V1 - u^2

A

d/dx sin^-1

25
1 / 1 + u^2
d/dx tan^-1
26
∫x
x^n + 1/ n + 1
27
∫cosx dx
sinx + C
28
∫sinx dx
-cosx + C
29
Trapezoid Approximation
(B+b)h / 2
30
Midpoint
Must have equal intervals
31
Right + Left Rimean sum
GRAPHHHH b-a / n = distance
32
Tan line approximation
Over f" (x) < 0 Under f" (x) > 0 CONCAVITY
33
Right Endpoint approximation
Over f' (x) > 0 Under f' (x) < 0
34
Left Endpoint approximation
Over f' (x) > 0 Under f' (x) < 0
35
Trapezoid Approximation
Over f" (x) > 0 Under f" (x) < 0
36
Limits exists if
Phineas and Ferb Roller coaster lim x -> a- f(x) = lim x -> a+ f(x)
37
Definition of continuous
connecting Roller coaster, you might not live - if limit exists - f(a) is defined - lim x ->a f(x) = f(a)
38
Definition of differentiable
functional Roller coaster - function is continuous -lim x -> a- f'(x) = lim x -> a+ f'(x)
39
Mean Value theorem
- f is continuous [a,b] - f is differential (a,b) TAKE THE SLOPEEE (f(b)-f(a)) / (b- a)
40
Mean Value theorem for integral
integral from a to b f(x) = f(c)(b-a)
41
Intermediate Value Theorem
- f is continuous - f(a) doesnt equal f(b) Result: 𝑖𝑓 𝑎 < 𝑐 < 𝑏, there must be a value such as 𝑓(𝑎) < 𝑓(𝑐) < 𝑓(𝑏)
42
Rolle's Value theorem
1. f is Continuous [a,b] 2. f is Differentiable (a,b) 3. f(a)=f(b) Result: There must be at least one c, such as 𝑓′(𝑐) = 0
43
Squeeze Theorem
1. 𝑓(𝑥) ≤ 𝑘(𝑥) ≤ 𝑔(𝑥) 2. lim𝑓(𝑥) = lim 𝑥→𝑎 𝑔(𝑥) = 𝐿 Result: lim x→𝑎 𝑘(x) = 𝐿
44
Total Distance
∫ |𝑣(𝑡)|𝑑𝑡 (a to b)
45
Total Displacement
∫ 𝑣(𝑡) 𝑑𝑡 (a to b)
46
Average Velocity
s(b) - s(a) / b - a or 1 / b - a integral (a to b) v(t) dt
47
Average Acceleration
v(b) - v(a) / b - a v(0) = particle changes direction
48
Speed
= |v(t)|
49
Speed increases or decreases
It increases when V(t) and Ac (t) have same signs It decreases when V(t) and Ac (t) have diff. signs
50
How far a part. TRAVELS
NO initial condi
51
How far the part. is FROM THE ORIGINAL POINT
Yes, initial condi.
52
Inverse Derivative
f-1 '(x) = 1/ f'(g(x))