Cheat Sheet

Question 1

Definition of Derivative with h

Answer 1

lim h-> 0 f(x+h) - f(x) / h = f’(x)

Question 2

Definition of Derivative with a

Answer 2

lim x-> a f(x) - f(a) / x-a = f’(x)

Question 3

Horizontal Tangent

Answer 3

f’(x) = 0
numerator of dy/dx = 0

Question 4

Vertical Tangent

Answer 4

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

Question 5

Horizontal asymptote

Answer 5

divide by denominator’s highest degree

Question 6

lim x-> ∞ e^x

Answer 6

∞

Question 7

lim x-> -∞ e^x

Answer 7

0

Question 8

Average rate of change

Answer 8

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

Question 9

Average value

Answer 9

1 / b - a ∫ (a to b) f(x)

Question 10

Tan line

Answer 10

y - y1 = m (x - x1)

Question 11

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

Answer 11

• 1 / m

Question 12

Critical Points

Answer 12

f’(x) = 0
f’(x) = undefined

Question 13

Absolute Min and Max FRQ
(3)

Answer 13

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

Question 14

Inflection Points

Answer 14

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

Question 15

F’ and F”

Answer 15

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

Question 16

Quoetient Rule

Answer 16

g(x)f ‘ (x) - f(x)g ‘ (x) / (g(x))^2

Question 17

d/du e^u

Answer 17

e^u du

Question 18

d/dx sin

Answer 18

cos

Question 19

d/dx cos

Answer 19

-sin

Question 20

d/dx tan

Answer 20

sec^2

Question 21

d/dx cotx

Answer 21

-csc^2

Question 22

d/dx sec

Answer 22

secxtanx

Question 23

d/dx csc

Answer 23

-cscxcotx

Question 24

1 / V1 - u^2

Answer 24

d/dx sin^-1

Question 25

1 / 1 + u^2

Answer 25

d/dx tan^-1

Question 26

∫x

Answer 26

x^n + 1/ n + 1

Question 27

∫cosx dx

Answer 27

sinx + C

Question 28

∫sinx dx

Answer 28

-cosx + C

Question 29

Trapezoid Approximation

Answer 29

(B+b)h / 2

Question 30

Midpoint

Answer 30

Must have equal intervals

Question 31

Right + Left Rimean sum

Answer 31

GRAPHHHH
b-a / n = distance

Question 32

Tan line approximation

Answer 32

Over f” (x) < 0
Under f” (x) > 0
CONCAVITY

Question 33

Right Endpoint approximation

Answer 33

Over f’ (x) > 0
Under f’ (x) < 0

Question 34

Left Endpoint approximation

Answer 34

Over f’ (x) > 0
Under f’ (x) < 0

Question 35

Trapezoid Approximation

Answer 35

Over f” (x) > 0
Under f” (x) < 0

Question 36

Limits exists if

Answer 36

Phineas and Ferb Roller coaster

lim x -> a- f(x) = lim x -> a+ f(x)

Question 37

Definition of continuous

Answer 37

connecting Roller coaster, you might not live

• if limit exists
• f(a) is defined
• lim x ->a f(x) = f(a)

Question 38

Definition of differentiable

Answer 38

functional Roller coaster

• function is continuous
  -lim x -> a- f’(x) = lim x -> a+ f’(x)

Question 39

Mean Value theorem

Answer 39

• f is continuous [a,b]
• f is differential (a,b)

TAKE THE SLOPEEE

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

Question 40

Mean Value theorem for integral

Answer 40

integral from a to b f(x) = f(c)(b-a)

Question 41

Intermediate Value Theorem

Answer 41

• 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)

Question 42

Rolle’s Value theorem

Answer 42

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

Question 43

Squeeze Theorem

Answer 43

1. 𝑓(𝑥) ≤ 𝑘(𝑥) ≤ 𝑔(𝑥)
2. lim𝑓(𝑥) = lim 𝑥→𝑎 𝑔(𝑥) = 𝐿

Result: lim x→𝑎 𝑘(x) = 𝐿

Question 44

Total Distance

Answer 44

∫ |𝑣(𝑡)|𝑑𝑡
(a to b)

Question 45

Total Displacement

Answer 45

∫ 𝑣(𝑡) 𝑑𝑡
(a to b)

Question 46

Average Velocity

Answer 46

s(b) - s(a) / b - a
or
1 / b - a integral (a to b) v(t) dt

Question 47

Average Acceleration

Answer 47

v(b) - v(a) / b - a

v(0) = particle changes direction

Question 48

Speed

Answer 48

= |v(t)|

Question 49

Speed increases or decreases

Answer 49

It increases when V(t) and Ac (t) have same signs

It decreases when V(t) and Ac (t) have diff. signs

Question 50

How far a part. TRAVELS

Answer 50

NO initial condi

Question 51

How far the part. is FROM THE ORIGINAL POINT

Answer 51

Yes, initial condi.

Question 52

Inverse Derivative

Answer 52

f-1 ‘(x) = 1/ f’(g(x))