Cheat Sheet Flashcards
Definition of Derivative with h
lim h-> 0 f(x+h) - f(x) / h = f’(x)
Definition of Derivative with a
lim x-> a f(x) - f(a) / x-a = f’(x)
Horizontal Tangent
f’(x) = 0
numerator of dy/dx = 0
Vertical Tangent
f’(x) = undefined
denominator of dy/dx = 0
Horizontal asymptote
divide by denominator’s highest degree
lim x-> ∞ e^x
∞
lim x-> -∞ e^x
0
Average rate of change
f(b) - f(a) / b - a
Average value
1 / b - a ∫ (a to b) f(x)
Tan line
y - y1 = m (x - x1)
slope (m) of tan line for perpendicular (normal line)
- 1 / m
Critical Points
f’(x) = 0
f’(x) = undefined
Absolute Min and Max FRQ
(3)
- Endpoints
- Critical values and evaluate
- Compare all evaluations with a TABLE
Inflection Points
f” (x) = 0
changing from f”(x) > 0 to f”(x)< 0 and vice versa
F’ and F”
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
Quoetient Rule
g(x)f ‘ (x) - f(x)g ‘ (x) / (g(x))^2
d/du e^u
e^u du
d/dx sin
cos
d/dx cos
-sin
d/dx tan
sec^2
d/dx cotx
-csc^2
d/dx sec
secxtanx
d/dx csc
-cscxcotx
1 / V1 - u^2
d/dx sin^-1
1 / 1 + u^2
d/dx tan^-1
∫x
x^n + 1/ n + 1
∫cosx dx
sinx + C
∫sinx dx
-cosx + C
Trapezoid Approximation
(B+b)h / 2
Midpoint
Must have equal intervals
Right + Left Rimean sum
GRAPHHHH
b-a / n = distance
Tan line approximation
Over f” (x) < 0
Under f” (x) > 0
CONCAVITY
Right Endpoint approximation
Over f’ (x) > 0
Under f’ (x) < 0
Left Endpoint approximation
Over f’ (x) > 0
Under f’ (x) < 0
Trapezoid Approximation
Over f” (x) > 0
Under f” (x) < 0
Limits exists if
Phineas and Ferb Roller coaster
lim x -> a- f(x) = lim x -> a+ f(x)
Definition of continuous
connecting Roller coaster, you might not live
- if limit exists
- f(a) is defined
- lim x ->a f(x) = f(a)
Definition of differentiable
functional Roller coaster
- function is continuous
-lim x -> a- f’(x) = lim x -> a+ f’(x)
Mean Value theorem
- f is continuous [a,b]
- f is differential (a,b)
TAKE THE SLOPEEE
(f(b)-f(a)) / (b- a)
Mean Value theorem for integral
integral from a to b f(x) = f(c)(b-a)
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)
Rolle’s Value theorem
- f is Continuous [a,b]
- f is Differentiable (a,b)
- f(a)=f(b)
Result: There must be at least one c, such as 𝑓′(𝑐) = 0
Squeeze Theorem
- 𝑓(𝑥) ≤ 𝑘(𝑥) ≤ 𝑔(𝑥)
- lim𝑓(𝑥) = lim 𝑥→𝑎 𝑔(𝑥) = 𝐿
Result: lim x→𝑎 𝑘(x) = 𝐿
Total Distance
∫ |𝑣(𝑡)|𝑑𝑡
(a to b)
Total Displacement
∫ 𝑣(𝑡) 𝑑𝑡
(a to b)
Average Velocity
s(b) - s(a) / b - a
or
1 / b - a integral (a to b) v(t) dt
Average Acceleration
v(b) - v(a) / b - a
v(0) = particle changes direction
Speed
= |v(t)|
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
How far a part. TRAVELS
NO initial condi
How far the part. is FROM THE ORIGINAL POINT
Yes, initial condi.
Inverse Derivative
f-1 ‘(x) = 1/ f’(g(x))
