Section 1
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cos(2x) = ?
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Cards (104)
Section 1
(50 cards)
cos(2x) = ?
2cos^2x-1 1-2sin^2x cos^2x-sin^2x
Position, Velocity, Acceleration
S V = S' A = V' = S''
d/dx cos^-1x
-x'/sqrt(1-x^2)
Relate f and f'
f inc f'>0 → above the x-axis f dec f'<0 → below the x-axis
L'Hôpital's Rule Conditions
0/0, ∞/∞, 0/∞, ∞/0
3 variables start with the...
question
#/∞
0
d/dx secx
secxtanx
d/dx cscx
-cscxcotx
lim (x → 0) sinx/x
1
secant line
m=average rate
y →slope y'→y values
m of f is the y of f' at same x
Derivative=?
Slope
Product Rule
uv' + vu'
dy/dx dy → ? ; dx → ?
dy function dx variable
m= lim (h → 0) (f(x+h)-f(x))/h = ?
f'(x)
d/dx tanx
sec^2x
lim (x → ∞) f(x)=constant CONSTANT=?
H.A.
sin^2x+cos^2x = ?
1
tan^-1x
x'/(1+x^2)
d/dx sec^-1x
x'/(|x|•sqrt(x^2-1))
d/dx (f^2)
2ff'
y' can be in terms of....
-y only -x only -x and y
lim (x → 0) cosx-1/x
0
Displacement = ?
Final position-Initial position b ∫ V a
Relate graphs of f and f'
graph of f •mountains and valleys graph of f' •zeros
Quotient Rule
(uv'-vu')/v²
d/dx eˣ
eˣ•x'
d/dx sin^-1x
x'/sqrt(1-x^2)
d/dx sinx
cosx
tangent line
m=rate at a point
lim (x → ∞) sinx/x
0
V=-
Left
d/dx cot^-1x
-x'/(1+x^2)
d/dx cosx
-sinx
avg f' = ?
(f(b)-f(a))/b-a
V=0
Stop
lim (x → c) f(x)= ∞ C=?
V.A.
Continuity
lim (x → c-) f(x)= lim (x → c+) f(x)= f(c)
lim (x → 0) 1-cosx/x
0
Speed
|V|
L'Hôpital's Rule
lim (x → c) f(x)/g(x) take derivative of TOP and BOTTOM separately
Intermediate Value Theorem (IVT) m's
Every m exists if 1) closed interval 2) differentiable
V=+
Right
Differentiability
1) continuous 2) mL=mR
Intermediate Value Theorem (IVT) ~y's
Every y exists if 1) closed interval 2) continuous
sin(2x) = ?
2sinxcosx
(+/- sqrt) only when
solving an equation
d/dx sqrt(x)
1/2sqrt(x)
d/dx cotx
-csc^2x
Section 2
(50 cards)
Mean Value Theorem (MVT)
there exists a number c such that f'(c )= (f(b) -f(a))/(b-a) if 1) closed interval 2) continuous
b ∫ f(x) dx = ? split the limit a
c. b ∫ f(x) dx + ∫ f(x) dx a. c
∫secxtanx dx
secx+C
Absolute Max & Min [x,x]
x - critical points f(x) end pt. plug y'=0 (cont.) y'=und. (D) (highest value->abs max condition (piecewise). lowest value->abs min) end pt.
Relate f, f', f''
f concave up f' inc f'' >0 f concave down f' dec f'' <0
∫cscxcotx dx
-cscx+C
When is a particle slowing down?
When v(t) and a(t) have opposite signs.
∫loga(x) dx
(x•ln(x)-x)/ln(x)+C
Related rates
-x^2+y^2=c2 -A=1/2xy -sinx=opp/hyp -cosx=adj/hyp -tanx=opp/adj
d/dx csc^-1x
-x'/(|x|•sqrt(x^2-1))
Increasing and concave up
RRAM. over LRAM. under MRAM. under TR. over tangent line. under
Total Distance = ?
b ∫ |V| a
d/dx a^x
a^x• ln(a)•x'
∫ #/linear
(ln|linear|)/derivative +C
sin^-1x range
[-pi/2, pi/2]
Decreasing and concave up
RRAM. under LRAM. over MRAM. over TR. under tangent line. under
inflection point
Switch between concave up and concave down (midway) •y''=0 or y''=und •change in sign of y''
Fundamental Theorem of Calculus (PART 2)
b ∫ derv = org(b)-org(a) a
∫ sinx dx
-cosx+C
Domain 1) Polynomial ? 2) sqrt(inside) ? 3) ln(inside) ? 4) rational ?
1) (-∞, ∞) 2) inside>_ 0 3) inside>0 4) bottom ≠ 0
inc. or dec.
[ , ] unless ∞
b ∫ k•f(x) dx = ? a
b k• ∫ f(x) dx a
Your Life Formula
b ∫ f(b)=f(a)+ ∫ f'(x) a
∫1/x dx
ln|x|+C
∫ cosx dx
sinx +C
Decreasing and concave down
RRAM. under LRAM. over MRAM. under TR. over tangent line. over
Extreme Value Theorem (EVT)
Must have abs min & abs max 1) closed interval 2) continuous
∫a^x dx
(a^x)/(ln(a))+C
d/dx loga(x)
1/(x•lna)•x'
Fundamental Theorem of Calculus (PART 1)
x d/dx ∫ f(t) dt =f(x) • x' a
d/dx ln(x)
1/x • x'
∫
antiderivative
∫ sec^2(x) dx
tanx + C
∫csc^2x dx
-cotx+C
When is a particle speeding up?
When v(t) and a(t) have the same signs.
lim (n → ∞)
n l w ∑ f(ck) Δx k=1
Increasing and concave down
RRAM. over LRAM. under MRAM. over TR. under tangent line. over
∫ x^3
x^4/4+C
AREA (x-style)
x2 AREA = ∫ TOP-BOTTOM x1
a ∫ f(x) dx = ? a
0
cos^-1(x) range
[0, pi]
sec^2(x)=
1+tan^2(x)
∫lnx dx
(x•lnx-x)/ln(a)+C
tan^-1(x) range
[-pi/2, pi/2]
b ∫ f(x) dx = - ? a
a ∫ f(x) dx = b
Slope Fields
m=0 - m=+ / m=- \ m=und. |
∫ e^x
e^x /x' + C
Concepts 1) min & max 2) decreasing 3) increasing 4) inflection pt 5) concave down 6) concave up
1) f'=0 2) f'<0 3) f'>0 4) f''=0 5) f''<0 6) f''>0
Area of a trapezoid
A=1/2(b1+b2)h
b ∫ [f(x)+/-g(x)] dx =? split the function a
b b ∫ f(x) dx +/- ∫ g(x) dx = ? a a
Section 3
(4 cards)