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AP Calculus BC
BC Calc Memory Quiz
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AP Calculus BC
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69 Terms
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1
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definition of continuity
lim (x→c⁻) f(x)=lim(x→c⁺)f(x)=f(c) OR lim(x→c)f(x)=f(c)
2
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Intermediate Value Theorem
(a) since f is continuous on \[a,b\] and
(b) f(a)
3
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chain rule
(d/dx)(f(g(x)))=f’(g(x))\*g’(x)
4
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product rule
(d/dx)\[f\*g\]= f’g+fg’
5
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quotient rule
(d/dx)\[f/g\]= (f’g-fg’)/g²
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(d/dx) sinu
cosu(u’)
7
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(d/dx) cosu
\-sinu(u’)
8
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(d/dx) tanu
sec²u(u’)
9
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(d/dx) cotu
\-csc²u(u’)
10
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(d/dx) secu
secutanu(u’)
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(d/dx) cscu
\-cscucotu(u’)
12
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extreme value theorem
since f is a continuous function on \[a,b\] must have a maximum and minimum value within that interval.
13
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relative/local extrema
a high/low point relative to the points around it; can only occur at a critical value.
14
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Absolute Extremum (Min/Max)
the highest/lowest point on a given interval; can occur at a critical value OR an endpoint.
15
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Mean Value Theorem(MVT)
Since f is continuous on \[a, b\] and differentiable on (a, b) MVT guarantees that there exists an x-value such that f’(x)=(f(b)-f(a))/(b-a)
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∫(cosu)du
sin u +C
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∫(sinu)du
\-cos u +C
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∫(sec²u)du
tan u +C
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∫(csc²u)du
\-cot u +C
20
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∫(secutantu)du
sec u +C
21
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∫(cscucotu)du
\-csc u +C
22
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1st FTC
∫(a to b) ƒ’(x)dx=f(b)-f(a)
23
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2nd FTC
(d/dx)∫(u to v) g(t)dt=g(v)(v’)-g(u)(u’)
24
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average value of a function on \[a,b\]
(1/(b-a))∫(a to b) f(x)dx
25
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displacement
∫(a to b) v(t)dt
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total distance
∫(a to b) |v(t)| dt
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∫(tanu)du
\-ln|cos u|+C
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∫(cotu)du
ln|sin u| +C
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∫(secu)du
ln|secu+tanu|+C
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∫(cscu)du
\-ln|cscu+cotu|+C
31
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inverse derivative
(f⁻¹)’(x)=1/(f’(f⁻¹(x))
32
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(d/dx)\[eⁿ\]
eⁿ+c
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(d/dx)\[aⁿ\]
aⁿ(ln a) n’
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(d/dx)ln u
u’/u
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(d/dx) logₙu
u’/(u (ln n))
36
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∫eⁿdn
eⁿ+C
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∫aⁿdn
aⁿ/lna +C
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∫(1/u)du
ln |u| +C
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(d/dx)\[arcsin u\]
u’/√(1-u²)
40
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(d/dx) \[arctan u\]
u’/(1+u²)
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(d/dx)\[arcsecu\]
u’/(u√(u²-1))
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(d/dx)\[arccosu\]
\-u’/√(1-u²)
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(d/dx)\[arccotu\]
\-u’/(1+u²)
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(d/dx) \[arccsc u\]
\-u’/(u√(u²-1))
45
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∫(1/√(a²-u²))du
arcsin (u/a) +C
46
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∫(1/(a²+u²))du
(1/a) arctan (u/a) +C
47
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exponential growth/decay
if dy/dt=ky, then y=c(e^kt)
48
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logistics equation
if dy/dt =ky(1-(y/L)), then y=L/(1+C(e^-kt))
49
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area between two curves
∫(a to b) (top-bottom)dx OR ∫(a to b) (right-left)dy
50
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volume disk method
v=π(∫(a to b) R²dx
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Volume washer method
v=π(∫(a to b) (R²-r²)dx
52
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Volume shell method
v=2π(∫(a to b) (ph)dx
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solids of known cross-section
v=(∫(a to b) A(x)dx
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arc length
L=∫(a to b) √(1+\[f’(x)\]²)dx
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Surface Area
S=∫(a to b) 2πr√(1+\[f’(x)\]²)dx
56
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integration by parts
∫u dv = uv-∫v du
57
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Taylor Series
f(x) = f(c)+f’(c)(x-c)+f’’(c)(x-c)²/2!+…+fⁿ(c)(n-c)ⁿ/n!
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Taylor series for e^x (centered at 0)
1+x+x²/2!+x³/3!+…+xⁿ/n! (for all real numbers)
59
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Taylor series for sin x (centered at 0)
x-x³/3!+x⁵/5!-x⁷/7!+…+(-1)ⁿx²ⁿ⁺¹/(2n+1)! (for all real numbers)
60
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taylor series for cos x (centered at 0)
1-x²/2!+x⁴/4!-x⁶/6!+…+(-1)ⁿx²ⁿ/(2n)! (for all real numbers)
61
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power series for 1/(1-x)
1+x+x²+x³+x⁴+…+xⁿ (for -1
62
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Alternating series error
error≤|Aₙ₊₁|
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Lagrange error
error≤|fⁿ⁺¹(max)|/(n+1)!×(x-c)ⁿ
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parametric equation slope
dy/dx= (dy/dt)/(dx/dt)
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Parametric 2nd derivative
d²y/dx²=(d/dt)\[dy/dx\]/(dx/dt)
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parametric speed
√\[(dx/dt)²+(dy/dt)²\]
67
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parametric arc length (aka total distance
L=∫(a to b) √\[(dx/dt)²+(dy/dt)²\] dt
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polar area
A=½∫(a to b) r²dθ
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Polar parametrics
x=r cosθ and y=r sinθ
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