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AP Calculus AB Review
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Gravity
Key Concepts:
Terms in this set (73)
Limit Definition of Derivative
limit (as h approaches 0)= F(x+h)-F(x)/h
Alternate Definition of Derivative
limit (as x approaches a number c)=
f(x)-f(c)/x-c x≠c
limit as x approaches 0: sinx/x
1
limit as x approaches 0:
1-cosx/x
0
Continuity Rule
If the limit exists (aka left limit and right limit are equal), and the limit equals the function at that point.
Basic Derivative
f(x^n)= nX^(n-1)
d/dx(sinx)
cosx
d/dx(cosx)
-sinx
d/dx(tanx)
sec²x
d/dx(cotx)
-csc²x
d/dx(secx)
secxtanx
d/dx(cscx)
-cscxcotx
d/dx(lnu)
u'/u
d/dx(e^u)
e^u(u')
d/dx(a^u)
a^u(lna)(u')
Chain rule of f(x)^n
nf(x)f'(x)
Product rule of f(x)g(x)
f'(x)g(x)+g'(x)f(x)
Quotient rule of f(x)/g(x)
g(x)f'(x)-f(x)g'(x)/g(x)²
Intermediate Value Theorem
if f(x) is continuous on [a,b], then there will be a point x=c that lies in between [a,b]
Extreme Value Theorem
if f(x) is continuous on [a,b], then f(x) has an absolute max or min on the interval
Rolle's Theorem
if f(x) is continuous on [a,b] and differentiable on (a,b), and if f(a)=f(b), then there is at least one point (x=c) on (a,b) [DON'T INCLUDE END POINTS] where f'(c)=0
Mean Value Theorem
if f(x) is continuous on [a,b] and differentiable on (a,b), there is at least one point (x=c) where f'(c)= F(b)-F(a)/b-a
If f'(x)=0
there is a max or min on f(x) [number line test]
If f'(x)>0
f(x) is increasing
If f'(x)<0
f(x) is decreasing
If f''(x)=0
f(x) has a point of inflection & f'(x) has a max or min
If f''(x)>0
f(x) is concave up & f'(x) is increasing
If f''(x)<0
f(x) is concave down & f'(x) is decreasing
p(t), x(t), s(t)
means position function
p'(t)
v(t)= velocity
p''(t) or v'(t)
a(t)= acceleration
v(t)=0
p(t) is at rest or changing direction
v(t)>0
p(t) is moving right
v(t)<0
p(t) is moving left
a(t)=0
v(t) not changing
a(t)>0
v(t) increasing
a(t)<0
v(t) decreasing
v(t) and a(t) has same signs
speed of particle increasing
v(t) and a(t) has different signs
speed of particle decreasing
∫(x^n)dx
x^(n+1)∕(n+1) +C
∫(1/x)dx
ln|x|+C
∫(e^kx)dx
ekx/k +C
∫sinx dx
-cosx+C
∫cosx dx
sinx+C
∫sec²x dx
tanx+C
∫csc²x dx
-cotx+C
∫secxtanx dx
secx+C
∫cscxcotx
-cscx+C
∫k dx [k IS A CONSTANT]
kx+C
1st fundamental theorem of calculus
(bounded by a to b) ∫f(x)dx= F(b)-F(a)
2nd fundamental theorem
(bounded by 1 to x)
d/dx[∫f(t)dt]= f(x)(x')
average value
(1/(b-a))[∫f(x)dx] [BOUNDED BY A TO B]
Area between curves
A=∫f(x)-g(x) dx
Volume (DISK)
V=π∫f(x)²dx
Volume (WASHER)
V=π∫f(x)²-g(x)²dx
∫f(x)dx [BOUNDS ARE SAME]
0
Displacement of particle
∫v(t)dt
total distance of particle
∫|v(t)|dt
position of particle at specific point
p(x)= initial condition + ∫v(t)dt (bounds are initial condition and p(x))
derivative of exponential growth equation:
P(t)=Pe^kt
dP/dt=kP
Cross section for volume: square [A=s²]
v=∫[f(x)-g(x)]²dx
Cross section for volume:
isosceles triangle [A=1/2s²]
v= 1/2∫[f(x)-g(x)]²dx
Cross section for volume:
equilateral triangle [A=√3/4s²]
v= √3/4∫[f(x)-g(x)]²dx
Cross section for volume:
semicircle [A=1/2πs²]
v= 1/2π∫[f(x)-g(x)]²dx
d/dx(sin⁻¹u)
u'/√(1-u²)
d/dx(cos⁻¹u)
-u'/√(1-u²)
d/dx(tan⁻¹u)
u'/(1+u²)
d/dx(cot⁻¹u)
-u'/(1+u²)
d/dx(sec⁻¹u)
u'/|u|√(u²-1)
d/dx(csc⁻¹u)
u'/|u|√(u²-1)
∫du/√(a²-u²)
(sin⁻¹u/a)+C
∫du/(a²+u²)
(1/a)(tan⁻¹u/a)+C
∫du/|u|√(u²-a²)
(1/a)(sec⁻¹u/a)+C
THIS SET IS OFTEN IN FOLDERS WITH...
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AP Calculus AB Exam
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