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BC Calculus AP Review
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Gravity
Terms in this set (204)
Double Angle Formula for cos²(θ)
Double Angle Formula for sin²(θ)
sin(0)=
sin(π/4)
sin⁻¹(-1)
tan⁻¹(-1)
1+cot²(θ)
1+tan²(θ)
sin(2θ)
cos(2θ)
log(AB)
log(A / B)
log(A) ^ x
e^(ln(x))
ln(x) / ln(a)
Simplify the expression into one log:
2 ln(x) + ln(x+1) - ln(x-1)
For what value of x is there a
hole, and for what value of x
is there a vertical asymptote?
f(x) = ((x - a)(x - b))/ ((x - a)(x - c))
Definition of the Derivative
(Using the limit as h→0)
lim x→₀ sin(x)/x
lim x→∞ tan⁻¹(x)
First derivative test for a local max of f at x = a
First derivative test for a local min of f at x = a
Second derivative test for a local max of f at x = a
Second derivative test for a local min of f at x = a
Test for max and mins of f on [a, b]
Inflection Points
ƒ'(x) < 0
ƒ''(x) < 0 or ƒ'(x) is decreasing
ƒ'(x) > 0
ƒ''(x) > 0 or ƒ'(x) is increasing
Intermediate Value Theorem (IVT)
Mean Value Theorem (MVT)
Rolle's Theorem
Squeeze Theorem
ƒ(x) is continuous at x = a if...
Extreme Value Theorem
Critical Points
Three types of discontinuities.
ƒ(x) is differentiable at x = a if...
Three conditions where ƒ(x) is not differentiable
Average rate of change of ƒ(x) over [a, b]
Instantaneous rate of change of ƒ(a)
d/dx ( tan⁻¹ ( x ) )
d/dx ( sin⁻¹ ( x ) )
d/dx ( e ^ x )
d/dx ( ln x )
d/dx ( a ^ x )
d/dx ( sin x )
d/dx ( cos x )
d/dx ( tan x )
d/dx ( sec x )
d/dx ( csc x )
d/dx ( cot x )
Product Rule
Quotient Rule
Chain Rule
d/dx (ƒ(x)³)
d/dx ( ln ƒ(x) )
d/dx (e ^ ƒ(x) )
Derivative of the Inverse of ƒ(x)
Implicit Differentiation
Find dy/dx:
x²/9+y²/4=1
Equation of a line in point-slope form
Equation of the tangent line
to y = ƒ(x)
at x = a
A normal line to a curve is...
Velocity of a point moving along a line with position at time t given by d(t)
Speed of a point moving along a line
Average velocity
of s over [a, b]
Average speed
of s over [a, b]
Average acceleration
given v over [a, b]
An object in motion is at rest when...
An object in motion reverses direction when...
Acceleration of a point moving along a line with position at time t given by d(t)
How to tell if a point moving along the x-axis with velocity v(t) is speeding up or slowing down at some time t?
Position at time t = b of a particle moving along a line given velocity v(t) and position s(t) at time t = a
Displacement of a particle moving along a line with velocity v(t) for a ≤ t ≤ b.
Total distance traveled by a particle moving along a line with velocity v(t) for a ≤ t ≤ b
...
The total change in ƒ(x) over [a, b] in terms of the rate of change, ƒ'(x)
Graph of y = 1/x
Graph of y = e ^ (kx)
Graph of y = ln x
Graph of y = sin x
Graph of y = cos x
Graph of y = tan x
Graph of y = tan⁻¹ x
Graph of y = √(1 - x²)
Graph of x²/a² + y²/b² = 1
L'Hopital's Rule
To find the limits of
indeterminate forms:
∞ × 0
To find the limits of
indeterminate forms:
0 ^ 0, 1 ^ ∞, ∞ ^ 0
If ƒ(x) is increasing, then a left Riemann sum ...
If ƒ(x) is decreasing, then a left Riemann sum ...
If ƒ(x) is increasing, then a right Riemann sum ...
If ƒ(x) is decreasing, then a right Riemann sum ...
If ƒ(x) is concave up, then the trapezoidal approximation of the integral...
If ƒ(x) is concave down, then the trapezoidal approximation of the integral...
If ƒ(x) is concave up, then a midpoint Riemann sum...
If ƒ(x) is concave down, then a midpoint Riemann sum...
Area of a trapezoid
If ƒ(x) is concave down then the linear approximation...
If ƒ(x) is concave up then the linear approximation...
The Fundamental Theorem of Calculus (Part I)
The Fundamental Theorem of Calculus (Part II)
∫ x ^ n dx =
∫ e ^ x dx =
∫ 1/x dx =
∫ sin x dx =
∫ cos x dx =
∫ sec² x dx =
∫ a ^ x dx =
∫ tan x dx =
∫ 1 / (x² + 1) dx =
∫ 1 / √(1 - x² ) dx =
The average value of f from x = a to x = b
(Mean Value Theorem for Integrals)
Integral equation for a horizontal shift of 1 unit to the right.
Adding adjacent integrals
Swapping the bounds of an integral
Exponential Growth
Solution of
dy/dt = kP
P(0) = P₀
lim n→∞ (1 + 1/n) ^ n
Steps to solve a differential equation
To find the area between 2 curves using vertical rectangles (dx)
To find the area between 2 curves using horizontal rectangles (dy)
Volume of a disc; rotated about a horizontal line
Volume of a washer; rotated about a horizontal line
Volume of a disc; rotated about a vertical line
Volume of a washer; rotated about a vertical line
Volume of solid if cross sections perpendicular to the
x-axis are squares
Volume of solid if cross sections perpendicular to the
x-axis are isosceles right triangles
Volume of solid if cross sections perpendicular to the
x-axis are equilateral triangles
Volume of solid if cross sections perpendicular to the
x-axis are semicircles
Volume of a prism
Volume of a cylinder
Volume of a pyramid
Volume of a cone
Volume of a sphere
Surface Area of a cylinder
Surface Area of a sphere
Area of a Sector
(in radians)
Slope of a parametric curve
x = x(t) and y = y(t)
Horizontal Tangent
of a parametric curve
Vertical Tangent
of a parametric curve
Second Derivative
of a parametric curve
Velocity vector of a particle moving in the plane x = x(t) and y = y(t)
Acceleration vector of a particle moving in the plane
x = x(t) and y = y(t)
Speed of a particle moving in the plane
x = x(t) and y = y(t)
Distance traveled (Arc Length) by a particle moving in the plane with a ≤ t ≤ b x = x(t) and y = y(t)
Position at time t = b of a particle moving in the plane given x(a), y(a), x′(t), and y′(t).
Magnitude of a vector in terms of the x and y components
Graph of
θ = c
(c is a constant)
Graph of
r = θ
Graphs of:
r = c
r = c sin(θ)
r = c cos(θ)
(c is a constant)
Graphs of:
r = sin(k θ)
r = cos(k θ)
(k is a constant)
Graph of:
r = 1 + cos(θ)
Graph of:
r = 1 + 2 cos(θ)
Slope of polar graph r (θ)
Area enclosed by r = f(θ),
α ≤ θ ≤ β
Double Angle Formula for cos²θ
Double Angle Formula for sin²θ
dx/dθ < 0
dx/dθ > 0
dy/dθ < 0
dy/dθ > 0
Convert from polar (r,θ) to rectangular (x,y)
Convert from rectangular (x,y) to polar (r,θ)
Horizontal Tangent of a Polar Graph
Vertical Tangent of a Polar Graph
Integration by Parts Formula
∫ lnx dx = ?
Improper Integral:
∫ 1/x² dx
bounds: [0,1]
Improper Integral:
∫ f(x) dx
bounds: [0,∞]
Arc length
of a function f(x) from
x = a to x = b
Arc length
of a polar graph r
0 ≤ θ ≤ π
Arc Length
of a graph defined parametrically with
a ≤ t ≤ b
x = x(t) and y = y(t)
Differential equation for exponential growth
dP/dt = ?
Solution of a differential equation for exponential growth
Differential equation for decay
dP/dt = ?
Solution of a differential equation for decay
Logistic differential equation
dP/dt = ?
Solution of a logistic differential equation
Graph of a Logistic Function
(include inflection pt.)
Euler's Method for solving
y' = F (x,y)
with initial point (x₀ , y₀)
Power Series for
f(x) = 1 / (1 - x)
(include IOC)
Power Series for
f(x) = tan⁻¹ x
(include IOC)
Power Series for
f(x) = ln (1 + x)
(include IOC)
Taylor Series for f(x) about x = 0
(Maclaurin Series)
Taylor Series for f(x) about x = c
Maclaurin Series for
f (x) = e∧x
(include IOC)
Maclaurin Series for
f (x) = sin x
(include IOC)
Maclaurin Series for
f (x) = cos x
(include IOC)
Error for the partial sum, Sn, of an infinite series S
Error bound of an alternating series
Lagrange error bound
Geometric sequence
(def. and conv. property)
Harmonic Series
(def. and conv. property)
p-series
(def. and conv. property)
Divergence Test
If lim n→∞ a(sub n) = 0,
then ∑ a(sub n) for n from 1 to ∞ ...
Integral Test
Alternating Series Test
Direct Comparison Test
Limit Comparison Test
Ratio Test
n-th Root Test
Interval of Convergence (IOC)
Radius of Convergence
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