Section 1
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log(A / B)
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Mar 1, 2020
Cards (233)
Section 1
(50 cards)
log(A / B)
log(AB)
First derivative test for a local max of f at x = a
tan⁻¹(-1)
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))
Rolle's Theorem
d/dx ( sin⁻¹ ( x ) )
ƒ''(x) > 0 or ƒ'(x) is increasing
Test for max and mins of f on [a, b]
sin(π/4)
lim x→₀ sin(x)/x
lim x→∞ (ax^n+...)/(bx^m+...)
Intermediate Value Theorem (IVT)
Inflection Points
Average rate of change of ƒ(x) over [a, b]
Instantaneous rate of change of ƒ(a)
Three types of discontinuities.
lim x→∞ tan⁻¹(x)
Three conditions where ƒ(x) is not differentiable
ƒ'(x) > 0
sin⁻¹(-1)
Second derivative test for a local max of f at x = a
First derivative test for a local min of f at x = a
e^(ln(x))
1+cot²(θ)
log(A ^ x)
ƒ(x) is continuous at x = a if...
d/dx ( tan⁻¹ ( x ) )
Mean Value Theorem (MVT)
Definition of the Derivative (Using the limit as h→0)
cos(-θ)
Alternate Definition of the Derivative (Using the limit as x→a)
ln(x) / ln(a)
SOH CAH TOA
ƒ(x) is differentiable at x = a if...
cos(2θ)
sin(-θ)
Squeeze Theorem
Extreme Value Theorem
ƒ''(x) < 0 or ƒ'(x) is decreasing
1+tan²(θ)
To evaluate a limit of the type: lim x→∞ x^2 sin(1/x) use:
Double Angle Formula for sin²(θ)
sin(0)=
ƒ'(x) < 0
Critical Points
Second derivative test for a local min of f at x = a
Double Angle Formula for cos²(θ)
sin(2θ)
Section 2
(50 cards)
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
Graph of y = ln x
Velocity of a point moving along a line with position at time t given by d(t)
To find the limits of indeterminate forms: 0 ^ 0, 1 ^ ∞, ∞ ^ 0
Average acceleration given v over [a, b]
d/dx ( csc x )
Graph of y = sin x
Average speed of s over [a, b]
Graph of y = √(1 - x²)
Graph of y = 1/x
d/dx ( a ^ x )
d/dx (ƒ(x)³)
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?
Speed of a point moving along a line
d/dx ( ln x )
The total change in ƒ(x) over [a, b] in terms of the rate of change, ƒ'(x)
d/dx (e ^ ƒ(x) )
Graph of x²/a² + y²/b² = 1
d/dx ( cos x )
A normal line to a curve is...
d/dx ( sin x )
Graph of y = tan⁻¹ x
d/dx ( ln ƒ(x) )
If ƒ(x) is decreasing, then a left Riemann sum ...
Graph of y = e ^ (kx)
Graph of y = cos x
Equation of the tangent line to y = ƒ(x) at x = a
Acceleration of a point moving along a line with position at time t given by d(t)
If ƒ(x) is increasing, then a right Riemann sum ...
Graph of y = tan x
Quotient Rule
Position at time t = b of a particle moving along a line given velocity v(t) and position s(t) at time t = a
d/dx ( log (base a) x )
If ƒ(x) is increasing, then a left Riemann sum ...
An object in motion reverses direction when...
To find the limits of indeterminate forms: ∞ × 0
An object in motion is at rest when...
L'Hopital's Rule
d/dx ( cos⁻¹ ( x ) )
Chain Rule
Average velocity of s over [a, b]
Total distance traveled by a particle moving along a line with velocity v(t) for a ≤ t ≤ b
...
Product Rule
d/dx ( tan x )
d/dx ( e ^ x )
d/dx ( cot x )
d/dx ( sec x )
Displacement of a particle moving along a line with velocity v(t) for a ≤ t ≤ b.
Section 3
(50 cards)
The Fundamental Theorem of Calculus (Part I)
Volume of a sphere
If ƒ(x) is decreasing, then a right Riemann sum ...
∫ 1/x dx =
Swapping the bounds of an integral
Volume of a pyramid
lim n→∞ (1 + 1/n) ^ n
∫ -1 / √(1 - x² ) dx =
To find the area between 2 curves using horizontal rectangles (dy)
Volume of a cylinder
Steps to solve a differential equation
If F(x) = ∫ (from h(x) to g(x)) ƒ(t) dt then F'(x) =
Exponential Growth Solution of dy/dt = kP P(0) = P₀
To find the area between 2 curves using vertical rectangles (dx)
Volume of solid if cross sections perpendicular to the x-axis are isosceles right triangles
If F(x) = ∫ (from a to g(x)) ƒ(t) dt then F'(x) =
Volume of solid if cross sections perpendicular to the x-axis are equilateral triangles
∫ sec² x dx =
The Fundamental Theorem of Calculus (Part II)
Volume of a washer; rotated about a horizontal line
If ƒ(x) is concave down then the linear approximation...
Area of a trapezoid
∫ a ^ x dx =
Surface Area of a cylinder
Volume of a prism
∫ cos x dx =
∫ 1 / √(1 - x² ) dx =
Area of an Equilateral Triangle
Volume of a disc; rotated about a horizontal line
Volume of solid if cross sections perpendicular to the x-axis are squares
∫ tan x dx =
If ƒ(x) is concave up, then the trapezoidal approximation of the integral...
Volume of a washer; rotated about a vertical line
If ƒ(x) is concave down, then the trapezoidal approximation of the integral...
Volume of solid if cross sections perpendicular to the x-axis are semicircles
Surface Area of a sphere
If F(x) = ∫ (from x to a) ƒ(t) dt then F'(x) =
Volume of a disc; rotated about a vertical line
Adding adjacent integrals
If ƒ(x) is concave up then the linear approximation...
Area of a Sector (in radians)
∫ sin x dx =
Integral equation for a horizontal shift of 1 unit to the right.
∫ x ^ n dx =
If ƒ(x) is concave down, then a midpoint Riemann sum...
∫ 1 / (x² + 1) dx =
If ƒ(x) is concave up, then a midpoint Riemann sum...
Volume of a cone
∫ e ^ x dx =
The average value of f from x = a to x = b (Mean Value Theorem for Integrals)
Section 4
(50 cards)
∫ lnx dx = ?
Horizontal Tangent of a parametric curve
WORK formula
Magnitude of a vector in terms of the x and y components
dy/dθ < 0
Differential equation for exponential growth dP/dt = ?
Solution of a differential equation for exponential growth
Second Derivative of a parametric curve
Arc Length of a graph defined parametrically with a ≤ t ≤ b x = x(t) and y = y(t)
Convert from polar (r,θ) to rectangular (x,y)
Double Angle Formula for sin²θ
Improper Integral: ∫ f(x) dx bounds: [0,∞]
dr/dθ > 0
Speed of a particle moving in the plane x = x(t) and y = y(t)
Solution of a differential equation for decay
Graphs of: r = sin(k θ) r = cos(k θ) (k is a constant)
Distance traveled (Arc Length) by a particle moving in the plane with a ≤ t ≤ b x = x(t) and y = y(t)
Graph of: r = 1 + 2 cos(θ)
Horizontal Asymptotes of a parametric curve
Arc length of a polar graph r 0 ≤ θ ≤ π
Position at time t = b of a particle moving in the plane given x(a), y(a), x′(t), and y′(t).
dx/dθ < 0
Arc length of a function f(x) from x = a to x = b
Graph of: r = 1 + cos(θ)
Double Angle Formula for cos²θ
dy/dθ > 0
Slope of polar graph r (θ)
Vertical Tangent of a parametric curve
Graph of r = θ
Improper Integral: ∫ 1/x² dx bounds: [0,1]
Velocity vector of a particle moving in the plane x = x(t) and y = y(t)
Graph of θ = c (c is a constant)
Differential equation for decay dP/dt = ?
Curve of: x = cos t y = sin t 0 ≤ t ≤ 2π
Vertical Asymptotes of a parametric curve
Figure out what values of t make y(t) go to positive and negative infinity and then compute x(t) for those values.
Double Angle Formula for sin²θ
Slope of a parametric curve x = x(t) and y = y(t)
Graphs of: r = c r = c sin(θ) r = c cos(θ) (c is a constant)
dx/dθ > 0
lim n→∞ (1 + 1/n)^n = ?
Convert from rectangular (x,y) to polar (r,θ)
Direction angle (θ) of a vector in terms of the x and y components
Double Angle Formula for cos²θ
Curve of: x = a cos t y = b sin t 0 ≤ t ≤ 2π
Area enclosed by r = f(θ), α ≤ θ ≤ β
Acceleration vector of a particle moving in the plane x = x(t) and y = y(t)
dr/dθ < 0
Integration by Parts Formula
Horizontal Tangent of a Polar Graph
Vertical Tangent of a Polar Graph
Section 5
(33 cards)