Basic Derivative of x to the power of n
๐/๐๐ฅ (๐ฅ^๐) = ๐๐ฅ^(๐โ1)
Basic Derivative of sin x
๐/๐๐ฅ (sin ๐ฅ) = cos ๐ฅ
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Flashcards of key calculus concepts and theorems.
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Basic Derivative of x to the power of n
๐/๐๐ฅ (๐ฅ^๐) = ๐๐ฅ^(๐โ1)
Basic Derivative of sin x
๐/๐๐ฅ (sin ๐ฅ) = cos ๐ฅ
Basic Derivative of cos x
๐/๐๐ฅ (cos ๐ฅ) = โ sin ๐ฅ
Basic Derivative of tan x
๐/๐๐ฅ (tan ๐ฅ) = secยฒ ๐ฅ
Basic Derivative of cot x
๐/๐๐ฅ (cot ๐ฅ) = โ cscยฒ ๐ฅ
Basic Derivative of sec x
๐/๐๐ฅ (sec ๐ฅ) = sec ๐ฅ tan ๐ฅ
Basic Derivative of csc x
๐/๐๐ฅ (csc ๐ฅ) = โ csc ๐ฅ cot ๐ฅ
Basic Derivative of ln u
๐/๐๐ฅ (ln ๐ข) = (1/๐ข) * du/dx
Basic Derivative of e to the power of u
๐/๐๐ฅ (๐^๐ข) = ๐^๐ข * du/dx
Chain Rule
๐/๐๐ฅ [๐(๐ข)] = ๐โฒ(๐ข) * du/dx
Product Rule
๐/๐๐ฅ (๐ข๐ฃ) = ๐ข * dv/dx + ๐ฃ * du/dx
Quotient Rule
๐/๐๐ฅ (๐ข/๐ฃ) = (๐ฃ * du/dx - ๐ข * dv/dx) / ๐ฃยฒ
Intermediate Value Theorem
If ๐(๐ฅ) is continuous on [๐, ๐], and y is a number between ๐(๐) and ๐(๐), then there exists at least one number ๐ฅ = ๐ in (๐, ๐) such that ๐(๐) = ๐ฆ
Mean Value Theorem
If ๐(๐ฅ) is continuous on [๐, ๐], AND the first derivative exists on the interval (๐, ๐) then there is at least one number ๐ฅ = ๐ in (๐, ๐) such that ๐โฒ(๐) = (๐(๐)โ๐(๐))/(๐โ๐)
Rolleโs Theorem
If ๐(๐ฅ) is continuous on [๐, ๐], AND the first derivative exists on the interval (๐, ๐) AND ๐(๐) = ๐(๐), then there is at least one number ๐ฅ = ๐ in (๐, ๐) such that ๐โฒ(๐) = 0
Extreme Value Theorem
If ๐(๐ฅ) is continuous on [๐, ๐], then the function is guaranteed to have an absolute maximum and an absolute minimum on the interval.
Alternate Definition of the Derivative
๐โฒ(๐) = lim ๐ฅโ๐ (๐(๐ฅ) โ ๐(๐)) / (๐ฅ โ ๐)
Critical Point
Point where dy/dx = 0 or is undefined.
Local Minimum
๐๐ฆ/๐๐ฅ goes from (โ, 0, +) or (โ, undefined, +) OR ๐ยฒ๐ฆ/๐๐ฅยฒ > 0
Local Maximum
๐๐ฆ/๐๐ฅ goes (+, 0, โ) or (+, undefined, โ) OR ๐ยฒ๐ฆ/๐๐ฅยฒ < 0
Point of Inflection
Concavity changes; ๐ยฒ๐ฆ/๐๐ฅยฒ goes from (+, 0, โ), (โ, 0, +), (+, undefined, โ), OR (โ, undefined, +)
Derivative of an Inverse Function
๐โฒ(๐ฅ) = 1 / ๐โฒ(๐(๐ฅ))
Implicit Differentiation
Remember that in implicit differentiation you will have a ๐๐ฆ/๐๐ฅ for each y in the original function or equation. Isolate the ๐๐ฆ/๐๐ฅ. If you are taking the second derivative ๐ยฒ๐ฆ/๐๐ฅยฒ, you will often substitute the expression you found for the first derivative somewhere in the process.
Average Rate of Change (ARoC)
๐_sec = (๐(๐) โ ๐(๐)) / (๐ โ ๐)
Instantaneous Rate of Change (IRoC)
๐_tan = ๐โฒ(๐ฅ) = lim โโ0 (๐(๐ฅ + โ) โ ๐(๐ฅ)) / โ
Equation of a Tangent Line
y - y1 = m(x - x1)
๐โฒ(๐ฅ) > 0
Function is increasing
๐โฒ(๐ฅ) < 0
Function is decreasing
๐โฒ(๐ฅ) = 0 or DNE
Critical Values at x.
Relative Maximum
๐โฒ(๐ฅ) = 0 or DNE and sign of ๐โฒ(๐ฅ) changes from + to โ
Relative Minimum
๐โฒ(๐ฅ) = 0 or DNE and sign of ๐โฒ(๐ฅ) changes from โ to +
Absolute Max or Min
Check Endpoints Also. The maximum value is a y-value.
๐โฒโฒ(๐ฅ) > 0
Function is concave up.
๐โฒโฒ(๐ฅ) < 0
Function is concave down.
Point of Inflection Second Derivative
๐โฒ(๐ฅ) = 0 and sign of ๐โฒโฒ(๐ฅ) changes, then there is a point of inflection at x.
Relative Maximum (Second derivative)
๐โฒโฒ(๐ฅ) < 0
Relative Minimum (Second derivative)
๐โฒโฒ(๐ฅ) > 0
Horizontal Asymptotes: Largest exponent in numerator < largest exponent in denominator
lim ๐ฅโยฑโ ๐(๐ฅ) = 0
Horizontal Asymptotes: Largest exponent in numerator > largest exponent in denominator
lim ๐ฅโยฑโ ๐(๐ฅ) = DNE
Horizontal Asymptotes: Largest exponent in numerator = largest exponent in denominator
lim ๐ฅโยฑโ ๐(๐ฅ) = a/b (quotient of leading coefficients)
The Fundamental Theorem of Calculus
โซ ๐(๐ฅ)๐๐ฅ = ๐น(๐) โ ๐น(๐) where ๐นโฒ(๐ฅ) = ๐(๐ฅ)
Corollary to FTC
๐/๐๐ฅ โซ ๐(๐ก)๐๐ก from a to g(x) = ๐(๐(๐ฅ)) * ๐โฒ(๐ฅ)
๐ฅ(๐ก)
Position Function
๐ฃ(๐ก)
Velocity Function
๐(๐ก)
Acceleration Function
Relationship between Position, Velocity, and Acceleration (Derivatives)
derivative of position (ft) is velocity (ft/sec); the derivative of velocity (ft/sec) is acceleration (ft/sec2)
Relationship between Position, Velocity, and Acceleration (Integrals)
The integral of acceleration (ft/sec2) is velocity (ft/sec) ; the integral of velocity (ft/sec) is position (ft).
Speed
| velocity |
If acceleration and velocity have the same sign
Then the speed is increasing
If the acceleration and velocity have different signs
Then the speed is decreasing.
The particle is moving right
When velocity is positive
the particle is moving left
When velocity is negative
Displacement
โซ ๐ฃ(๐ก) ๐๐ก from ๐ก0 to ๐ก๐
Total Distance
โซ | ๐ฃ(๐ก)| ๐๐ก from ๐ก0 to ๐ก๐
Average Velocity
(final position โ initial position) / (total time) = ฮ๐ฅ / ฮ๐ก
Accumulation
๐ฅ(0) + โซ ๐ฃ(๐ก) ๐๐ก from ๐ก0 to ๐ก๐
Four things on a calculator that need no work shown
Graphing a function within an arbitrary view window, Finding the zeros of a function, Computing the derivative of a function numerically, Computing the definite integral of a function numerically.
Definition of Logarithms
๐๐ ๐ = ๐ โ ๐^๐ = ๐
Logarithm Product Rule
๐๐ ๐๐ = ln ๐ + ln ๐
Logarithm Quotient Rule
ln (๐/๐) = ln ๐ โ ln ๐
Logarithm Power Rule
๐ โ ln ๐ = ln ๐^๐
Exponential Growth and Decay Formula
๐ฆ = ๐ถ๐^(๐๐ก)
โthe rate of change of y is proportional to yโ
๐ฆโฒ = ๐๐ฆ
When solving a differential equation
Separate variables, Integrate, Add +C to one side, Use initial conditions to find โCโ, Write the equation if the form of ๐ฆ = ๐(๐ฅ)
Mean Value Theorem for Integrals: The Average Value
๐_avg = (1 / (๐ โ ๐)) โซ ๐(๐ฅ) ๐๐ฅ from a to b
Riemann Sums
A rectangular approximation. Do NOT EVALUATE THE INTEGRAL; you add up the areas of the rectangles.
Area of a Trapezoid
๐ด_๐ = (1/2) โ[๐1 + ๐2]
Trapezoidal Rule (for even intervals)
โซ ๐(๐ฅ) ๐๐ฅ from a to b = (๐ โ ๐) / (2๐) [๐ฆ0 + 2๐ฆ1 + 2๐ฆ2 + โฆ +2๐ฆ๐โ1 + ๐ฆ๐]
sin(โ๐ฅ) (ODD)
sin(โ๐ฅ) = โ sin ๐ฅ
cos(โ๐ฅ) (EVEN)
cos(โ๐ฅ) = cos ๐ฅ
Pythagorean Identities
๐ ๐ ๐ ๐ ๐๐ยฒ๐ + ๐๐๐๐๐ ๐ ยฒ๐ = 1
Double Angle Formulas (sin 2x)
sin 2๐ฅ = 2 sin ๐ฅ cos ๐ฅ
Double Angle Formula (cos 2x)
cos 2๐ฅ = cosยฒ๐ฅ โ sinยฒ๐ฅ = 1 โ 2 sinยฒ๐ฅ
Power-Reducing Formulas (cosยฒx)
cosยฒ๐ฅ = (1/2) (1 + cos 2๐ฅ)
Power-Reducing Formulas (sinยฒx)
sinยฒ๐ฅ = (1/2) (1 โ cos 2๐ฅ)
Area Between Two Curves
๐lices โฅ to x-axis: ๐ด = โซ [๐(๐ฅ) โ ๐(๐ฅ)] ๐๐ฅ from a to b. Slices โฅ to y-axis: ๐ด = โซ [๐(๐ฆ) โ ๐(๐ฆ)] ๐๐ฆ from c to d
Volume By Disk Method
About x-axis: ๐ = ๐ โซ [๐ (๐ฅ)]ยฒ ๐๐ฅ from a to b. About y-axis: ๐ = ๐ โซ [๐ (๐ฆ)]ยฒ ๐๐ฆ from c to d
Volume By Washer Method
About x-axis: ๐ = ๐ โซ ([๐ (๐ฅ)]ยฒ โ [๐(๐ฅ)]ยฒ) ๐๐ฅ from a to b. About y-axis: ๐ = ๐ โซ ([๐ (๐ฆ)]ยฒ โ [๐(๐ฆ)]ยฒ) ๐๐ฆ from c to d
Volume By Shell Method
About x-axis: ๐ = 2 ๐ โซ ๐ฆ [๐ (๐ฆ)] ๐๐ฆ from c to d. About y-axis: ๐ = 2 ๐ โซ ๐ฅ [๐ (๐ฅ)] ๐๐ฅ from a to b
General Equations for Known Cross Section
SQUARES: ๐ = โซ (๐๐๐ ๐)ยฒ ๐๐ฅ from a to b. TRIANGLES EQUILATERAL: ๐ = (โ3 / 4) โซ (๐๐๐ ๐)ยฒ ๐๐ฅ from a to b. ISOSCELES RIGHT: ๐ = (1 / 4) โซ (๐๐๐ ๐)ยฒ ๐๐ฅ from a to b. RECTANGLES: ๐ = โซ (๐๐๐ ๐) โ โ ๐๐ฅ from a to b where h is the height of the rectangles. SEMI-CIRCLES: ๐ = โซ (๐๐๐๐๐ข๐ )ยฒ ๐๐ฅ from a to b where radius is ยฝ distance between the two curves.
โซ ๐๐ข / ๐ข
ln| ๐ข | + ๐ถ
โซ ๐ข^๐ ๐๐ข
๐ข^(๐+1) / (๐ + 1) + ๐ถ (๐ โ โ1)
โซ ๐^๐ข ๐๐ข
๐^๐ข + ๐ถ
โซ ๐^๐ข ๐๐ข
๐^๐ข / ln ๐ + ๐ถ
โซ sin ๐ข ๐๐ข
โ cos ๐ข + ๐ถ
โซ cos ๐ข ๐๐ข
sin ๐ข + ๐ถ
โซ tan ๐ข ๐๐ข
โ ln|cos ๐ข + ๐ถ|
โซ cot ๐ข ๐๐ข
ln|sin ๐ข| + ๐ถ
โซ sec ๐ข ๐๐ข
ln|sec ๐ข + tan ๐ข| + ๐ถ
โซ csc ๐ข ๐๐ข
โ ln|csc ๐ข + cot ๐ข| + ๐ถ
โซ secยฒ๐ข ๐๐๐๐
tan ๐ข + ๐ถ
โซ cscยฒ๐ข ๐๐๐๐
โ cot ๐ข + ๐ถ
โซ sec ๐ข tan ๐ข ๐๐๐๐
sec ๐ข + ๐ถ
โซ csc ๐ข cot ๐ข ๐๐๐๐
โ csc ๐ข + ๐ถ
๐/๐๐ฅ (sinโปยน(๐ข/๐))
1 / โ(๐ยฒโ ๐ขยฒ) * du/dx
๐/๐๐ฅ [cotโปยน ๐ฅ]
-1 / (1 +xยฒ)
๐/๐๐ฅ (secโปยน(๐ข/๐))
๐ / (|๐ข|โ(๐ขยฒโ ๐ยฒ)) * du/dx
๐/๐๐ฅ (๐^๐ข)
๐^๐ข ln ๐ * du/dx
๐/๐๐ฅ [logโ ๐ฅ]
1 / (๐ฅ ln ๐)
โซ ๐๐ฅ / โ(๐ยฒ โ ๐ขยฒ)
sinโปยน(๐ข/๐) + ๐ถ