Calculus Review

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Flashcards of key calculus concepts and theorems.

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102 Terms

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Basic Derivative of x to the power of n

𝑑/𝑑𝑥 (𝑥^𝑛) = 𝑛𝑥^(𝑛−1)

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Basic Derivative of sin x

𝑑/𝑑𝑥 (sin 𝑥) = cos 𝑥

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Basic Derivative of cos x

𝑑/𝑑𝑥 (cos 𝑥) = − sin 𝑥

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Basic Derivative of tan x

𝑑/𝑑𝑥 (tan 𝑥) = sec² 𝑥

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Basic Derivative of cot x

𝑑/𝑑𝑥 (cot 𝑥) = − csc² 𝑥

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Basic Derivative of sec x

𝑑/𝑑𝑥 (sec 𝑥) = sec 𝑥 tan 𝑥

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Basic Derivative of csc x

𝑑/𝑑𝑥 (csc 𝑥) = − csc 𝑥 cot 𝑥

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Basic Derivative of ln u

𝑑/𝑑𝑥 (ln 𝑢) = (1/𝑢) * du/dx

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Basic Derivative of e to the power of u

𝑑/𝑑𝑥 (𝑒^𝑢) = 𝑒^𝑢 * du/dx

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Chain Rule

𝑑/𝑑𝑥 [𝑓(𝑢)] = 𝑓′(𝑢) * du/dx

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Product Rule

𝑑/𝑑𝑥 (𝑢𝑣) = 𝑢 * dv/dx + 𝑣 * du/dx

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Quotient Rule

𝑑/𝑑𝑥 (𝑢/𝑣) = (𝑣 * du/dx - 𝑢 * dv/dx) / 𝑣²

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Intermediate Value Theorem

If 𝑓(𝑥) is continuous on [𝑎, 𝑏], and y is a number between 𝑓(𝑎) and 𝑓(𝑏), then there exists at least one number 𝑥 = 𝑐 in (𝑎, 𝑏) such that 𝑓(𝑐) = 𝑦

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Mean Value Theorem

If 𝑓(𝑥) is continuous on [𝑎, 𝑏], AND the first derivative exists on the interval (𝑎, 𝑏) then there is at least one number 𝑥 = 𝑐 in (𝑎, 𝑏) such that 𝑓′(𝑐) = (𝑓(𝑏)−𝑓(𝑎))/(𝑏−𝑎)

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Rolle’s Theorem

If 𝑓(𝑥) is continuous on [𝑎, 𝑏], AND the first derivative exists on the interval (𝑎, 𝑏) AND 𝑓(𝑎) = 𝑓(𝑏), then there is at least one number 𝑥 = 𝑐 in (𝑎, 𝑏) such that 𝑓′(𝑐) = 0

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Extreme Value Theorem

If 𝑓(𝑥) is continuous on [𝑎, 𝑏], then the function is guaranteed to have an absolute maximum and an absolute minimum on the interval.

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Alternate Definition of the Derivative

𝑓′(𝑐) = lim 𝑥→𝑐 (𝑓(𝑥) − 𝑓(𝑐)) / (𝑥 − 𝑐)

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Critical Point

Point where dy/dx = 0 or is undefined.

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Local Minimum

𝑑𝑦/𝑑𝑥 goes from (−, 0, +) or (−, undefined, +) OR 𝑑²𝑦/𝑑𝑥² > 0

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Local Maximum

𝑑𝑦/𝑑𝑥 goes (+, 0, −) or (+, undefined, −) OR 𝑑²𝑦/𝑑𝑥² < 0

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Point of Inflection

Concavity changes; 𝑑²𝑦/𝑑𝑥² goes from (+, 0, −), (−, 0, +), (+, undefined, −), OR (−, undefined, +)

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Derivative of an Inverse Function

𝑔′(𝑥) = 1 / 𝑓′(𝑔(𝑥))

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Implicit Differentiation

Remember that in implicit differentiation you will have a 𝑑𝑦/𝑑𝑥 for each y in the original function or equation. Isolate the 𝑑𝑦/𝑑𝑥. If you are taking the second derivative 𝑑²𝑦/𝑑𝑥², you will often substitute the expression you found for the first derivative somewhere in the process.

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Average Rate of Change (ARoC)

𝑚_sec = (𝑓(𝑏) − 𝑓(𝑎)) / (𝑏 − 𝑎)

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Instantaneous Rate of Change (IRoC)

𝑚_tan = 𝑓′(𝑥) = lim ℎ→0 (𝑓(𝑥 + ℎ) − 𝑓(𝑥)) / ℎ

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Equation of a Tangent Line

y - y1 = m(x - x1)

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𝑓′(𝑥) > 0

Function is increasing

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𝑓′(𝑥) < 0

Function is decreasing

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𝑓′(𝑥) = 0 or DNE

Critical Values at x.

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Relative Maximum

𝑓′(𝑥) = 0 or DNE and sign of 𝑓′(𝑥) changes from + to −

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Relative Minimum

𝑓′(𝑥) = 0 or DNE and sign of 𝑓′(𝑥) changes from − to +

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Absolute Max or Min

Check Endpoints Also. The maximum value is a y-value.

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𝑓′′(𝑥) > 0

Function is concave up.

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𝑓′′(𝑥) < 0

Function is concave down.

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Point of Inflection Second Derivative

𝑓′(𝑥) = 0 and sign of 𝑓′′(𝑥) changes, then there is a point of inflection at x.

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Relative Maximum (Second derivative)

𝑓′′(𝑥) < 0

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Relative Minimum (Second derivative)

𝑓′′(𝑥) > 0

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Horizontal Asymptotes: Largest exponent in numerator < largest exponent in denominator

lim 𝑥→±∞ 𝑓(𝑥) = 0

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Horizontal Asymptotes: Largest exponent in numerator > largest exponent in denominator

lim 𝑥→±∞ 𝑓(𝑥) = DNE

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Horizontal Asymptotes: Largest exponent in numerator = largest exponent in denominator

lim 𝑥→±∞ 𝑓(𝑥) = a/b (quotient of leading coefficients)

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The Fundamental Theorem of Calculus

∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎) where 𝐹′(𝑥) = 𝑓(𝑥)

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Corollary to FTC

𝑑/𝑑𝑥 ∫ 𝑓(𝑡)𝑑𝑡 from a to g(x) = 𝑓(𝑔(𝑥)) * 𝑔′(𝑥)

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𝑥(𝑡)

Position Function

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𝑣(𝑡)

Velocity Function

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𝑎(𝑡)

Acceleration Function

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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)

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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).

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Speed

| velocity |

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If acceleration and velocity have the same sign

Then the speed is increasing

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If the acceleration and velocity have different signs

Then the speed is decreasing.

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The particle is moving right

When velocity is positive

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the particle is moving left

When velocity is negative

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Displacement

∫ 𝑣(𝑡) 𝑑𝑡 from 𝑡0 to 𝑡𝑓

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Total Distance

∫ | 𝑣(𝑡)| 𝑑𝑡 from 𝑡0 to 𝑡𝑓

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Average Velocity

(final position − initial position) / (total time) = Δ𝑥 / Δ𝑡

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Accumulation

𝑥(0) + ∫ 𝑣(𝑡) 𝑑𝑡 from 𝑡0 to 𝑡𝑓

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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.

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Definition of Logarithms

𝑙𝑛 𝑁 = 𝑝 ↔ 𝑒^𝑝 = 𝑁

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Logarithm Product Rule

𝑙𝑛 𝑀𝑁 = ln 𝑀 + ln 𝑁

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Logarithm Quotient Rule

ln (𝑀/𝑁) = ln 𝑀 − ln 𝑁

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Logarithm Power Rule

𝑝 ∙ ln 𝑀 = ln 𝑀^𝑝

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Exponential Growth and Decay Formula

𝑦 = 𝐶𝑒^(𝑘𝑡)

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“the rate of change of y is proportional to y”

𝑦′ = 𝑘𝑦

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When solving a differential equation

Separate variables, Integrate, Add +C to one side, Use initial conditions to find “C”, Write the equation if the form of 𝑦 = 𝑓(𝑥)

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Mean Value Theorem for Integrals: The Average Value

𝑓_avg = (1 / (𝑏 − 𝑎)) ∫ 𝑓(𝑥) 𝑑𝑥 from a to b

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Riemann Sums

A rectangular approximation. Do NOT EVALUATE THE INTEGRAL; you add up the areas of the rectangles.

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Area of a Trapezoid

𝐴_𝑇 = (1/2) ℎ[𝑏1 + 𝑏2]

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Trapezoidal Rule (for even intervals)

∫ 𝑓(𝑥) 𝑑𝑥 from a to b = (𝑏 − 𝑎) / (2𝑛) [𝑦0 + 2𝑦1 + 2𝑦2 + … +2𝑦𝑛−1 + 𝑦𝑛]

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sin(−𝑥) (ODD)

sin(−𝑥) = − sin 𝑥

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cos(−𝑥) (EVEN)

cos(−𝑥) = cos 𝑥

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Pythagorean Identities

𝑠𝑠𝑠𝑠𝑛𝑛²𝜃 + 𝑐𝑐𝑐𝑐𝑠𝑠²𝜃 = 1

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Double Angle Formulas (sin 2x)

sin 2𝑥 = 2 sin 𝑥 cos 𝑥

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Double Angle Formula (cos 2x)

cos 2𝑥 = cos²𝑥 − sin²𝑥 = 1 − 2 sin²𝑥

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Power-Reducing Formulas (cos²x)

cos²𝑥 = (1/2) (1 + cos 2𝑥)

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Power-Reducing Formulas (sin²x)

sin²𝑥 = (1/2) (1 − cos 2𝑥)

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Area Between Two Curves

𝑆lices ⊥ to x-axis: 𝐴 = ∫ [𝑓(𝑥) − 𝑔(𝑥)] 𝑑𝑥 from a to b. Slices ⊥ to y-axis: 𝐴 = ∫ [𝑓(𝑦) − 𝑔(𝑦)] 𝑑𝑦 from c to d

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Volume By Disk Method

About x-axis: 𝑉 = 𝜋 ∫ [𝑅(𝑥)]² 𝑑𝑥 from a to b. About y-axis: 𝑉 = 𝜋 ∫ [𝑅(𝑦)]² 𝑑𝑦 from c to d

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Volume By Washer Method

About x-axis: 𝑉 = 𝜋 ∫ ([𝑅(𝑥)]² − [𝑟(𝑥)]²) 𝑑𝑥 from a to b. About y-axis: 𝑉 = 𝜋 ∫ ([𝑅(𝑦)]² − [𝑟(𝑦)]²) 𝑑𝑦 from c to d

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Volume By Shell Method

About x-axis: 𝑉 = 2 𝜋 ∫ 𝑦 [𝑅(𝑦)] 𝑑𝑦 from c to d. About y-axis: 𝑉 = 2 𝜋 ∫ 𝑥 [𝑅(𝑥)] 𝑑𝑥 from a to b

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General Equations for Known Cross Section

SQUARES: 𝑉 = ∫ (𝑏𝑎𝑠𝑒)² 𝑑𝑥 from a to b. TRIANGLES EQUILATERAL: 𝑉 = (√3 / 4) ∫ (𝑏𝑎𝑠𝑒)² 𝑑𝑥 from a to b. ISOSCELES RIGHT: 𝑉 = (1 / 4) ∫ (𝑏𝑎𝑠𝑒)² 𝑑𝑥 from a to b. RECTANGLES: 𝑉 = ∫ (𝑏𝑎𝑠𝑒) ∙ ℎ 𝑑𝑥 from a to b where h is the height of the rectangles. SEMI-CIRCLES: 𝑉 = ∫ (𝑟𝑎𝑑𝑖𝑢𝑠)² 𝑑𝑥 from a to b where radius is ½ distance between the two curves.

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∫ 𝑑𝑢 / 𝑢

ln| 𝑢 | + 𝐶

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∫ 𝑢^𝑛 𝑑𝑢

𝑢^(𝑛+1) / (𝑛 + 1) + 𝐶 (𝑛 ≠ −1)

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∫ 𝑒^𝑢 𝑑𝑢

𝑒^𝑢 + 𝐶

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∫ 𝑎^𝑢 𝑑𝑢

𝑎^𝑢 / ln 𝑎 + 𝐶

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∫ sin 𝑢 𝑑𝑢

− cos 𝑢 + 𝐶

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∫ cos 𝑢 𝑑𝑢

sin 𝑢 + 𝐶

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∫ tan 𝑢 𝑑𝑢

− ln|cos 𝑢 + 𝐶|

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∫ cot 𝑢 𝑑𝑢

ln|sin 𝑢| + 𝐶

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∫ sec 𝑢 𝑑𝑢

ln|sec 𝑢 + tan 𝑢| + 𝐶

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∫ csc 𝑢 𝑑𝑢

− ln|csc 𝑢 + cot 𝑢| + 𝐶

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∫ sec²𝑢 𝑑𝑑𝑑𝑑

tan 𝑢 + 𝐶

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∫ csc²𝑢 𝑑𝑑𝑑𝑑

− cot 𝑢 + 𝐶

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∫ sec 𝑢 tan 𝑢 𝑑𝑑𝑑𝑑

sec 𝑢 + 𝐶

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∫ csc 𝑢 cot 𝑢 𝑑𝑑𝑑𝑑

− csc 𝑢 + 𝐶

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𝑑/𝑑𝑥 (sin⁻¹(𝑢/𝑎))

1 / √(𝑎²− 𝑢²) * du/dx

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𝑑/𝑑𝑥 [cot⁻¹ 𝑥]

-1 / (1 +x²)

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𝑑/𝑑𝑥 (sec⁻¹(𝑢/𝑎))

𝑎 / (|𝑢|√(𝑢²− 𝑎²)) * du/dx

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𝑑/𝑑𝑥 (𝑎^𝑢)

𝑎^𝑢 ln 𝑎 * du/dx

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𝑑/𝑑𝑥 [logₐ 𝑥]

1 / (𝑥 ln 𝑎)

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∫ 𝑑𝑥 / √(𝑎² − 𝑢²)

sin⁻¹(𝑢/𝑎) + 𝐶