Calculus Review

Flashcards of key calculus concepts and theorems.

Basic Derivative of x to the power of n

𝑑/𝑑𝑥 (𝑥^𝑛) = 𝑛𝑥^(𝑛−1)

Basic Derivative of sin x

𝑑/𝑑𝑥 (sin 𝑥) = cos 𝑥

Basic Derivative of cos x

𝑑/𝑑𝑥 (cos 𝑥) = − sin 𝑥

Basic Derivative of tan x

𝑑/𝑑𝑥 (tan 𝑥) = sec² 𝑥

Basic Derivative of cot x

𝑑/𝑑𝑥 (cot 𝑥) = − csc² 𝑥

Basic Derivative of sec x

𝑑/𝑑𝑥 (sec 𝑥) = sec 𝑥 tan 𝑥

Basic Derivative of csc x

𝑑/𝑑𝑥 (csc 𝑥) = − csc 𝑥 cot 𝑥

Basic Derivative of ln u

𝑑/𝑑𝑥 (ln 𝑢) = (1/𝑢) * du/dx

Basic Derivative of e to the power of u

𝑑/𝑑𝑥 (𝑒^𝑢) = 𝑒^𝑢 * du/dx

Chain Rule

𝑑/𝑑𝑥 [𝑓(𝑢)] = 𝑓′(𝑢) * du/dx

Product Rule

𝑑/𝑑𝑥 (𝑢𝑣) = 𝑢 * dv/dx + 𝑣 * du/dx

Quotient Rule

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

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

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)

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

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.

Alternate Definition of the Derivative

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

Critical Point

Point where dy/dx = 0 or is undefined.

Local Minimum

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

Local Maximum

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

Point of Inflection

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

Derivative of an Inverse Function

𝑔′(𝑥) = 1 / 𝑓′(𝑔(𝑥))

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.

Average Rate of Change (ARoC)

𝑚_sec = (𝑓(𝑏) − 𝑓(𝑎)) / (𝑏 − 𝑎)

Instantaneous Rate of Change (IRoC)

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

Equation of a Tangent Line

y - y1 = m(x - x1)

𝑓′(𝑥) > 0

Function is increasing

𝑓′(𝑥) < 0

Function is decreasing

𝑓′(𝑥) = 0 or DNE

Critical Values at x.

Relative Maximum

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

Relative Minimum

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

Absolute Max or Min

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

𝑓′′(𝑥) > 0

Function is concave up.

𝑓′′(𝑥) < 0

Function is concave down.

Point of Inflection Second Derivative

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

Relative Maximum (Second derivative)

𝑓′′(𝑥) < 0

Relative Minimum (Second derivative)

𝑓′′(𝑥) > 0

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

lim 𝑥→±∞ 𝑓(𝑥) = 0

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

lim 𝑥→±∞ 𝑓(𝑥) = DNE

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

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

The Fundamental Theorem of Calculus

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

Corollary to FTC

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

𝑥(𝑡)

Position Function

𝑣(𝑡)

Velocity Function

𝑎(𝑡)

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

∫ 𝑣(𝑡) 𝑑𝑡 from 𝑡0 to 𝑡𝑓

Total Distance

∫ | 𝑣(𝑡)| 𝑑𝑡 from 𝑡0 to 𝑡𝑓

Average Velocity

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

Accumulation

𝑥(0) + ∫ 𝑣(𝑡) 𝑑𝑡 from 𝑡0 to 𝑡𝑓

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

𝑙𝑛 𝑁 = 𝑝 ↔ 𝑒^𝑝 = 𝑁

Logarithm Product Rule

𝑙𝑛 𝑀𝑁 = ln 𝑀 + ln 𝑁

Logarithm Quotient Rule

ln (𝑀/𝑁) = ln 𝑀 − ln 𝑁

Logarithm Power Rule

𝑝 ∙ ln 𝑀 = ln 𝑀^𝑝

Exponential Growth and Decay Formula

𝑦 = 𝐶𝑒^(𝑘𝑡)

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

𝑦′ = 𝑘𝑦

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)

Mean Value Theorem for Integrals: The Average Value

𝑓_avg = (1 / (𝑏 − 𝑎)) ∫ 𝑓(𝑥) 𝑑𝑥 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

𝐴_𝑇 = (1/2) ℎ[𝑏1 + 𝑏2]

Trapezoidal Rule (for even intervals)

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

sin(−𝑥) (ODD)

sin(−𝑥) = − sin 𝑥

cos(−𝑥) (EVEN)

cos(−𝑥) = cos 𝑥

Pythagorean Identities

𝑠𝑠𝑠𝑠𝑛𝑛²𝜃 + 𝑐𝑐𝑐𝑐𝑠𝑠²𝜃 = 1

Double Angle Formulas (sin 2x)

sin 2𝑥 = 2 sin 𝑥 cos 𝑥

Double Angle Formula (cos 2x)

cos 2𝑥 = cos²𝑥 − sin²𝑥 = 1 − 2 sin²𝑥

Power-Reducing Formulas (cos²x)

cos²𝑥 = (1/2) (1 + cos 2𝑥)

Power-Reducing Formulas (sin²x)

sin²𝑥 = (1/2) (1 − cos 2𝑥)

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

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

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

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

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.

∫ 𝑑𝑢 / 𝑢

ln| 𝑢 | + 𝐶

∫ 𝑢^𝑛 𝑑𝑢

𝑢^(𝑛+1) / (𝑛 + 1) + 𝐶 (𝑛 ≠ −1)

∫ 𝑒^𝑢 𝑑𝑢

𝑒^𝑢 + 𝐶

∫ 𝑎^𝑢 𝑑𝑢

𝑎^𝑢 / ln 𝑎 + 𝐶

∫ sin 𝑢 𝑑𝑢

− cos 𝑢 + 𝐶

∫ cos 𝑢 𝑑𝑢

sin 𝑢 + 𝐶

∫ tan 𝑢 𝑑𝑢

− ln|cos 𝑢 + 𝐶|

∫ cot 𝑢 𝑑𝑢

ln|sin 𝑢| + 𝐶

∫ sec 𝑢 𝑑𝑢

ln|sec 𝑢 + tan 𝑢| + 𝐶

∫ csc 𝑢 𝑑𝑢

− ln|csc 𝑢 + cot 𝑢| + 𝐶

∫ sec²𝑢 𝑑𝑑𝑑𝑑

tan 𝑢 + 𝐶

∫ csc²𝑢 𝑑𝑑𝑑𝑑

− cot 𝑢 + 𝐶

∫ sec 𝑢 tan 𝑢 𝑑𝑑𝑑𝑑

sec 𝑢 + 𝐶

∫ csc 𝑢 cot 𝑢 𝑑𝑑𝑑𝑑

− csc 𝑢 + 𝐶

𝑑/𝑑𝑥 (sin⁻¹(𝑢/𝑎))

1 / √(𝑎²− 𝑢²) * du/dx

𝑑/𝑑𝑥 [cot⁻¹ 𝑥]

-1 / (1 +x²)

𝑑/𝑑𝑥 (sec⁻¹(𝑢/𝑎))

𝑎 / (|𝑢|√(𝑢²− 𝑎²)) * du/dx

𝑑/𝑑𝑥 (𝑎^𝑢)

𝑎^𝑢 ln 𝑎 * du/dx

𝑑/𝑑𝑥 [logₐ 𝑥]

1 / (𝑥 ln 𝑎)

∫ 𝑑𝑥 / √(𝑎² − 𝑢²)

sin⁻¹(𝑢/𝑎) + 𝐶