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Flashcards of key calculus concepts and theorems.
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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