Harold’s Calculus Notes Cheat Sheet AP Calculus Limits

Harold’s Calculus Notes Cheat Sheet 17 November 2017

AP Calculus Limits Definition of Limit Let f be a function defined on an open interval containing c and let L be a real number. The statement: lim 𝑓(𝑥) = 𝐿 𝑥→𝑎

means that for each 𝜖 > 0 there exists a 𝛿 > 0 such that if 0 < |𝑥 − 𝑎| < 𝛿, then |𝑓(𝑥) − 𝐿| < 𝜖 Tip : Direct substitution: Plug in 𝑓(𝑎) and see if it provides a legal answer. If so then L = 𝑓(𝑎). The Existence of a Limit The limit of 𝑓(𝑥) as 𝑥 approaches a is L if and only if:

lim 𝑓(𝑥) = 𝐿

𝑥→𝑎−

lim 𝑓(𝑥) = 𝐿

𝑥→𝑎+ 𝟐

Prove that 𝒇(𝒙) = 𝒙 − 𝟏 is a continuous function.

Definition of Continuity A function f is continuous at c if for every 𝜀 > 0 there exists a 𝛿 > 0 such that |𝑥 − 𝑐| < 𝛿 and |𝑓(𝑥) − 𝑓(𝑐)| < 𝜀. Tip: Rearrange |𝑓(𝑥) − 𝑓(𝑐)| to have |(𝑥 − 𝑐)| as a factor. Since |𝑥 − 𝑐| < 𝛿 we can find an equation that relates both 𝛿 and 𝜀 together.

|𝑓(𝑥) − 𝑓(𝑐)| = |(𝑥 2 − 1) − (𝑐 2 − 1)| = |𝑥 2 − 1 − 𝑐 2 + 1| = |𝑥 2 − 𝑐 2 | = |(𝑥 + 𝑐)(𝑥 − 𝑐)| = |(𝑥 + 𝑐)| |(𝑥 − 𝑐)| Since |(𝑥 + 𝑐)| ≤ |2𝑐| |𝑓(𝑥) − 𝑓(𝑐)| ≤ |2𝑐||(𝑥 − 𝑐)| < 𝜀 𝟏

So given 𝜀 > 0, we can choose 𝜹 = | | 𝜺 > 𝟎 in the 𝟐𝒄 Definition of Continuity. So substituting the chosen 𝛿 for |(𝑥 − 𝑐)| we get: 1 |𝑓(𝑥) − 𝑓(𝑐)| ≤ |2𝑐| (| | 𝜀) = 𝜀 2𝑐 Since both conditions are met, 𝑓(𝑥) is continuous.

Two Special Trig Limits 𝑙𝑖𝑚

𝑥→0

𝑠𝑖𝑛 𝑥 =1 𝑥

1 − 𝑐𝑜𝑠 𝑥 =0 𝑥→0 𝑥 𝑙𝑖𝑚

Copyright © 2015-2017 by Harold Toomey, WyzAnt Tutor 1

Derivatives Definition of a Derivative of a Function Slope Function Notation for Derivatives

0. The Chain Rule

1. The Constant Multiple Rule 2. The Sum and Difference Rule 3. The Product Rule 4. The Quotient Rule 5. The Constant Rule 6a. The Power Rule 6b. The General Power Rule 7. The Power Rule for x 8. Absolute Value 9. Natural Logorithm 10. Natural Exponential 11. Logorithm 12. Exponential 13. Sine 14. Cosine 15. Tangent 16. Cotangent 17. Secant

(See Larson’s 1-pager of common derivatives) 𝑓(𝑥 + ℎ) − 𝑓(𝑥) 𝑓 ′ (𝑥) = lim ℎ→0 ℎ 𝑓(𝑥) − 𝑓(𝑐) 𝑓 ′ (𝑐) = lim 𝑥→𝑐 𝑥−𝑐 𝑑𝑦 𝑑 𝑓 ′ (𝑥), 𝑓 (𝑛) (𝑥), , 𝑦 ′ , [𝑓(𝑥)], 𝐷𝑥 [𝑦] 𝑑𝑥 𝑑𝑥 𝑑 [𝑓(𝑔(𝑥))] = 𝑓 ′ (𝑔(𝑥))𝑔′ (𝑥) 𝑑𝑥 𝑑𝑦 𝑑𝑦 𝑑𝑡 = · 𝑑𝑥 𝑑𝑡 𝑑𝑥 𝑑 [𝑐𝑓(𝑥)] = 𝑐𝑓 ′ (𝑥) 𝑑𝑥 𝑑 [𝑓(𝑥) ± 𝑔(𝑥)] = 𝑓 ′ (𝑥) ± 𝑔′ (𝑥) 𝑑𝑥 𝑑 [𝑓𝑔] = 𝑓𝑔′ + 𝑔 𝑓 ′ 𝑑𝑥 𝑑 𝑓 𝑔𝑓 ′ − 𝑓𝑔′ [ ]= 𝑑𝑥 𝑔 𝑔2 𝑑 [𝑐 ] = 0 𝑑𝑥 𝑑 𝑛 [𝑥 ] = 𝑛𝑥 𝑛−1 𝑑𝑥 𝑑 𝑛 [𝑢 ] = 𝑛𝑢𝑛−1 𝑢′ 𝑤ℎ𝑒𝑟𝑒 𝑢 = 𝑢(𝑥) 𝑑𝑥 𝑑 [𝑥 ] = 1 (𝑡ℎ𝑖𝑛𝑘 𝑥 = 𝑥 1 𝑎𝑛𝑑 𝑥 0 = 1) 𝑑𝑥 𝑑 𝑥 [|𝑥|] = |𝑥| 𝑑𝑥 𝑑 1 [ln 𝑥 ] = 𝑑𝑥 𝑥 𝑑 𝑥 [𝑒 ] = 𝑒 𝑥 𝑑𝑥 𝑑 1 [log 𝑎 𝑥 ] = 𝑑𝑥 (ln 𝑎) 𝑥 𝑑 𝑥 [𝑎 ] = (ln 𝑎) 𝑎 𝑥 𝑑𝑥 𝑑 [𝑠𝑖𝑛(𝑥)] = cos(𝑥) 𝑑𝑥 𝑑 [𝑐𝑜𝑠(𝑥)] = −𝑠𝑖𝑛(𝑥) 𝑑𝑥 𝑑 [𝑡𝑎𝑛(𝑥)] = 𝑠𝑒𝑐 2 (𝑥) 𝑑𝑥 𝑑 [𝑐𝑜𝑡(𝑥)] = −𝑐𝑠𝑐 2 (𝑥) 𝑑𝑥 𝑑 [𝑠𝑒𝑐(𝑥)] = 𝑠𝑒𝑐(𝑥) 𝑡𝑎𝑛(𝑥) 𝑑𝑥

Copyright © 2015-2017 by Harold Toomey, WyzAnt Tutor

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Derivatives 18. Cosecant 19. Arcsine 20. Arccosine 21. Arctangent 22. Arccotangent 23. Arcsecant 24. Arccosecant 25. Hyperbolic Sine

(

26. Hyperbolic Cosine

(

𝒆𝒙 −𝒆−𝒙 𝟐

𝒆𝒙 +𝒆−𝒙 𝟐

27. Hyperbolic Tangent 28. Hyperbolic Cotangent 29. Hyperbolic Secant 30. Hyperbolic Cosecant 31. Hyperbolic Arcsine 32. Hyperbolic Arccosine 33. Hyperbolic Arctangent 34. Hyperbolic Arccotangent 35. Hyperbolic Arcsecant 36. Hyperbolic Arccosecant Position Function Velocity Function Acceleration Function Jerk Function

)

)

(See Larson’s 1-pager of common derivatives) 𝑑 [𝑐𝑠𝑐(𝑥)] = − 𝑐𝑠𝑐(𝑥) 𝑐𝑜𝑡(𝑥) 𝑑𝑥 𝑑 1 [sin−1 (𝑥)] = 𝑑𝑥 √1 − 𝑥 2 𝑑 −1 [cos −1 (𝑥)] = 𝑑𝑥 √1 − 𝑥 2 𝑑 1 [tan−1 (𝑥)] = 𝑑𝑥 1 + 𝑥2 𝑑 −1 [cot −1 (𝑥)] = 𝑑𝑥 1 + 𝑥2 𝑑 1 [sec −1 (𝑥)] = 𝑑𝑥 |𝑥| √𝑥 2 − 1 𝑑 −1 [csc −1 (𝑥)] = 𝑑𝑥 |𝑥| √𝑥 2 − 1 𝑑 [𝑠𝑖𝑛ℎ(𝑥)] = cosh(𝑥) 𝑑𝑥 𝑑 [𝑐𝑜𝑠ℎ(𝑥)] = 𝑠𝑖𝑛ℎ(𝑥) 𝑑𝑥 𝑑 [𝑡𝑎𝑛ℎ(𝑥)] = 𝑠𝑒𝑐ℎ2 (𝑥) 𝑑𝑥 𝑑 [𝑐𝑜𝑡ℎ(𝑥)] = −𝑐𝑠𝑐ℎ2 (𝑥) 𝑑𝑥 𝑑 [𝑠𝑒𝑐ℎ(𝑥)] = − 𝑠𝑒𝑐ℎ(𝑥) 𝑡𝑎𝑛ℎ(𝑥) 𝑑𝑥 𝑑 [𝑐𝑠𝑐ℎ(𝑥)] = − 𝑐𝑠𝑐ℎ(𝑥) 𝑐𝑜𝑡ℎ(𝑥) 𝑑𝑥 𝑑 1 [sinh−1 (𝑥)] = 𝑑𝑥 √𝑥 2 + 1 𝑑 1 [cosh−1 (𝑥)] = 2 𝑑𝑥 √𝑥 − 1 𝑑 1 [tanh−1 (𝑥)] = 𝑑𝑥 1 − 𝑥2 𝑑 1 [coth−1 (𝑥)] = 𝑑𝑥 1 − 𝑥2 𝑑 −1 [sech−1 (𝑥)] = 𝑑𝑥 𝑥 √1 − 𝑥 2 𝑑 −1 [csch−1 (𝑥)] = 𝑑𝑥 |𝑥| √1 + 𝑥 2 1 𝑠(𝑡) = 𝑔𝑡 2 + 𝑣0 𝑡 + 𝑠0 2 𝑣(𝑡) = 𝑠 ′ (𝑡) = 𝑔𝑡 + 𝑣0 𝑎(𝑡) = 𝑣 ′ (𝑡) = 𝑠 ′′ (𝑡) 𝑗(𝑡) = 𝑎′ (𝑡) = 𝑣 ′′ (𝑡) = 𝑠 (3) (𝑡)


3

Analyzing the Graph of a Function x-Intercepts (Zeros or Roots) y-Intercept Domain Range Continuity Vertical Asymptotes (VA) Horizontal Asymptotes (HA) Infinite Limits at Infinity Differentiability Relative Extrema Concavity Points of Inflection

(See Harold’s Illegals and Graphing Rationals Cheat Sheet) f(x) = 0 f(0) = y Valid x values Valid y values No division by 0, no negative square roots or logs x = division by 0 or undefined lim− 𝑓(𝑥) → 𝑦 and lim+ 𝑓(𝑥) → 𝑦 𝑥→∞

𝑥→∞

lim− 𝑓(𝑥) → ∞ and lim+ 𝑓(𝑥) → ∞

𝑥→∞

𝑥→∞

Limit from both directions arrives at the same slope Create a table with domains, f(x), f ’(x), and f ”(x) If 𝑓 ”(𝑥) → +, then cup up ⋃ If 𝑓 ”(𝑥) → −, then cup down ⋂ f ”(x) = 0 (concavity changes)

Graphing with Derivatives Test for Increasing and Decreasing Functions

The First Derivative Test

The Second Deriviative Test Let f ’(c)=0, and f ”(x) exists, then Test for Concavity Points of Inflection Change in concavity

1. If f ’(x) > 0, then f is increasing (slope up) ↗ 2. If f ’(x) < 0, then f is decreasing (slope down) ↘ 3. If f ’(x) = 0, then f is constant (zero slope) → 1. If f ’(x) changes from – to + at c, then f has a relative minimum at (c, f(c)) 2. If f ’(x) changes from + to - at c, then f has a relative maximum at (c, f(c)) 3. If f ’(x), is + c + or - c -, then f(c) is neither 1. If f ”(x) > 0, then f has a relative minimum at (c, f(c)) 2. If f ”(x) < 0, then f has a relative maximum at (c, f(c)) 3. If f ’(x) = 0, then the test fails (See 1𝑠𝑡 derivative test) 1. If f ”(x) > 0 for all x, then the graph is concave up ⋃ 2. If f ”(x) < 0 for all x, then the graph is concave down ⋂ If (c, f(c)) is a point of inflection of f, then either 1. f ”(c) = 0 or 2. f ” does not exist at x = c.

Tangent Lines Genreal Form Slope-Intercept Form Point-Slope Form Calculus Form Slope


𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 𝑦 = 𝑚𝑥 + 𝑏 𝑦 − 𝑦0 = 𝑚(𝑥 − 𝑥0 ) 𝑦 = 𝑓 ′ (𝑐)(𝑥 − 𝑐) + 𝑓(𝑐) 𝑟𝑖𝑠𝑒 ∆𝑦 𝑦2 − 𝑦1 𝑑𝑦 𝑚= = = = = 𝑓 ′ (𝑥) 𝑟𝑢𝑛 ∆𝑥 𝑥2 − 𝑥1 𝑑𝑥

4

Differentiation & Differentials Rolle’s Theorem f is continuous on the closed interval [a,b], and f is differentiable on the open interval (a,b). Mean Value Theorem If f meets the conditions of Rolle’s Theorem, then

If f(a) = f(b), then there exists at least one number c in (a,b) such that f’(c) = 0. 𝑓(𝑏) − 𝑓(𝑎) 𝑏−𝑎 𝑓(𝑏) = 𝑓(𝑎) + (𝑏 − 𝑎)𝑓′(𝑐) Find ‘c’. If f takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval. 𝑓 ′ (𝑐) =

Intermediate Value Therem f is a continuous function with an interval, [a, b], as its domain.

Calculating Differentials Tanget line approximation Newton’s Method Finds zeros of f, or finds c if f(c) = 0.

𝑓(𝑥 + ∆𝑥) ≈ 𝑓(𝑥) + 𝑑𝑦 = 𝑓(𝑥) + 𝑓 ′ (𝑥) 𝑑𝑥 4

4

Example: √82 → 𝑓(𝑥) = √𝑥 , 𝑓(𝑥 + ∆𝑥) = 𝑓(81 + 1) 𝑓(𝑥𝑛 ) 𝑥𝑛+1 = 𝑥𝑛 − ′ 𝑓 (𝑥𝑛 ) 4

Related Rates

Example: √82 → 𝑓(𝑥) = 𝑥 4 − 82 = 0, 𝑥𝑛 = 3 Steps to solve: 1. Identify the known variables and rates of change. 𝑚 (𝑥 = 15 𝑚; 𝑦 = 20 𝑚; 𝑥 ′ = 2 ; 𝑦 ′ = ? ) 𝑠 2. Construct an equation relating these quantities. (𝑥 2 + 𝑦 2 = 𝑟 2 ) 3. Differentiate both sides of the equation. (2𝑥𝑥 ′ + 2𝑦𝑦 ′ = 0) 4. Solve for the desired rate of change. 𝑥 (𝑦 ′ = − 𝑥 ′ ) 𝑦 5. Substitute the known rates of change and quantities into the equation. 15 3𝑚 (𝑦 ′ = − ⦁ 2 = ) 20 2 𝑠 𝑃(𝑥) = 𝑥→𝑐 𝑄(𝑥)

𝐼𝑓 lim 𝑓(𝑥) = lim 𝑥→𝑐

L’Hôpital’s Rule

0 ∞ { , , 0 • ∞, 1∞ , 00 , ∞0 , ∞ − ∞} , 𝑏𝑢𝑡 𝑛𝑜𝑡 {0∞ }, 0 ∞ 𝑃(𝑥) 𝑃′ (𝑥) 𝑃′′ (𝑥) 𝑡ℎ𝑒𝑛 lim = lim ′ = lim ′′ =⋯ 𝑥→𝑐 𝑄(𝑥) 𝑥→𝑐 𝑄 (𝑥) 𝑥→𝑐 𝑄 (𝑥)


5

Summation Formulas

𝑛

∑ 𝑐 = 𝑐𝑛 𝑖=1 𝑛

∑𝑖 = 𝑖=1 𝑛

𝑛(𝑛 + 1) 𝑛2 𝑛 = + 2 2 2

∑ 𝑖2 = 𝑖=1 𝑛

𝑛(𝑛 + 1)(2𝑛 + 1) 𝑛3 𝑛2 𝑛 = + + 6 3 2 6 2

𝑛 3

∑ 𝑖 = (∑ 𝑖 ) = 𝑖=1 𝑛

Sum of Powers

𝑖=1

𝑛2 (𝑛 + 1)2 𝑛4 𝑛3 𝑛2 = + + 4 4 2 4

𝑛(𝑛 + 1)(2𝑛 + 1)(3𝑛2 + 3𝑛 − 1) 𝑛5 𝑛4 𝑛3 𝑛 ∑ 𝑖4 = = + + − 30 5 2 3 30 𝑖=1 𝑛

∑ 𝑖5 = 𝑖=1 𝑛

∑ 𝑖6 = 𝑖=1 𝑛

𝑛2 (𝑛 + 1)2 (2𝑛2 + 2𝑛 − 1) 𝑛6 𝑛5 5𝑛4 𝑛2 = + + − 12 6 2 12 12 𝑛(𝑛 + 1)(2𝑛 + 1)(3𝑛4 + 6𝑛3 − 3𝑛 + 1) 42

𝑛2 (𝑛 + 1)2 (3𝑛4 + 6𝑛3 − 𝑛2 − 4𝑛 + 2) ∑𝑖 = 24 7

𝑖=1

𝑛

𝑘−1

(𝑛 + 1)𝑘+1 1 𝑘+1 ) 𝑆𝑟 (𝑛) 𝑆𝑘 (𝑛) = ∑ 𝑖 = − ∑( 𝑘+1 𝑘+1 𝑟 𝑘

𝑖=1

𝑛

𝑟=0

𝑛

𝑛 2

∑ 𝑖(𝑖 + 1) = ∑ 𝑖 + ∑ 𝑖 = 𝑖=1 𝑛

Misc. Summation Formulas

𝑖=1

𝑖=1

𝑛(𝑛 + 1)(𝑛 + 2) 3

1 𝑛 ∑ = 𝑖(𝑖 + 1) 𝑛 + 1 𝑖=1 𝑛

∑ 𝑖=1

1 𝑛(𝑛 + 3) = 𝑖(𝑖 + 1)(𝑖 + 2) 4(𝑛 + 1)(𝑛 + 2)


6

Numerical Methods

𝑛

𝑏

𝑃0 (𝑥) = ∫ 𝑓(𝑥) 𝑑𝑥 = lim ∑ 𝑓(𝑥𝑖∗ ) ∆𝑥𝑖 ‖𝑃‖→0

𝑎

𝑖=1

where 𝑎 = 𝑥0 < 𝑥1 < 𝑥2 < ⋯ < 𝑥𝑛 = 𝑏 and ∆𝑥𝑖 = 𝑥𝑖 − 𝑥𝑖−1 and ‖𝑃‖ = 𝑚𝑎𝑥 {∆𝑥𝑖 } Types: • Left Sum (LHS) • Middle Sum (MHS) • Right Sum (RHS)

Riemann Sum

𝑛

𝑏

𝑃0 (𝑥) = ∫ 𝑓(𝑥) 𝑑𝑥 ≈ ∑ 𝑓(𝑥̅𝑖 ) ∆𝑥 = 𝑎

𝑖=1

∆𝑥 [𝑓(𝑥̅1 ) + 𝑓(𝑥̅2 ) + 𝑓(𝑥̅3 ) + ⋯ + 𝑓(𝑥̅𝑛 )] 𝑏−𝑎 where ∆𝑥 =

Midpoint Rule (Middle Sum)

𝑛

1

and 𝑥̅𝑖 = (𝑥𝑖−1 + 𝑥𝑖 ) = 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 [𝑥𝑖−1 , 𝑥𝑖 ] 2

Error Bounds: |𝐸𝑀 | ≤ 𝑏

𝐾(𝑏−𝑎)3 24𝑛2

𝑃1 (𝑥) = ∫ 𝑓(𝑥) 𝑑𝑥 ≈ 𝑎

∆𝑥 [𝑓(𝑥0 ) + 2𝑓(𝑥1 ) + 2𝑓(𝑥3 ) + ⋯ + 2𝑓(𝑥𝑛−1 ) 2 + 𝑓(𝑥𝑛 )] 𝑏−𝑎 where ∆𝑥 = 𝑛 and 𝑥𝑖 = 𝑎 + 𝑖∆𝑥

Trapezoidal Rule

Error Bounds: |𝐸𝑇 | ≤ 𝑏

𝐾(𝑏−𝑎)3 12𝑛2

𝑃2 (𝑥) = ∫ 𝑓(𝑥)𝑑𝑥 ≈ 𝑎

Simpson’s Rule

∆𝑥 [𝑓(𝑥0 ) + 4𝑓(𝑥1 ) + 2𝑓(𝑥2 ) + 4𝑓(𝑥3 ) + ⋯ 3 + 2𝑓(𝑥𝑛−2 ) + 4𝑓(𝑥𝑛−1 ) + 𝑓(𝑥𝑛 )] Where n is even 𝑏−𝑎 and ∆𝑥 = 𝑛 and 𝑥𝑖 = 𝑎 + 𝑖∆𝑥 Error Bounds: |𝐸𝑆 | ≤

𝐾(𝑏−𝑎)5 180𝑛4

[MATH] fnInt(f(x),x,a,b), [MATH] [1] [ENTER] TI-84 Plus

TI-Nspire CAS

Example: [MATH] fnInt(x^2,x,0,1) 1 1 ∫ 𝑥 2 𝑑𝑥 = 3 0 [MENU] [4] Calculus [3] Integral [TAB] [TAB] [X] [^] [2] [TAB] [TAB] [X] [ENTER] Shortcut: [ALPHA] [WINDOWS] [4]


7

Integration

(See Harold’s Fundamental Theorem of Calculus Cheat Sheet) ∫ 𝑓 ′ (𝑥) 𝑑𝑥 = 𝑓(𝑥) + 𝐶

Basic Integration Rules Integration is the “inverse” of differentiation, and vice versa.

𝑑 ∫ 𝑓(𝑥) 𝑑𝑥 = 𝑓(𝑥) 𝑑𝑥

𝑓(𝑥) = 0

∫ 0 𝑑𝑥 = 𝐶

𝑓(𝑥) = 𝑘 = 𝑘𝑥 0

∫ 𝑘 𝑑𝑥 = 𝑘𝑥 + 𝐶

1. The Constant Multiple Rule 2. The Sum and Difference Rule The Power Rule 𝑓(𝑥) = 𝑘𝑥 𝑛

∫ 𝑘 𝑓(𝑥) 𝑑𝑥 = 𝑘 ∫ 𝑓(𝑥) 𝑑𝑥 ∫[𝑓(𝑥) ± 𝑔(𝑥)] 𝑑𝑥 = ∫ 𝑓(𝑥) 𝑑𝑥 ± ∫ 𝑔(𝑥) 𝑑𝑥 ∫ 𝑥 𝑛 𝑑𝑥 =

𝑥 𝑛+1 + 𝐶, 𝑤ℎ𝑒𝑟𝑒 𝑛 ≠ −1 𝑛+1

𝐼𝑓 𝑛 = −1, 𝑡ℎ𝑒𝑛 ∫ 𝑥 −1 𝑑𝑥 = ln|𝑥| + 𝐶 𝑑

The General Power Rule

If 𝑢 = 𝑔(𝑥), 𝑎𝑛𝑑 𝑢′ = 𝑔(𝑥) then 𝑑𝑥 𝑢𝑛+1 𝑛 ′ ∫ 𝑢 𝑢 𝑑𝑥 = + 𝐶, 𝑤ℎ𝑒𝑟𝑒 𝑛 ≠ −1 𝑛+1 𝑛

Reimann Sum

∑ 𝑓(𝑐𝑖 ) ∆𝑥𝑖 ,

𝑤ℎ𝑒𝑟𝑒 𝑥𝑖−1 ≤ 𝑐𝑖 ≤ 𝑥𝑖

𝑖=1

‖∆‖ = ∆𝑥 = 𝑛

Definition of a Definite Integral Area under curve

𝑏

lim ∑ 𝑓(𝑐𝑖 ) ∆𝑥𝑖 = ∫ 𝑓(𝑥) 𝑑𝑥

‖∆‖→0

𝑏

Swap Bounds Additive Interval Property

𝑏−𝑎 𝑛 𝑎

𝑖=1

𝑎

∫ 𝑓(𝑥) 𝑑𝑥 = − ∫ 𝑓(𝑥) 𝑑𝑥 𝑎

𝑏

𝑐

𝑏

𝑏

∫ 𝑓(𝑥) 𝑑𝑥 = ∫ 𝑓(𝑥) 𝑑𝑥 + ∫ 𝑓(𝑥) 𝑑𝑥 𝑎

𝑎

𝑏

The Fundamental Theorem of Calculus

𝑐

∫ 𝑓(𝑥) 𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎) 𝑎

𝑥

𝑑 ∫ 𝑓(𝑡) 𝑑𝑡 = 𝑓(𝑥) 𝑑𝑥 𝑔(𝑥)

𝑎

𝑑 ∫ 𝑓(𝑡) 𝑑𝑡 = 𝑓(𝑔(𝑥))𝑔′ (𝑥) 𝑑𝑥

The Second Fundamental Theorem of Calculus

𝑎

ℎ(𝑥)

𝑑 ∫ 𝑓(𝑡) 𝑑𝑡 = 𝑓(ℎ(𝑥))ℎ′ (𝑥) − 𝑓(𝑔(𝑥))𝑔′(𝑥) 𝑑𝑥 𝑔(𝑥)

Mean Value Theorem for Integrals The Average Value for a Function


𝑏

∫ 𝑓(𝑥) 𝑑𝑥 = 𝑓(𝑐)(𝑏 − 𝑎) Find ‘𝑐’. 𝑎

𝑏 1 ∫ 𝑓(𝑥) 𝑑𝑥 𝑏−𝑎 𝑎

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Integration Methods 1. Memorized

See Larson’s 1-pager of common integrals ∫ 𝑓(𝑔(𝑥))𝑔′ (𝑥)𝑑𝑥 = 𝐹(𝑔(𝑥)) + 𝐶 Set 𝑢 = 𝑔(𝑥), then 𝑑𝑢 = 𝑔′ (𝑥) 𝑑𝑥

2. U-Substitution

∫ 𝑓(𝑢) 𝑑𝑢 = 𝐹(𝑢) + 𝐶 𝑢 = _____

𝑑𝑢 = _____ 𝑑𝑥

∫ 𝑢 𝑑𝑣 = 𝑢𝑣 − ∫ 𝑣 𝑑𝑢 𝑢 = _____ 𝑑𝑢 = _____

3. Integration by Parts

𝑣 = _____ 𝑑𝑣 = _____

Pick ‘𝑢’ using the LIATED Rule: L – Logarithmic : ln 𝑥 , log 𝑏 𝑥 , 𝑒𝑡𝑐. I – Inverse Trig.: tan−1 𝑥 , sec −1 𝑥 , 𝑒𝑡𝑐. A – Algebraic: 𝑥 2 , 3𝑥 60 , 𝑒𝑡𝑐. T – Trigonometric: sin 𝑥 , tan 𝑥 , 𝑒𝑡𝑐. E – Exponential: 𝑒 𝑥 , 19𝑥 , 𝑒𝑡𝑐. D – Derivative of: 𝑑𝑦⁄𝑑𝑥 𝑃(𝑥) ∫ 𝑑𝑥 𝑄(𝑥) where 𝑃(𝑥) 𝑎𝑛𝑑 𝑄(𝑥) are polynomials

4. Partial Fractions

Case 1: If degree of 𝑃(𝑥) ≥ 𝑄(𝑥) then do long division first Case 2: If degree of 𝑃(𝑥) < 𝑄(𝑥) then do partial fraction expansion ∫ √𝑎2 − 𝑥 2 𝑑𝑥 Substutution: 𝑥 = 𝑎 sin 𝜃 Identity: 1 − 𝑠𝑖𝑛2 𝜃 = 𝑐𝑜𝑠 2 𝜃

5a. Trig Substitution for √𝒂𝟐 − 𝒙𝟐

∫ √𝑥 2 − 𝑎2 𝑑𝑥 Substutution: 𝑥 = 𝑎 sec 𝜃 Identity: 𝑠𝑒𝑐 2 𝜃 − 1 = 𝑡𝑎𝑛2 𝜃

5b. Trig Substitution for √𝒙𝟐 − 𝒂𝟐

∫ √𝑥 2 + 𝑎2 𝑑𝑥 5c. Trig Substitution for √𝒙𝟐 + 𝒂𝟐 6. Table of Integrals 7. Computer Algebra Systems (CAS) 8. Numerical Methods 9. WolframAlpha

Substutution: 𝑥 = 𝑎 tan 𝜃 Identity: 𝑡𝑎𝑛2 𝜃 + 1 = 𝑠𝑒𝑐 2 𝜃 CRC Standard Mathematical Tables book TI-Nspire CX CAS Graphing Calculator TI –Nspire CAS iPad app Riemann Sum, Midpoint Rule, Trapezoidal Rule, Simpson’s Rule, TI-84 Google of mathematics. Shows steps. Free. www.wolframalpha.com


9

Partial Fractions Condition

Example Expansion

Typical Solution

(See Harold’s Partial Fractions Cheat Sheet) 𝑃(𝑥) 𝑓(𝑥) = 𝑄(𝑥) where 𝑃(𝑥) 𝑎𝑛𝑑 𝑄(𝑥) are polynomials and degree of 𝑃(𝑥) < 𝑄(𝑥) If degree of 𝑃(𝑥) ≥ 𝑄(𝑥) 𝑡ℎ𝑒𝑛 𝑑𝑜 𝑙𝑜𝑛𝑔 𝑑𝑖𝑣𝑖𝑠𝑖𝑜𝑛 𝑓𝑖𝑟𝑠𝑡 𝑃(𝑥) (𝑎𝑥 + 𝑏)(𝑐𝑥 + 𝑑)2 (𝑒𝑥 2 + 𝑓𝑥 + 𝑔) 𝐴 𝐵 𝐶 𝐷𝑥 + 𝐸 = + + + 2 2 (𝑎𝑥 + 𝑏) (𝑐𝑥 + 𝑑) (𝑐𝑥 + 𝑑) (𝑒𝑥 + 𝑓𝑥 + 𝑔) 𝑎 ∫ 𝑑𝑥 = 𝑎 𝑙𝑛|𝑥 + 𝑏| + 𝐶 𝑥+𝑏

Sequences & Series

(See Harold’s Series Cheat Sheet) lim 𝑎𝑛 = 𝐿 (Limit) 𝑛→∞

Sequence

Example: (𝑎𝑛 , 𝑎𝑛+1 , 𝑎𝑛+2 , …) 𝑎(1 − 𝑟 𝑛 ) 𝑎 = 𝑛→∞ 1−𝑟 1−𝑟

𝑆 = lim Geometric Series

Convergence Tests

Series Convergence Tests

Taylor Series

only if |𝑟| < 1 where 𝑟 is the radius of convergence and (−𝑟, 𝑟) is the interval of convergence (See Harold’s Series Convergence Tests Cheat Sheet) 1. Divergence or 𝑛𝑡ℎ Term

6. Ratio

2. Geometric Series

7. Root

3. p-Series

8. Direct Comparison

4. Alternating Series

9. Limit Comparison

5. Integral

10. Telescoping

(See Harold’s Taylor Series Cheat Sheet) 𝑓(𝑥) = 𝑃𝑛 (𝑥) + 𝑅𝑛 (𝑥) +∞

Taylor Series

=∑ 𝑛=0

𝑓 (𝑛) (𝑐) 𝑓 (𝑛+1) (𝑥 ∗ ) (𝑥 − 𝑐)𝑛 + (𝑥 − 𝑐)𝑛+1 𝑛! (𝑛 + 1)!

where 𝑥 ≤ 𝑥 ∗ ≤ 𝑐 (worst case scenario 𝑥 ∗ ) and lim 𝑅𝑛 (𝑥) = 0 𝑥→+∞


10

Harold’s Calculus Notes Cheat Sheet AP Calculus Limits

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