Copyright Β© 2015-2017 by Harold Toomey, WyzAnt Tutor 1
Haroldβs 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:
limπ₯βπ
π(π₯) = πΏ
means that for each π > 0 there exists a
πΏ > 0 such that
if 0 < |π₯ β π| < πΏ, then |π(π₯) β πΏ| < π Tip : Direct substitution: Plug in π(π) and see if it provides a legal answer. If so then L = π(π).
The Existence of a Limit The limit of π(π₯) as π₯ approaches a is L if and only if:
limπ₯βπβ
π(π₯) = πΏ
limπ₯βπ+
π(π₯) = πΏ
Definition of Continuity A function f is continuous at c if for every ν > 0 there exists a πΏ > 0 such that |π₯ β π| < πΏ and |π(π₯) β π(π)| < ν. Tip: Rearrange |π(π₯) β π(π)| to have |(π₯ β π)| as a factor. Since |π₯ β π| < πΏ we can find an equation that relates both πΏ and ν together.
Prove that π(π) = ππ β π is a continuous function.
|π(π₯) β π(π)| = |(π₯2 β 1) β (π2 β 1)| = |π₯2 β 1 β π2 + 1| = |π₯2 β π2| = |(π₯ + π)(π₯ β π)| = |(π₯ + π)| |(π₯ β π)|
Since |(π₯ + π)| β€ |2π| |π(π₯) β π(π)| β€ |2π||(π₯ β π)| < ν
So given ν > 0, we can choose πΉ = |π
ππ| πΊ > π in the
Definition of Continuity. So substituting the chosen πΏ for |(π₯ β π)| we get:
|π(π₯) β π(π)| β€ |2π| (|1
2π| ν) = ν
Since both conditions are met, π(π₯) is continuous. Two Special Trig Limits
ππππ₯β0
π ππ π₯
π₯= 1
ππππ₯β0
1 β πππ π₯
π₯= 0
Copyright Β© 2015-2017 by Harold Toomey, WyzAnt Tutor 2
Derivatives (See Larsonβs 1-pager of common derivatives)
Definition of a Derivative of a Function Slope Function
πβ²(π₯) = limββ0
π(π₯ + β) β π(π₯)
β
πβ²(π) = limπ₯βπ
π(π₯) β π(π)
π₯ β π
Notation for Derivatives πβ²(π₯), π(π)(π₯),ππ¦
ππ₯, π¦β²,
π
ππ₯[π(π₯)], π·π₯[π¦]
0. The Chain Rule
π
ππ₯[π(π(π₯))] = πβ²(π(π₯))πβ²(π₯)
ππ¦
ππ₯=
ππ¦
ππ‘Β·
ππ‘
ππ₯
1. The Constant Multiple Rule π
ππ₯[ππ(π₯)] = ππβ²(π₯)
2. The Sum and Difference Rule π
ππ₯[π(π₯) Β± π(π₯)] = πβ²(π₯) Β± πβ²(π₯)
3. The Product Rule π
ππ₯[ππ] = ππβ² + π πβ²
4. The Quotient Rule π
ππ₯[π
π] =
ππβ² β ππβ²
π2
5. The Constant Rule π
ππ₯[π] = 0
6a. The Power Rule π
ππ₯[π₯π] = ππ₯πβ1
6b. The General Power Rule π
ππ₯[π’π] = ππ’πβ1 π’β² π€βπππ π’ = π’(π₯)
7. The Power Rule for x π
ππ₯[π₯] = 1 (π‘βπππ π₯ = π₯1 πππ π₯0 = 1)
8. Absolute Value π
ππ₯[|π₯|] =
π₯
|π₯|
9. Natural Logorithm π
ππ₯[ln π₯] =
1
π₯
10. Natural Exponential π
ππ₯[ππ₯] = ππ₯
11. Logorithm π
ππ₯[logπ π₯] =
1
(ln π) π₯
12. Exponential π
ππ₯[ππ₯] = (ln π) ππ₯
13. Sine π
ππ₯[π ππ(π₯)] = cos(π₯)
14. Cosine π
ππ₯[πππ (π₯)] = βπ ππ(π₯)
15. Tangent π
ππ₯[π‘ππ(π₯)] = π ππ2(π₯)
16. Cotangent π
ππ₯[πππ‘(π₯)] = βππ π2(π₯)
17. Secant π
ππ₯[π ππ(π₯)] = π ππ(π₯) π‘ππ(π₯)
Copyright Β© 2015-2017 by Harold Toomey, WyzAnt Tutor 3
Derivatives (See Larsonβs 1-pager of common derivatives)
18. Cosecant π
ππ₯[ππ π(π₯)] = β ππ π(π₯) πππ‘(π₯)
19. Arcsine π
ππ₯[sinβ1(π₯)] =
1
β1 β π₯2
20. Arccosine π
ππ₯[cosβ1(π₯)] =
β1
β1 β π₯2
21. Arctangent π
ππ₯[tanβ1(π₯)] =
1
1 + π₯2
22. Arccotangent π
ππ₯[cotβ1(π₯)] =
β1
1 + π₯2
23. Arcsecant π
ππ₯[secβ1(π₯)] =
1
|π₯| βπ₯2 β 1
24. Arccosecant π
ππ₯[cscβ1(π₯)] =
β1
|π₯| βπ₯2 β 1
25. Hyperbolic Sine ( ππβπβπ
π)
π
ππ₯[π ππβ(π₯)] = cosh(π₯)
26. Hyperbolic Cosine (ππ+πβπ
π)
π
ππ₯[πππ β(π₯)] = π ππβ(π₯)
27. Hyperbolic Tangent π
ππ₯[π‘ππβ(π₯)] = π ππβ2(π₯)
28. Hyperbolic Cotangent π
ππ₯[πππ‘β(π₯)] = βππ πβ2(π₯)
29. Hyperbolic Secant π
ππ₯[π ππβ(π₯)] = β π ππβ(π₯) π‘ππβ(π₯)
30. Hyperbolic Cosecant π
ππ₯[ππ πβ(π₯)] = β ππ πβ(π₯) πππ‘β(π₯)
31. Hyperbolic Arcsine π
ππ₯[sinhβ1(π₯)] =
1
βπ₯2 + 1
32. Hyperbolic Arccosine π
ππ₯[coshβ1(π₯)] =
1
βπ₯2 β 1
33. Hyperbolic Arctangent π
ππ₯[tanhβ1(π₯)] =
1
1 β π₯2
34. Hyperbolic Arccotangent π
ππ₯[cothβ1(π₯)] =
1
1 β π₯2
35. Hyperbolic Arcsecant π
ππ₯[sechβ1(π₯)] =
β1
π₯ β1 β π₯2
36. Hyperbolic Arccosecant π
ππ₯[cschβ1(π₯)] =
β1
|π₯| β1 + π₯2
Position Function π (π‘) =1
2ππ‘2 + π£0π‘ + π 0
Velocity Function π£(π‘) = π β²(π‘) = ππ‘ + π£0
Acceleration Function π(π‘) = π£β²(π‘) = π β²β²(π‘)
Jerk Function π(π‘) = πβ²(π‘) = π£β²β²(π‘) = π (3)(π‘)
Copyright Β© 2015-2017 by Harold Toomey, WyzAnt Tutor 4
Analyzing the Graph of a Function
(See Haroldβs Illegals and Graphing Rationals Cheat Sheet)
x-Intercepts (Zeros or Roots) f(x) = 0 y-Intercept f(0) = y Domain Valid x values Range Valid y values Continuity No division by 0, no negative square roots or logs Vertical Asymptotes (VA) x = division by 0 or undefined
Horizontal Asymptotes (HA) limπ₯βββ
π(π₯) β π¦ and limπ₯ββ+
π(π₯) β π¦
Infinite Limits at Infinity limπ₯βββ
π(π₯) β β and limπ₯ββ+
π(π₯) β β
Differentiability Limit from both directions arrives at the same slope Relative Extrema Create a table with domains, f(x), f β(x), and f β(x)
Concavity If π β(π₯) β +, then cup up β
If π β(π₯) β β, then cup down β Points of Inflection f β(x) = 0 (concavity changes)
Graphing with Derivatives
Test for Increasing and Decreasing Functions
1. If f β(x) > 0, then f is increasing (slope up) β 2. If f β(x) < 0, then f is decreasing (slope down) β 3. If f β(x) = 0, then f is constant (zero slope) β
The First Derivative Test
1. If f β(x) changes from β to + at c, then f has a relative minimum at (c, f(c)) 2. If f β(x) changes from + to - at c, then f has a relative maximum at (c, f(c)) 3. If f β(x), is + c + or - c -, then f(c) is neither
The Second Deriviative Test Let f β(c)=0, and f β(x) exists, then
1. If f β(x) > 0, then f has a relative minimum at (c, f(c)) 2. If f β(x) < 0, then f has a relative maximum at (c, f(c)) 3. If f β(x) = 0, then the test fails (See 1π π‘ derivative test)
Test for Concavity 1. If f β(x) > 0 for all x, then the graph is concave up β 2. If f β(x) < 0 for all x, then the graph is concave down β
Points of Inflection Change in concavity
If (c, f(c)) is a point of inflection of f, then either 1. f β(c) = 0 or 2. f β does not exist at x = c.
Tangent Lines Genreal Form ππ₯ + ππ¦ + π = 0 Slope-Intercept Form π¦ = ππ₯ + π Point-Slope Form π¦ β π¦0 = π(π₯ β π₯0) Calculus Form π¦ = πβ²(π)(π₯ β π) + π(π)
Slope π =πππ π
ππ’π=
βπ¦
βπ₯=
π¦2 β π¦1
π₯2 β π₯1=
ππ¦
ππ₯= πβ²(π₯)
Copyright Β© 2015-2017 by Harold Toomey, WyzAnt Tutor 5
Differentiation & Differentials Rolleβs Theorem f is continuous on the closed interval [a,b], and f is differentiable on the open interval (a,b).
If f(a) = f(b), then there exists at least one number c in (a,b) such that fβ(c) = 0.
Mean Value Theorem If f meets the conditions of Rolleβs Theorem, then
πβ²(π) =π(π) β π(π)
π β π
π(π) = π(π) + (π β π)πβ²(π) Find βcβ.
Intermediate Value Therem f is a continuous function with an interval, [a, b], as its domain.
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.
Calculating Differentials Tanget line approximation
π(π₯ + βπ₯) β π(π₯) + ππ¦ = π(π₯) + πβ²(π₯) ππ₯
Example: β824
β π(π₯) = βπ₯4
, π(π₯ + βπ₯) = π(81 + 1)
Newtonβs Method Finds zeros of f, or finds c if f(c) = 0.
π₯π+1 = π₯π βπ(π₯π)
πβ²(π₯π)
Example: β824
β π(π₯) = π₯4 β 82 = 0, π₯π = 3
Related Rates
Steps to solve: 1. Identify the known variables and rates of change.
(π₯ = 15 π; π¦ = 20 π; π₯β² = 2π
π ; π¦β² = ? )
2. Construct an equation relating these quantities. (π₯2 + π¦2 = π2)
3. Differentiate both sides of the equation. (2π₯π₯β² + 2π¦π¦β² = 0)
4. Solve for the desired rate of change.
(π¦β² = βπ₯
π¦ π₯β²)
5. Substitute the known rates of change and quantities into the equation.
(π¦β² = β15
20β¦ 2 =
3
2 π
π )
LβHΓ΄pitalβs Rule
πΌπ limπ₯βπ
π(π₯) = limπ₯βπ
π(π₯)
π(π₯)=
{0
0,β
β, 0 β’ β, 1β, 00, β0, β β β} , ππ’π‘ πππ‘ {0β},
π‘βππ limπ₯βπ
π(π₯)
π(π₯)= lim
π₯βπ
πβ²(π₯)
πβ²(π₯)= lim
π₯βπ
πβ²β²(π₯)
πβ²β²(π₯)= β―
Copyright Β© 2015-2017 by Harold Toomey, WyzAnt Tutor 6
Summation Formulas
Sum of Powers
β π
π
π=1
= ππ
β π
π
π=1
=π(π + 1)
2=
π2
2+
π
2
β π2
π
π=1
=π(π + 1)(2π + 1)
6=
π3
3+
π2
2+
π
6
β π3
π
π=1
= (β π
π
π=1
)
2
=π2(π + 1)2
4=
π4
4+
π3
2+
π2
4
β π4
π
π=1
=π(π + 1)(2π + 1)(3π2 + 3π β 1)
30=
π5
5+
π4
2+
π3
3β
π
30
β π5
π
π=1
=π2(π + 1)2(2π2 + 2π β 1)
12=
π6
6+
π5
2+
5π4
12β
π2
12
β π6
π
π=1
=π(π + 1)(2π + 1)(3π4 + 6π3 β 3π + 1)
42
β π7
π
π=1
=π2(π + 1)2(3π4 + 6π3 β π2 β 4π + 2)
24
ππ(π) = β ππ
π
π=1
=(π + 1)π+1
π + 1β
1
π + 1β (
π + 1
π) ππ(π)
πβ1
π=0
Misc. Summation Formulas
β π(π + 1)
π
π=1
= β π2
π
π=1
+ β π
π
π=1
=π(π + 1)(π + 2)
3
β1
π(π + 1)
π
π=1
=π
π + 1
β1
π(π + 1)(π + 2)
π
π=1
=π(π + 3)
4(π + 1)(π + 2)
Copyright Β© 2015-2017 by Harold Toomey, WyzAnt Tutor 7
Numerical Methods
Riemann Sum
π0(π₯) = β« π(π₯) ππ₯π
π
= limβπββ0
β π(π₯πβ) βπ₯π
π
π=1
where π = π₯0 < π₯1 < π₯2 < β― < π₯π = π and βπ₯π = π₯π β π₯πβ1
and βπβ = πππ₯{βπ₯π} Types:
β’ Left Sum (LHS) β’ Middle Sum (MHS) β’ Right Sum (RHS)
Midpoint Rule (Middle Sum)
π0(π₯) = β« π(π₯) ππ₯π
π
β β π(οΏ½Μ οΏ½π) βπ₯ =
π
π=1
βπ₯[π(οΏ½Μ οΏ½1) + π(οΏ½Μ οΏ½2) + π(οΏ½Μ οΏ½3) + β― + π(οΏ½Μ οΏ½π)]
where βπ₯ =πβπ
π
and οΏ½Μ οΏ½π =1
2(π₯πβ1 + π₯π) = ππππππππ‘ ππ [π₯πβ1, π₯π]
Error Bounds: |πΈπ| β€ πΎ(πβπ)3
24π2
Trapezoidal Rule
π1(π₯) = β« π(π₯) ππ₯π
π
β
βπ₯
2[π(π₯0) + 2π(π₯1) + 2π(π₯3) + β― + 2π(π₯πβ1)
+ π(π₯π)]
where βπ₯ =πβπ
π
and π₯π = π + πβπ₯
Error Bounds: |πΈπ| β€ πΎ(πβπ)3
12π2
Simpsonβs Rule
π2(π₯) = β« π(π₯)ππ₯π
π
β
βπ₯
3[π(π₯0) + 4π(π₯1) + 2π(π₯2) + 4π(π₯3) + β―
+ 2π(π₯πβ2) + 4π(π₯πβ1) + π(π₯π)] Where n is even
and βπ₯ =πβπ
π
and π₯π = π + πβπ₯
Error Bounds: |πΈπ| β€ πΎ(πβπ)5
180π4
TI-84 Plus
[MATH] fnInt(f(x),x,a,b), [MATH] [1] [ENTER]
Example: [MATH] fnInt(x^2,x,0,1)
β« π₯2 ππ₯1
0
=1
3
TI-Nspire CAS
[MENU] [4] Calculus [3] Integral [TAB] [TAB]
[X] [^] [2] [TAB] [TAB] [X] [ENTER]
Shortcut: [ALPHA] [WINDOWS] [4]
Copyright Β© 2015-2017 by Harold Toomey, WyzAnt Tutor 8
Integration (See Haroldβs Fundamental Theorem of Calculus Cheat Sheet)
Basic Integration Rules Integration is the βinverseβ of differentiation, and vice versa.
β« πβ²(π₯) ππ₯ = π(π₯) + πΆ
π
ππ₯β« π(π₯) ππ₯ = π(π₯)
π(π₯) = 0 β« 0 ππ₯ = πΆ
π(π₯) = π = ππ₯0 β« π ππ₯ = ππ₯ + πΆ
1. The Constant Multiple Rule β« π π(π₯) ππ₯ = π β« π(π₯) ππ₯
2. The Sum and Difference Rule β«[π(π₯) Β± π(π₯)] ππ₯ = β« π(π₯) ππ₯ Β± β« π(π₯) ππ₯
The Power Rule π(π₯) = ππ₯π
β« π₯π ππ₯ =π₯π+1
π + 1+ πΆ, π€βπππ π β β1
πΌπ π = β1, π‘βππ β« π₯β1 ππ₯ = ln|π₯| + πΆ
The General Power Rule
If π’ = π(π₯), πππ π’β² =π
ππ₯π(π₯) then
β« π’π π’β²ππ₯ =π’π+1
π + 1+ πΆ, π€βπππ π β β1
Reimann Sum β π(ππ)
π
π=1
βπ₯π , π€βπππ π₯πβ1 β€ ππ β€ π₯π
βββ = βπ₯ =π β π
π
Definition of a Definite Integral Area under curve
limββββ0
β π(ππ)
π
π=1
βπ₯π = β« π(π₯) ππ₯π
π
Swap Bounds β« π(π₯) ππ₯π
π
= β β« π(π₯) ππ₯π
π
Additive Interval Property β« π(π₯) ππ₯π
π
= β« π(π₯) ππ₯π
π
+ β« π(π₯) ππ₯π
π
The Fundamental Theorem of Calculus
β« π(π₯) ππ₯π
π
= πΉ(π) β πΉ(π)
The Second Fundamental Theorem of Calculus
π
ππ₯ β« π(π‘) ππ‘
π₯
π
= π(π₯)
π
ππ₯ β« π(π‘) ππ‘
π(π₯)
π
= π(π(π₯))πβ²(π₯)
π
ππ₯β« π(π‘) ππ‘
β(π₯)
π(π₯)
= π(β(π₯))ββ²(π₯) β π(π(π₯))πβ²(π₯)
Mean Value Theorem for Integrals
β« π(π₯) ππ₯π
π
= π(π)(π β π) Find βπβ.
The Average Value for a Function
1
π β πβ« π(π₯) ππ₯
π
π
Copyright Β© 2015-2017 by Harold Toomey, WyzAnt Tutor 9
Integration Methods 1. Memorized See Larsonβs 1-pager of common integrals
2. U-Substitution
β« π(π(π₯))πβ²(π₯)ππ₯ = πΉ(π(π₯)) + πΆ
Set π’ = π(π₯), then ππ’ = πβ²(π₯) ππ₯
β« π(π’) ππ’ = πΉ(π’) + πΆ
π’ = _____ ππ’ = _____ ππ₯
3. Integration by Parts
β« π’ ππ£ = π’π£ β β« π£ ππ’
π’ = _____ π£ = _____ ππ’ = _____ ππ£ = _____
Pick βπ’β using the LIATED Rule:
L β Logarithmic : ln π₯ , logπ π₯ , ππ‘π.
I β Inverse Trig.: tanβ1 π₯ , secβ1 π₯ , ππ‘π.
A β Algebraic: π₯2, 3π₯60, ππ‘π.
T β Trigonometric: sin π₯ , tan π₯ , ππ‘π.
E β Exponential: ππ₯ , 19π₯ , ππ‘π.
D β Derivative of: ππ¦ππ₯
β
4. Partial Fractions
β«π(π₯)
π(π₯) ππ₯
where π(π₯) πππ π(π₯) are polynomials
Case 1: If degree of π(π₯) β₯ π(π₯) then do long division first
Case 2: If degree of π(π₯) < π(π₯) then do partial fraction expansion
5a. Trig Substitution for βππ β ππ
β« βπ2 β π₯2 ππ₯
Substutution: π₯ = π sin π Identity: 1 β π ππ2 π = πππ 2 π
5b. Trig Substitution for βππ β ππ
β« βπ₯2 β π2 ππ₯
Substutution: π₯ = π sec π Identity: π ππ2 π β 1 = π‘ππ2 π
5c. Trig Substitution for βππ + ππ
β« βπ₯2 + π2 ππ₯
Substutution: π₯ = π tan π Identity: π‘ππ2 π + 1 = π ππ2 π
6. Table of Integrals CRC Standard Mathematical Tables book
7. Computer Algebra Systems (CAS) TI-Nspire CX CAS Graphing Calculator
TI βNspire CAS iPad app
8. Numerical Methods Riemann Sum, Midpoint Rule, Trapezoidal Rule, Simpsonβs
Rule, TI-84
9. WolframAlpha Google of mathematics. Shows steps. Free.
www.wolframalpha.com
Copyright Β© 2015-2017 by Harold Toomey, WyzAnt Tutor 10
Partial Fractions (See Haroldβs Partial Fractions Cheat Sheet)
Condition
π(π₯) =π(π₯)
π(π₯)
where π(π₯) πππ π(π₯) are polynomials and degree of π(π₯) < π(π₯)
If degree of π(π₯) β₯ π(π₯) π‘βππ ππ ππππ πππ£ππ πππ ππππ π‘
Example Expansion
π(π₯)
(ππ₯ + π)(ππ₯ + π)2(ππ₯2 + ππ₯ + π)
=π΄
(ππ₯ + π)+
π΅
(ππ₯ + π)+
πΆ
(ππ₯ + π)2+
π·π₯ + πΈ
(ππ₯2 + ππ₯ + π)
Typical Solution β«π
π₯ + π ππ₯ = π ππ|π₯ + π| + πΆ
Sequences & Series (See Haroldβs Series Cheat Sheet)
Sequence
limπββ
ππ = πΏ (Limit)
Example: (ππ, ππ+1, ππ+2, β¦)
Geometric Series
π = limπββ
π(1 β ππ)
1 β π =
π
1 β π
only if |π| < 1
where π is the radius of convergence and (βπ, π) is the interval of convergence
Convergence Tests (See Haroldβs Series Convergence Tests Cheat Sheet)
Series Convergence Tests
1. Divergence or ππ‘β Term
2. Geometric Series
3. p-Series
4. Alternating Series
5. Integral
6. Ratio
7. Root
8. Direct Comparison
9. Limit Comparison
10. Telescoping
Taylor Series (See Haroldβs Taylor Series Cheat Sheet)
Taylor Series
π(π₯) = ππ(π₯) + π π(π₯)
= βπ(π)(π)
π!
+β
π=0
(π₯ β π)π + π(π+1)(π₯β)
(π + 1)!(π₯ β π)π+1
where π₯ β€ π₯β β€ π (worst case scenario π₯β)
and limπ₯β+β
π π(π₯) = 0