2. Calculus Cheat Sheet
Definition. L is called a limit of a function ƒ(x) at a point x=a when
∀ε>0∃δ: |x-a|<δ,|ƒ(x)-L|<ε
Notations. (lim_x→⋯)(ƒ(x))=L
Rules.
| (lim_x→a)(ƒ(x)+g(x))=(lim_x→a)(ƒ(x))+(lim_x→a)(g(x)) |
| (lim_x→a)(ƒ(x)*g(x))=(lim_x→a)(ƒ(x))⋅(lim_x→a)(g(x)) |
| (lim_x→a)(ƒ(x)/g(x))=(lim_x→a)(ƒ(x))/(lim_x→a)(g(x))↔(lim_x→a)(g(x))≠0 |
| (lim_x→a)(ƒ(x)n)=((lim_x→a)(ƒ(x)))n |
Common limits.
(lim_x→∞)(ℇx)=∞, (lim_x→-∞)(ℇx)=0
(lim_x→0)(sin(x)/x)=1
(lim_x→∞)(1+1/x)x=ℇ
(lim_x→∞)(1+x)(1/x)=ℇ
(lim_x→0)((1-cos(x))/x)=0
Continuity
Definition. A function ƒ is continuous at x=a if:
(lim_x→a)(ƒ(x))=ƒ(a)
Types of discontinuities:
Definition. ƒ(x)′ is a derivative for ƒ(x) when ƒ(x)′=(lim_Δ(x)→0)(ƒ(x+Δ(x))-ƒ(x))/Δ(x), when this limit exists.
Notations. ƒ(x)′ - Lagrange notation, d(ƒ)/d(x) - Leibniz notation.
Derivation rules.
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| (ƒ(x)⋅g(x))′=ƒ(x)′*g(x)+ƒ(x)g(x)′ |
| (ƒ(x)/g(x))′=ƒ(x)′*g(x)-ƒ(x)g(x)′*(g(x)) |
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Known Derivatives.
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| | d((log_a)(x))/d(x)=1/(x*ln(a)), x>0 |
| | d(asin(x))/d(x)=1/√(,1-x2) |
| | d(acos(x))/d(x)=1/√(,1-x2) |
| d(tan(x))/d(x)=1/(cos^2)(x) | |
Definition.
Indefinite Integral. A function F(x) is called an antiderivative of a continuous function ƒ(x) if F(x)′=ƒ(x). Indefinite integral of ƒ(x) represents a family of all functions F(x)+C, where C is a constant.
Definite Integral. Real number I is called a definite integral of a continuous ƒ(x) on the interval [a,b] if I=(lim_n→∞)((∑_i=0^N)(ƒ((x_i))*Δ(x))).
Notations.
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Evaluation of an
indefinite integral: | |
Rules.
| ∫(ƒ(x)+g(x))*d(x)=∫(ƒ(x)*d(x))+∫(g(x)*d(x)) |
| ∫(ƒ(x)*g(x)*d(x))=∫(ƒ(x))*d(x)⋅g(x)+∫(g(x)*d(x))⋅ƒ(x) |
Integration on empty interval | |
| (∫_a^b)(ƒ(x)*d(x))=-(∫_b^a)(ƒ(x)*d(x)) |
Known Indefinite Integrals.
(∫^)(xn*d(x))=(x(n+1))/(n+1)+C , n≠-1 | (∫_^)(1/x*d(x))=ln(|x|)+C |
| (∫_^)(sin(x)*d(x))=-cos(x)+C |
(∫_^)(cos(x)*d(x))=sin(x)+C | |
Integration Techniques
Fundamental Theorem of Calculus
Part I
If F(x)=(∫_a^x)(ƒ(t)*d(t)), then (F^′)(x)=ƒ(x)
Part II
If F is an antiderivative of ƒ, then
(∫_a^b)(ƒ(x)*d(x))=F(b)-F(a)
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