Analysis I EPFL

Theorems and results covering real numbers, functions, sequences,convergent/divergent sequences, limits, subsequences, continuous functions, series of real numbers, convergent/divergent series, absolute convergence, derivatives, class C^k, mean value theorem, Taylor's theorem, Taylor series, Riemann integrals, indefinite integrals, intermediate value theorem. Based on Analysis I notes by Roberto Svaldi, as taught at EPFL in 2020-2021.

109 flashcards · Mathematics · by Athena Papkouf

upper/lower bound
a∈Ra \in \mathbb{R}a∈R is an upper (resp. lower ) bound of S if x ≤ a (resp. x ≥ a) holds for all x ∈ S. The upper and lower bounds are not unique.
Maximum/minimum
The maximum (resp. minimum) of S denoted by max S (resp. min S) is an upper (resp. lower) bound of S which is contained in S
Supremum/infimum
The supremum Sup S (resp. infimum Inf S) is the smallest upper bound (resp. largest lower bound), if it exists at all.
Infimum axiom
Each non-empty subset of R+∗\mathbb{R}^*_+R+∗ admits an infimum
bounded set
If S has an upper/lower bound then it is called bounded from above/from below. If S is bounded both from above and below, then it is called bounded.
the sequence xnx_{n}xn is bounded from below bounded from above bounded \begin{array}{c}\text { bounded from below } \\ \text { bounded from above } \\ \text { bounded }\end{array} bounded from below bounded from above bounded
if the set xnx_{n}xn of its values is bounded from below bounded from above bounded \begin{array}{c}\text { bounded from below } \\ \text { bounded from above } \\ \text { bounded }\end{array} bounded from below bounded from above bounded
A sequence (xn)(x_{n})(xn) is constant increasing strictly increasing decreasing strictly decreasing \begin{array}{c}\text { constant } \\ \text { increasing } \\ \text { strictly increasing } \\ \text { decreasing } \\ \text { strictly decreasing }\end{array} constant increasing strictly increasing decreasing strictly decreasing
if for each n∈Nn \in \mathbb{N}n∈N x_{n+1}\end{array}"> xn=xn+1xn≤xn+1xn<xn+1xn≥xn+1xn>xn+1\begin{array}{l}x_{n}=x_{n+1} \\ x_{n} \leq x_{n+1} \\ x_{n}<x_{n+1} \\ x_{n} \geq x_{n+1} \\ x_{n}>x_{n+1}\end{array}xn=xn+1xn≤xn+1xn<xn+1xn≥xn+1xn>xn+1
Bernoulli inequality
If 1"> q>1q>1q>1 is a real number, then qn≥1+n(q−1)q^{n} \geq 1+n(q-1)qn≥1+n(q−1)
Limit of a sequence
Let (xn)(x_{n})(xn) be a sequence. We say that (xn)(x_{n})(xn) is convergent to some x∈Rx \in \mathbb{R}x∈R if for each 0<ε∈R0<\varepsilon \in \mathbb{R}0<ε∈R there exists nε∈Nn_{\varepsilon} \in \mathbb{N}nε∈N such that for all n≥nε,∣xn−x∣≤ε.n \geq n_{\varepsilon},\left|x_{n}-x\right| \leq \varepsilon .n≥nε,∣xn−x∣≤ε. If the xxx in the definition exists then it is called the limit of xnx_{n}xn and we denote it by lim⁡n→∞xn\lim _{n \rightarrow \infty} x_{n}limn→∞xn . If (xn)(x_{n})…
Squeeze Theorem
Let (xn),(yn),(zn) (x_{n}), (y_{n}), (z_{n}) (xn),(yn),(zn) be three sequences satisfying: (1) (xn)(x_n)(xn) and (zn)(z_n)(zn) are convergent, with the same limit, say a, (2) there is an n0∈Nn_0 \in \Nn0∈N , such that for all integers n≥n0,xn≤yn≤znn ≥ n_0 , x_n ≤ y_n ≤ z_nn≥n0,xn≤yn≤zn then (yn)(y_n) (yn) is also convergent, with limit a.
Quotient criterion
Let (xn)(x_n)(xn) be a sequence such that q=lim⁡n→∞∣xn+1∣∣xn∣q=\lim _{n \rightarrow \infty} \frac{\left|x_{n+1}\right|}{\left|x_{n}\right|}q=limn→∞∣xn∣∣xn+1∣ exists. If q<1q < 1q<1 , then (xn)(x_n) (xn) converges to 0, and if 1"> q>1q > 1q>1 , then (xn)(x_n) (xn) diverges.
finding the limit of a recursive sequence
(1) assuming that there exists a limit, computing what it can be using the algebraic rules of limits (2) showing some kind of upper or/and lower bounds that exclude all but one of the possibilities for the limit (obtained in the previous point), (3) showing that lim⁡n→∞∣xn−x∣=0\lim _{n \rightarrow \infty}\left|x_{n}-x\right|=0limn→∞∣xn−x∣=0 , where xxx is the only possibility left for the limit by the above points.
Bolzano-Weierstrass
Every bounded sequence contains a convergent sub- sequence.
Cauchy convergence
A sequence (xn)(x_n)(xn) is Cauchy convergent (or equivalently it is a Cauchy sequence), if for every 0<ϵ∈R0 < \epsilon ∈ \R0<ϵ∈R there is an nε∈Nn_ε ∈ \N nε∈N such that for every integer n,m≥nε,∣xn−xm∣≤εn, m ≥ n_ε , |x_n − x_m | ≤ εn,m≥nε,∣xn−xm∣≤ε .
Completeness Property
A sequence which is bounded and monotone for >1"> n>>1n>>1n>>1 has a limit.
infimum alternative definition
S⊆R,a∈RS \subseteq \R, a \in \RS⊆R,a∈R aaa is the infimum of SSS if : aaa is a lower bound of SSS 0~\exists s_{\epsilon} \in S"> ∀ϵ>0 ∃sϵ∈S\forall \epsilon>0~\exists s_{\epsilon} \in S∀ϵ>0 ∃sϵ∈S s.t. sϵ<a+ϵs_{\epsilon}<a+\epsilonsϵ<a+ϵ
integral part of xxx
The integral part [x][x][x] of xxx is the largest integer that is ≤x\le x≤x .
fractional part of xxx
{x}:=x−[x]\{x\}:=x-[x]{x}:=x−[x]
Archimede’s property
Let x,yx,yx,y be real numbers, with 0,~y\ge 0"> x>0, y≥0x>0,~y\ge 0x>0, y≥0 . Then there exists n∈Nn \in \Nn∈N such that y"> nx>ynx>ynx>y .
Binomial formula
(x+y)n=∑j=0n(nj)xjyn−j(x+y)^{n}=\sum_{j=0}^{n}\left(\begin{array}{l} n \\ j \end{array}\right) x^{j} y^{n-j}(x+y)n=∑j=0n(nj)xjyn−j
Binomial coefficient
(nj):=n!j!(n−j)!\left(\begin{array}{l} n \\ j \end{array}\right) := \frac{n !}{j !(n-j) !}(nj):=j!(n−j)!n!
lim⁡n→∞xn=0&(yn) bounded \lim _{n \rightarrow \infty} x_{n}=0 \quad \& \quad\left(y_{n}\right) \text { bounded }limn→∞xn=0&(yn) bounded
⇒lim⁡n→∞xn⋅yn=0\Rightarrow \lim _{n \rightarrow \infty} x_{n} \cdot y_{n}=0⇒limn→∞xn⋅yn=0
lim⁡n→∞xn=∞\lim _{n \rightarrow \infty} x_{n} = \inftylimn→∞xn=∞
∀C∈R∃nC∈N s.t. ∀n⩾nc⇒xn⩾C\forall C \in \mathbb{R} \quad \exists n_{C} \in \mathbb{N} \text { s.t. } \\ \forall n \geqslant n_{c} \Rightarrow x_{n} \geqslant C∀C∈R∃nC∈N s.t. ∀n⩾nc⇒xn⩾C
Squeeze theorem at ∞\infty∞
xn≤yn∀n≥n0lim⁡n→∞xn=+∞⇒lim⁡n→∞yn=+∞lim⁡n→∞yn=−∞⇒lim⁡n→∞xn=−∞\begin{array}{l} x_{n} \leq y_{n} \quad \forall n \geq n_{0} \\ \lim _{n \rightarrow \infty} x_{n}=+\infty \Rightarrow \lim _{n \rightarrow \infty} y_{n}=+\infty \\ \lim _{n \rightarrow \infty} y_{n}=-\infty \Rightarrow \lim _{n \rightarrow \infty} x_{n}=-\infty \end{array}xn≤yn∀n≥n0limn→∞xn=+∞⇒limn→∞yn=+∞limn→∞yn=−∞⇒limn→∞xn=−∞
A series Sn=∑k=0nxkS_{n}=\sum_{k=0}^{n} x_{k}Sn=∑k=0nxk is
convergent divergent absolute convergent }, if {(Sn) is convergent (Sn) is divergent (Sn′) is convergent, where Sn′=∑k=0n∣xk∣\left.\begin{array}{r} \text { convergent } \\ \text { divergent } \\ \text { absolute convergent } \end{array}\right\}, \text { if }\left\{\begin{array}{l} \left(S_{n}\right) \text { is convergent } \\ \left(S_{n}\right) \text { is divergent } \\ \left(S_{n}^{\prime}\right) \text { is convergent, where } S_{n}^{\prime}=\sum_{k=0}^{n}\left|x_{k}\right| \end{array}\right.…
Cauchy convergence for series
Sn=∑k=0nxkS_{n}=\sum_{k=0}^{n} x_{k}Sn=∑k=0nxk is convergent if and only if for every 0<ε∈R0 < ε \in \R0<ε∈R , there is an nϵ∈Nn_\epsilon \in \Nnϵ∈N such that for every integer n ≥ n_ε"> m>n≥nε m > n ≥ n_εm>n≥nε ∣∑k=n+1mxk∣<ε\left|\sum_{k=n+1}^{m} x_{k}\right|<\varepsilon∣∣∑k=n+1mxk∣∣<ε
Squeeze theorem for series
Assume there is an n0∈Nn_0 \in \Nn0∈N such that 0≤xn≤yn0 ≤ x_n ≤ y_n0≤xn≤yn for every integer n≥n0n ≥ n_0n≥n0 . Then: (1) If ∑k=0∞yk\sum_{k=0}^{\infty} y_{k}∑k=0∞yk is convergent, then ∑k=0∞xk\sum_{k=0}^{\infty} x_{k}∑k=0∞xk is also convergent. (2) If ∑k=0∞xk\sum_{k=0}^{\infty} x_{k}∑k=0∞xk is divergent, then ∑k=0∞yk\sum_{k=0}^{\infty} y_{k}∑k=0∞yk is also divergent.
boundedness and convergence of series
∑k=0∞xk={ convergent if Sn=∑k=0nxk is bounded +∞(divergent) if Sn is not bounded.\sum_{k=0}^{\infty} x_{k}=\left\{\begin{array}{ll} \text { convergent } & \text { if } S_{n}=\sum_{k=0}^{n} x_{k} \text { is bounded } \\ +\infty(\text {divergent}) & \text { if } S_{n} \text { is not bounded.} \end{array}\right.∑k=0∞xk={ convergent +∞(divergent) if Sn=∑k=0nxk is bounded if Sn is not bounded.
Leibniz criterion
if (xk)(x_k)(xk) is decreasing and lim⁡n→∞xk=0\lim _{n \rightarrow \infty} x_{k}=0limn→∞xk=0 , then ∑k=0∞(−1)kxk\sum_{k=0}^{\infty}(-1)^{k} x_{k}∑k=0∞(−1)kxk is convergent.
Cauchy convergence/divergence criterion
(1) If (∣xn∣n)\left(\sqrt[n]{\left|x_{n}\right|}\right)(n∣xn∣ ) is bounded and lim⁡n→∞sup⁡∣xn∣n<1,\lim _{n \rightarrow \infty} \sup \sqrt[n]{\left|x_{n}\right|}<1,limn→∞supn∣xn∣ <1, then ∑k=0∞xn\sum_{k=0}^{\infty} x_{n}∑k=0∞xn is absolutely convergent. (2) If (∣xn∣n)\left(\sqrt[n]{\left|x_{n}\right|}\right)(n∣xn∣ ) is bounded and 1"> lim⁡n→∞sup⁡∣xn∣n>1\lim _{n \rightarrow \infty} \sup \sqrt[n]{\left|x_{n}\right|}>1limn→∞supn∣xn∣ >1 or if (∣xn∣n)\left(\sqrt[n]{\left|x_{n}…
Alembert's criterion
Let (xn)\left(x_{n}\right)(xn) be a sequence such that (∣xn+1∣∣xn∣)\left(\frac{\left|x_{n+1}\right|}{\left|x_{n}\right|}\right)(∣xn∣∣xn+1∣) is convergent and let the limit be ρ\rhoρ . (1) if ρ<1,\rho<1,ρ<1, then ∑k=0∞xn\sum_{k=0}^{\infty} x_{n}∑k=0∞xn is absolutely convergent. (2) if 1"> ρ>1\rho > 1ρ>1 then ∑k=0∞xn\sum_{k=0}^{\infty} x_{n}∑k=0∞xn is divergent.
Even function
a function f:R→Rf: \mathbb{R} \rightarrow \mathbb{R}f:R→R is even if its graph can be reflected across the y-axis, i.e. f(x)=f(−x),∀x∈Rf(x)=f(-x), \forall x \in \mathbb{R}f(x)=f(−x),∀x∈R .
Odd function
a function f:R→Rf: \mathbb{R} \rightarrow \mathbb{R}f:R→R is even if its graph can be reflected across the origin, i.e. f(x)=−f(−x),∀x∈Rf(x)=-f(-x), \forall x \in \mathbb{R}f(x)=−f(−x),∀x∈R .
periodic function
a function f:R→Rf: \mathbb{R} \rightarrow \mathbb{R}f:R→R is periodic if there exists a real number 0"> P>0P >0P>0 , which we call the period, such that f(x)=f(x+P)∀x∈Rf(x) =f(x+P) \forall x \in \Rf(x)=f(x+P)∀x∈R .
Upper and lower bound of a function
A number y0y_0y0 is an upper (resp. lower) bound for fff if y0y_0y0 is an upper (resp. lower) bound for the range R(f)R(f)R(f) of fff , R(f)⊆R.R(f)⊆R.R(f)⊆R.
Bounded function
fff is bounded (resp. bounded from above, from below) if R(f)R(f)R(f) is bounded (resp. bounded from above, from below).
Supremum and infimum of function
A number y0y_0y0 is a supremum (resp. infimum) for fff , if y0y_0y0 is the supremum (resp. infimum) of the range R(f)R(f) R(f) of fff .
Increasing function
The function fff is increasing (resp. strictly increasing) if ∀x,y ∈E\forall x,y ~ \in E∀x,y ∈E such that x<yx < yx<y , then f(x)≤f(y)f(x)≤f(y)f(x)≤f(y) (resp. f(x)<f(y)f(x)< f(y)f(x)<f(y) ).
Decreasing function
The function fff is decreasing (resp. strictly decreasing) if ∀x,y ∈E\forall x,y ~ \in E∀x,y ∈E such that x<yx < yx<y , then f(x)≥f(y)f(x) \geq f(y)f(x)≥f(y) (resp. f(y)"> f(x)>f(y)f(x) > f(y)f(x)>f(y) ).
A function f:E→Rf : E → Rf:E→R is defined on a pointed neighborhood of x0∈Rx_0 \in \Rx0∈R
if there is an interval of the form ]x0−α,x0+α[]x_0 − α, x_0 + α[]x0−α,x0+α[ contained in E∪x0E \cup x_0E∪x0 .
limit of a function
Assume f:E→Rf : E → R f:E→R is defined on a pointed neighborhood of x0∈Rx_0 \in Rx0∈R . Then, lim⁡x→x0f(x)=l\lim _{x \rightarrow x_{0}} f(x)=llimx→x0f(x)=l if one of the following two equivalent definitions holds: (1) For every 0<ε∈R0<\varepsilon \in \mathbb{R}0<ε∈R there is a 0<δ∈R0<\delta \in \mathbb{R}0<δ∈R such that: ∀x∈E:0<∣x−x0∣≤δ⇒∣f(x)−l∣≤ε\forall x \in E: 0<\left|x-x_{0}\right| \leq \delta \Rightarrow|f(x)-l| \leq \varepsilon∀x∈E:0<∣x−x0∣≤δ⇒∣f(x)−l∣≤ε (2) For every…
Function continuous at x0x_0x0
Assume f:E→Rf: E \rightarrow \mathbb{R}f:E→R is defined on a ball around x0∈Rx_{0} \in \mathbb{R}x0∈R . Then, fff is continuous at x0x_{0}x0 if and only if one of the following two equivalent definitions hold: (1) For every 0<ε∈R0<\varepsilon \in \mathbb{R}0<ε∈R there is a 0<δ∈R0<\delta \in \mathbb{R}0<δ∈R such that: ∀x∈E:∣x−x0∣≤δ⇒∣f(x)−f(x0)∣≤ε\forall x \in E:\left|x-x_{0}\right| \leq \delta \Rightarrow\left|f(x)-f\left(x_{0}\right)\right| \leq \varepsilon∀x∈E:∣x−x0∣≤δ⇒∣f(x)−f(x…
Squeeze theorem for functions
Let f and g be two functions such that a pointed neighborhood of x0x_0x0 is in the domain of both f and g. Assume that the limits of f and g at x0x_0x0 exist and they are l and k, respectively. Then, if there is a third function h(x) such that there is also a pointed neighborhood of x0x_0x0 in the domain of h, and: (i) on some pointed neighborhood of x0x_0x0 we have f(x)≤h(x)≤g(x)f (x) ≤ h(x) ≤ g(x)f(x)≤h(x)≤g(x) , and (ii) l=kl = kl=k , then lim⁡h(x)=l\lim h(x) = llimh(x)=l .
If f,g:E→Rf, g : E → Rf,g:E→R are continuous functions at x0∈Ex_0 \in Ex0∈E , then so are (8 cases)
1) αf+βgαf + βgαf+βg for any α,β∈Rα, β \in \Rα,β∈R , (2) f⋅gf · gf⋅g , and (3) if g∣Eg|_Eg∣E is nowhere zero (meaning that for all x∈E:g(x)≠0x \in E : g(x) \neq 0x∈E:g(x)̸=0 ), then fg\frac{f}{g}gf too. (4) ∣f∣|f|∣f∣ , (5) max⁡{f,g},\max \{f, g\},max{f,g}, where max⁡{f,g}(x):=max⁡{f(x),g(x)} \max \{f, g\}(x):=\max \{f(x), g(x)\}max{f,g}(x):=max{f(x),g(x)} (6) min⁡{f,g}\min \{f, g\}min{f,g} ( defined similarly) (7) f+:=max⁡{f,0}f^{+}:=\max \{f, 0\}f+:=max{f,0} (8) f−:=min⁡{f,0}f^{-}:=\min \{f…
continuity of composition of functions at a point x0x_0x0
If f:E→Rf: E \rightarrow \mathbb{R}f:E→R and g:G→Rg: G \rightarrow \mathbb{R}g:G→R be two functions, such that (1) f(E)⊆Gf(E) \subseteq Gf(E)⊆G , (2) fff is continuous at x0x_{0}x0 , (3) ggg is continuous at y0:=f(x0)y_{0}:=f\left(x_{0}\right)y0:=f(x0) , then g∘fg \circ fg∘f is continuous at x0x_{0}x0 .
limit of composition of functions
Let f:E→Rf: E \rightarrow \mathbb{R}f:E→R and g:G→Rg: G \rightarrow \mathbb{R}g:G→R be functions and let x0∈Ex_{0} \in Ex0∈E be a point such that (1) f(E)⊂Gf(E) \subset Gf(E)⊂G (2) lim⁡x→x0f(x0)=y0\lim _{x \rightarrow x_{0}} f\left(x_{0}\right)=y_{0}limx→x0f(x0)=y0 (3) lim⁡y→y0g(y0)=l\lim _{y \rightarrow y_{0}} g\left(y_{0}\right)=llimy→y0g(y0)=l (4) there is a pointed neighborhood B(x0,r)\{x0}⊂EB\left(x_{0}, r\right) \backslash\left\{x_{0}\right\} \subset EB(x0,r)\{x0}⊂E such that fo…
Addition rule for infinite limits
Let x0x_0x0 ∈R\in \R∈R , or ±∞\pm \infty±∞ , and let f,g:E→Rf, g : E \to Rf,g:E→R be functions. If ∘lim⁡x→x0f(x)=+∞(resp.−∞), and ∘g(x) is bounded from below (resp. above) then lim⁡x→x0(f+g)(x)=+∞( resp. −∞)\begin{aligned} &\circ \lim _{x \rightarrow x_{0}} f(x)=+\infty(\text {resp.}-\infty), \text { and }\\ &\circ g(x) \text { is bounded from below (resp. above) }\\ &\text { then } \lim _{x \rightarrow x_{0}}(f+g)(x)=+\infty(\text { resp. }-\infty) \end{aligned}∘x→x0limf(x)=+∞(…
Product rule for infinite limits
Let x0x_0x0 ∈R\in \R∈R , or ±∞\pm \infty±∞ , and let f,g:E→Rf, g : E \to Rf,g:E→R be functions. If ∘lim⁡x→x0∣f(x)∣=+∞\circ \lim _{x \rightarrow x_{0}}|f(x)|=+\infty∘limx→x0∣f(x)∣=+∞ ∘∣g(x)∣\circ|g(x)|∘∣g(x)∣ is bounded from below by a positive number (that is, there is a 0"> δ>0\delta>0δ>0 such that ∣g(x)∣≥δ|g(x)| \geq \delta∣g(x)∣≥δ ∀x∈E\forall x \in E∀x∈E , and 0"> ∘f(x)g(x)>0\circ f(x) g(x)>0∘f(x)g(x)>0 (resp. <0<0<0 ) ∀x∈E\forall x \in E∀x∈E , then lim⁡x→x0f(x)g(x)=+∞\l…
First division rule for infinite limits
∘f(x)\circ f(x)∘f(x) is bounded, ∘g(x)\circ g(x)∘g(x) is nowhere zero, and ∘lim⁡x→x0∣g(x)∣=+∞\circ \lim _{x \rightarrow x_{0}}|g(x)|=+\infty∘limx→x0∣g(x)∣=+∞ Then lim⁡x→x0f(x)g(x)=0\lim _{x \rightarrow x_{0}} \frac{f(x)}{g(x)}=0limx→x0g(x)f(x)=0
Second division rule for infinite limits
If lim⁡x→x0g(x)=0\lim _{x \rightarrow x_{0}} g(x)=0limx→x0g(x)=0 ∣f(x)∣|f(x)|∣f(x)∣ is bounded from below by a positive number ( 0 ~ s.t. ~ |f(x)| \geq \delta ~ \forall x \in E"> ∃δ>0 s.t. ∣f(x)∣≥δ ∀x∈E\exists \delta > 0 ~ s.t. ~ |f(x)| \geq \delta ~ \forall x \in E∃δ>0 s.t. ∣f(x)∣≥δ ∀x∈E ), and 0"> ∘f(x)/g(x)>0\circ f(x) / g(x)>0∘f(x)/g(x)>0 (resp. <0<0<0 ) for all x∈Ex \in Ex∈E , then lim⁡x→x0f(x)g(x)=+∞\lim _{x \rightarrow x_{0}} \frac{f(x)}{g(x)}=+\inftylimx→x0g(x)f(x)…
Squeeze for infinite limits
If f(x)≤g(x)f(x) \leq g(x)f(x)≤g(x) and ∘\circ∘ if lim⁡x→x0f(x)=+∞\lim _{x \rightarrow x_{0}} f(x)=+\inftylimx→x0f(x)=+∞ , then lim⁡x→x0g(x)=+∞\lim _{x \rightarrow x_{0}} g(x)=+\inftylimx→x0g(x)=+∞ ∘\circ∘ if lim⁡x→x0g(x)=−∞\lim _{x \rightarrow x_{0}} g(x)=-\inftylimx→x0g(x)=−∞ , then lim⁡x→x0f(x)=−∞\lim _{x \rightarrow x_{0}} f(x)=-\inftylimx→x0f(x)=−∞
One sided limits
Let f:E→Rf: E \rightarrow \mathbb{R}f:E→R be a function, such that x0∈E,x_{0} \in E,x0∈E, and fff is defined on the left (resp. right) of x0x_{0}x0 . Let lll be either a real number or ±∞\pm \infty±∞ . Then, lim⁡x→x0−f(x)=l\lim _{x \rightarrow x_{0}^{-}} f(x)=llimx→x0−f(x)=l (resp. lim⁡x→x0+f(x)=l\lim _{x \rightarrow x_{0}^{+}} f(x)=llimx→x0+f(x)=l ) if for all sequences (xn)⊂{x∈E∣x<x0}(\left(x_{n}\right) \subset\left\{x \in E \mid x<x_{0}\right\}\left(\right.(xn)⊂{x∈E∣x<x0}( (xn)…
Limits of monotone functions
If f:E→Rf : E \to Rf:E→R is monotone, then at each point x0∈Ex_0 \in E x0∈E (1) if fff is defined on the left of x0,lim⁡x→x0−f(x)x_{0}, \lim _{x \rightarrow x_{0}^{-}} f(x)x0,limx→x0−f(x) exists, (2) if fff is defined on the right of x0,lim⁡x→x0+f(x)x_{0}, \lim _{x \rightarrow x_{0}^{+}} f(x)x0,limx→x0+f(x) exists, and (3) if fff is defined in a neighborhood of ±∞\pm \infty±∞ , then lim⁡x→±∞f(x)\lim _{x \rightarrow \pm \infty} f(x)limx→±∞f(x) exists.
Uniform continuity
A function f:E→Rf: E \rightarrow \mathbb{R}f:E→R is uniformly continuous if 0 ~ \exists \delta>0 ~s.t.~ \forall x,y \in E"> ∀ε>0 ∃δ>0 s.t. ∀x,y∈E\forall \varepsilon>0 ~ \exists \delta>0 ~s.t.~ \forall x,y \in E∀ε>0 ∃δ>0 s.t. ∀x,y∈E we have: ∣x−y∣≤δ⇒∣f(x)−f(y)∣≤ε|x-y| \leq \delta \Rightarrow|f(x)-f(y)| \leq \varepsilon∣x−y∣≤δ⇒∣f(x)−f(y)∣≤ε If f is uniformly continuous then it is continuous.
Left and right continuity
Let f:E→Rf: E \rightarrow \mathbb{R}f:E→R be a function, and x0∈Ex_{0} \in Ex0∈E (1) fff is left continuous at x0x_{0}x0 if lim⁡x→x0−f(x)=f(x0)\lim _{x \rightarrow x_{0}^{-}} f(x)=f\left(x_{0}\right)limx→x0−f(x)=f(x0) . (2) fff is right continuous at x0x_{0}x0 if lim⁡x→x0+f(x)=f(x0)\lim _{x \rightarrow x_{0}^{+}} f(x)=f\left(x_{0}\right)limx→x0+f(x)=f(x0) .
continuous function over an interval
A function f:[a,b]→Rf : [a, b] \to Rf:[a,b]→R is continuous if it is continuous at ccc for all a<c<ba < c < ba<c<b , it is left continuous at bbb and right continuous at aaa .
Minima and maxima for continuous functions
If f:[a,b]→Rf:[a, b] \rightarrow \mathbb{R}f:[a,b]→R is continuous for some a,b∈Ra, b \in \mathbb{R}a,b∈R then there are c,d∈[a,b]c, d \in[a, b]c,d∈[a,b] such that M:=Sup⁡x∈[a,b]f(x)=f(c)m:=Inf⁡x∈[a,b]f(x)=f(d)M:=\operatorname{Sup}_{x \in[a, b]} f(x)=f(c) \quad \\ m:=\operatorname{Inf}_{x \in[a, b]} f(x)=f(d)M:=Supx∈[a,b]f(x)=f(c)m:=Infx∈[a,b]f(x)=f(d) In particular the above Sup and Inf is Max and Min, respectively.
uniform continuity of a function over an interval
If f:[a,b]→Rf : [a, b] \to Rf:[a,b]→R is continuous for some a,b∈R,a, b \in R,a,b∈R, then fff is uniformly continuous.
Intermediate value theorem
If f:[a,b]→Rf:[a, b] \rightarrow \mathbb{R}f:[a,b]→R is continuous then it takes each value between M:=max⁡x∈[a,b]f(x)M:=\max _{x \in[a, b]} f(x)M:=maxx∈[a,b]f(x) and m:=min⁡x∈[a,b]f(x)m:=\min _{x \in[a, b]} f(x)m:=minx∈[a,b]f(x) at least once. That is, for each c∈[m,M]c \in[m, M]c∈[m,M] ∃d∈[a,b] s.t. f(d)=c\exists d \in [a,b]~s.t.~f(d)=c∃d∈[a,b] s.t. f(d)=c .
Banach fixed point theorem for closed intervals
If f:[a,b]→[a,b] is a continuous function (where a,b∈R), then it has a fixed point. That is, there is an x∈[a,b] such that f(x)=x\begin{aligned} &\text { If } f:[a, b] \rightarrow[a, b] \text { is a continuous function (where } a, b \in \mathbb{R}), \text { then it has a fixed point. That }\\ &\text {is, there is an } x \in[a, b] \text { such that } f(x)=x \end{aligned} If f:[a,b]→[a,b] is a continuous function (where a,b∈R), then it has a fixed point. That is, there is an x∈[a,b] such…
Range of a continuous strictly monotone function over and open interval
If f:]a,b[→Rf :]a, b[\to Rf:]a,b[→R is strictly monotone and continuous (where a,b∈R a, b \in Ra,b∈R ), then the range R(f)R(f)R(f) is an open interval.
Monotonicity and injectivity of continuous functions
Let f:E→Ff : E \to Ff:E→F be a continuous function on an interval. Then, f is strictly monotone if and only if it is injective.
Monotonicity and surjectivity of continuous functions
If f:E→Ff : E \to Ff:E→F is continuous, strictly monotone and surjective function between intervals, then f−1f^{−1}f−1 (which exists by Theorem 6.72) is also continuous.
Linear approximation of a function
f(x)=f(x0)+a(x−x0)+r(x)f(x)=f\left(x_{0}\right)+a\left(x-x_{0}\right)+r(x)f(x)=f(x0)+a(x−x0)+r(x) with lim⁡x→x0r(x)x−x0=0\lim _{x \rightarrow x_{0}} \frac{r(x)}{x-x_{0}}=0limx→x0x−x0r(x)=0
Approximation of function using derivative
f(x)=f(x0)+(x−x0)f′(x0)+r(x)f(x)=f\left(x_{0}\right)+\left(x-x_{0}\right) f^{\prime}\left(x_{0}\right)+r(x)f(x)=f(x0)+(x−x0)f′(x0)+r(x) with lim⁡x→x0r(x)x−x0=0\lim _{x \rightarrow x_{0}} \frac{r(x)}{x-x_{0}}=0limx→x0x−x0r(x)=0
Differentiability and continuity
If f:E→Rf : E \to Rf:E→R is differentiable at x0x_0x0 , then it is continuous at x0x_0x0 .
Differentiability of inverse functions
Let f:I=]a,b[→Ff: I=] a, b[\rightarrow Ff:I=]a,b[→F be a bijective continuous function (so fff is strictly monotone, and f−1f^{-1}f−1 exists and is continuous by Theorem 6.73 ), and let x0∈Ix_{0} \in Ix0∈I be such that f ′(x0) ≠0f^{~\prime}\left(x_{0}\right) ~~\neq 0f ′(x0) ̸=0 Then f−1f^{-1}f−1 is differentiable at y0:=f(x0),y_{0}:=f\left(x_{0}\right),y0:=f(x0), and we have (f−1)′(y0)=1f ′(x0)=1f ′(f−1(y0))\left(f^{-1}\right)^{\prime}\left(y_{0}\right)=\frac{1}{f^{~~\prime}\left(x_{0}\rig…
Derivative of composition
If f:E→Ff: E \rightarrow Ff:E→F is differentiable at x0∈E,g:G→Hx_{0} \in E, g: G \rightarrow Hx0∈E,g:G→H is differentiable at f(x0),f\left(x_{0}\right),f(x0), and f(E)⊂G,f(E) \subset G,f(E)⊂G, then g∘f:E→Hg \circ f: E \rightarrow Hg∘f:E→H is differentiable at x0,x_{0},x0, and (g∘f) ′(x0)=((g ′∘f)⋅f ′)(x0)(g \circ f)^{~~\prime}\left(x_{0}\right)=\left(\left(g^{~~\prime} \circ f\right) \cdot f^{~~\prime}\right)\left(x_{0}\right)(g∘f) ′(x0)=((g ′∘f)⋅f ′)(x0)
exe^xex as a summation
ex:=∑k=0∞xkk!e^{x}:=\sum_{k=0}^{\infty} \frac{x^{k}}{k !}ex:=∑k=0∞k!xk
Local extrema and derivatives
If f:E→Rf : E \to R f:E→R is differentiable at x0x_0 x0 , and fff admits a local extremum at x0x_0 x0 , then f ′f~~'f ′ (x0)=0(x_0)=0(x0)=0 .
Rolle's theorem
If f:[a,b]→Rf : [a, b] \to Rf:[a,b]→R is a continuous function (where a and b are real numbers), such that f∣]a,b[f |_{ ]a,b[}f∣]a,b[ is differentiable, and f(a)=f(b)f (a) = f (b)f(a)=f(b) , then there is a c∈]a,b[c ∈]a, b[c∈]a,b[ for which f ′(c)=0 f~~'(c) = 0f ′(c)=0 .
Mean value theorem
If f:[a,b]→Rf : [a, b] \to Rf:[a,b]→R is a continuous function (where a and b are real numbers), such that f∣]a,b[f |_{]a,b[}f∣]a,b[ is differentiable, then there is a c∈]a,b[c ∈]a, b[ c∈]a,b[ for which f ′(c)(b−a)=f(b)−f(a)f~~'(c)(b − a) = f (b) − f (a)f ′(c)(b−a)=f(b)−f(a) .
two functions with f ′(x)=g ′(x) f~~' (x) = g~~'(x)f ′(x)=g ′(x)
If f,g:[a,b]→Rf, g : [a, b] \to Rf,g:[a,b]→R are continuous functions (where a and b are real numbers) that are differentiable over ]a,b[]a, b[ ]a,b[ such that f ′(x)=g ′(x) f~~' (x) = g~~'(x)f ′(x)=g ′(x) for each x∈]a,b[x ∈]a, b[x∈]a,b[ , then there is a real number c such that f(x)=g(x)+cf (x) = g(x) + cf(x)=g(x)+c .
Monotone functions and differentials
If f:[a,b]→Rf : [a, b] → Rf:[a,b]→R is a continuous function (where a and b are real numbers), such that f∣]a,b[ f |_{]a,b[}f∣]a,b[ is differentiable, then fff is increasing (resp. decreasing) if and only if f ′(x)≥0f~~'(x) ≥ 0f ′(x)≥0 (resp. ≤0 ≤ 0≤0 ) if 0"> f ′(x)>0 f~~' (x) > 0f ′(x)>0 (resp. <0< 0<0 ), then f is strictly increasing (resp. strictly decreasing)
L’Hôpital rule
For lim⁡x→a(f(x)g(x)) if lim⁡x→a(f(x)g(x))=00 or lim⁡x→a(f(x)g(x))=±∞±∞ then lim⁡x→a(f(x)g(x))=lim⁡x→a(f ′(x)g ′(x))\begin{array}{l} \text { For } \lim _{x \rightarrow a}\left(\frac{f(x)}{g(x)}\right) \\ \text { if } \lim _{x \rightarrow a}\left(\frac{f(x)}{g(x)}\right)=\frac{0}{0} \text { or } \lim _{x \rightarrow a}\left(\frac{f(x)}{g(x)}\right)=\frac{\pm \infty}{\pm \infty} \\ \text { then } \lim _{x \rightarrow a}\left(\frac{f(x)}{g(x)}\right)=\lim _{x \rightarrow a}\left(\frac{f~~^{\prime}…
The n-th order expansion of a function f
Let f:E→R f : E → R f:E→R be a function such that there is a neighborhood of a∈Ea ∈ Ea∈E which is contained in the domain (so in E). The n-th order expansion of f around a is an equality f(x)=a0+a1(x−a)+a2(x−a)2+⋯+an(x−a)n+(x−a)nϵ(x)f(x)=a_{0}+a_{1}(x-a)+a_{2}(x-a)^{2}+\cdots+a_{n}(x-a)^{n}+(x-a)^{n} \epsilon(x)f(x)=a0+a1(x−a)+a2(x−a)2+⋯+an(x−a)n+(x−a)nϵ(x) where aia_iai are real numbers, and ϵ(x)\epsilon (x)ϵ(x) is a function E→RE → RE→R such that lim⁡x→aϵ(x)=0\lim _{x \rightarrow a} \eps…
Taylor expansion
Let n≥0n ≥ 0n≥0 be an integer. Let f:I→Rf : I → Rf:I→R be a function on an open interval III , which is n+1n + 1n+1 times differentiable on III , and let a∈Ia ∈ Ia∈I be an arbitrary real number. Then, for each x∈Ix ∈ Ix∈I there is an x ′x~~'x ′ (strictly) between a and x such that f(x)=(∑i=0nf(i)(a)i!(x−a)i)+f(n+1)(x′)(x−a)n+1(n+1)!f(x)=\left(\sum_{i=0}^{n} \frac{f^{(i)}(a)}{i !}(x-a)^{i}\right)+f^{(n+1)}\left(x^{\prime}\right) \frac{(x-a)^{n+1}}{(n+1) !}f(x)=(∑i=0ni!f(i)(a)(x−a)i)+f(n+1…
n-th order expansion of function f summation
Let n≥0n ≥ 0n≥0 be an integer. Let f:I→Rf : I → Rf:I→R be a function on an open interval III , which is n+1n + 1n+1 times differentiable on III , and let a∈Ia ∈ Ia∈I be an arbitrary real number. Then, the n-th order expansion of f around a exists and is ∑j=0nf(j)(a)j!(x−a)j\sum_{j=0}^{n} \frac{f^{(j)}(a)}{j !}(x-a)^{j}∑j=0nj!f(j)(a)(x−a)j
Higher derivatives and local extrema
Let n≥2n ≥ 2 n≥2 be an even integer. Let f:I→Rf : I → Rf:I→R be a function on an open interval III , which is n times differentiable on I, and let a∈Ia ∈ I a∈I be an arbitrary real number. If f ′(a)=⋅⋅⋅=f(n−1)(a)=0f~~'(a) = · · · = f^{(n−1)} (a) = 0f ′(a)=⋅⋅⋅=f(n−1)(a)=0 and (1) if 0,"> f(n)>0,f^{(n)}>0,f(n)>0, then fff has a local minimum, and (2) if f(n)<0,f^{(n)}<0,f(n)<0, then fff has a local maximum.
Inflection points of a function
If f:E→Rf : E → Rf:E→R is a function such that fff is differentiable at a∈Ea ∈ Ea∈E , then we say that f has an inflection point at a if there is an 0"> δ>0δ > 0δ>0 such that either 0, \text { and }\\ &\{x \in E \mid a-\delta (1) {x∈E∣a<x<a+δ}⇒f(x)−f(a)−f ′(a)(x−a)>0, and {x∈E∣a−δ<x<a}⇒f(x)−f(a)−f ′(a)(x−a)<0, or \begin{aligned} &\text { (1) }\{x \in E \mid a<x<a+\delta\} \Rightarrow f(x)-f(a)-f~~^{\prime}(a)(x-a)>0, \text { and }\\ &\{x \in E \mid a-\de…
Higher derivatives and inflection points
Let n≥3n ≥ 3n≥3 be an odd integer. Let f:I→Rf : I → Rf:I→R be a function on an open interval III , which is nnn times differentiable on III , and let a∈Ia ∈ Ia∈I be an arbitrary real number. If f ′′(a)=⋯=f(n−1)(a)=0f~~^{\prime \prime}(a)=\cdots=f^{(n-1)}(a)=0f ′′(a)=⋯=f(n−1)(a)=0 and f(n)(a) ≠0f^{(n)}(a) ~\neq 0f(n)(a) ̸=0 , then fff has an inflection point at aaa .
Convex and concave functions
A function f:I→Rf: I \rightarrow \mathbb{R}f:I→R on an open interval is called convex (resp. concave) if for every a,b∈Ia, b \in Ia,b∈I and every λ∈[0,1]\lambda \in[0,1]λ∈[0,1] we have: f(λa+(1−λ)b)≤λf(a)+(1−λ)f(b)f(\lambda a+(1-\lambda) b) \leq \lambda f(a)+(1-\lambda) f(b)f(λa+(1−λ)b)≤λf(a)+(1−λ)f(b) (resp. f(λa+(1−λ)b)≥λf(a)+(1−λ)f(b))f(\lambda a+(1-\lambda) b) \geq \lambda f(a)+(1-\lambda) f(b))f(λa+(1−λ)b)≥λf(a)+(1−λ)f(b))
Convex and concave functions and derivatives
Let f:I→Rf : I → Rf:I→R be a differentiable function on an open interval. Then fff is convex (resp. concave) if and only if f ′:I→Rf~~': I → Rf ′:I→R is an increasing (resp. decreasing) function.
Convex and concave functions and second derivatives
Let f:I→Rf : I → R f:I→R be a two times differentiable function on an open interval. Then fff is convex (resp. concave) if and only if f ′′(x)≥0 (resp. f ′′(x)≤0 ) for all x∈I . f~~^{\prime \prime}(x) \geq 0 \text { (resp. } f~~^{\prime \prime}(x) \leq 0 \text { ) for all } x \in I \text { . }f ′′(x)≥0 (resp. f ′′(x)≤0 ) for all x∈I .
Lipschitz continuity
Let 0"> k>0k>0k>0 be a real number. A function f:E→Rf: E \rightarrow \mathbb{R}f:E→R is kkk - Lipschitz if for every x,y∈E,∣f(x)−f(y)∣≤k∣x−y∣x, y \in E,|f(x)-f(y)| \leq k|x-y| x,y∈E,∣f(x)−f(y)∣≤k∣x−y∣ . If k<1k<1k<1 then we call kkk -Lipschitz functions k-contractions. If f:I→Rf: I \rightarrow \mathbb{R}f:I→R a function on an interval is k-Lipschitz (for any 0"> k>0k>0k>0 ), then it is uniformly continuous on III (and hence also continuous on III ).
Banach fixed point theorem - contraction version
Let III be either R\mathbb{R}R or a closed interval, and let 0<k<10<k<10<k<1 be a real number. If f:I→If: I \rightarrow If:I→I is a k-contraction, then there is a unique c∈Ic \in Ic∈I such that f(c)=cf(c)=cf(c)=c (so fff has a fixed point). Furthermore, c=lim⁡n→∞xnc=\lim _{n \rightarrow \infty} x_{n}c=limn→∞xn where xnx_{n}xn is a recursive sequence defined by xn+1=f(xn)x_{n+1}=f\left(x_{n}\right)xn+1=f(xn) with any choice of x1∈Ix_{1} \in Ix1∈I .
Asymptotes
(1) If for some c∈R,lim⁡x→c−f(x)=±∞c \in \mathbb{R}, \lim _{x \rightarrow c^{-}} f(x)=\pm \inftyc∈R,limx→c−f(x)=±∞ or lim⁡x→c+f(x)=±∞,\lim _{x \rightarrow c^{+}} f(x)=\pm \infty,limx→c+f(x)=±∞, then fff has a vertical asymptote. (2) If for some c∈R,lim⁡x→+∞f(x)=cc \in \mathbb{R}, \lim _{x \rightarrow+\infty} f(x)=cc∈R,limx→+∞f(x)=c or lim⁡x→−∞f(x)=c,\lim _{x \rightarrow-\infty} f(x)=c,limx→−∞f(x)=c, then fff has a horizontal asymptote. (3) If for some a≠0,b∈R,lim⁡x→+∞f(x)−ax=ba \neq 0, b \i…
oblique/ slant asymptote
if lim⁡x→∞=±∞\lim_{x\to\infty}=\pm\inftylimx→∞=±∞ m:=lim⁡x→∞f(x)xm:=\lim_{x\to\infty}\frac{f(x)}{x}m:=limx→∞xf(x) exists h:=lim⁡x→∞f(x)−mxh:=\lim_{x\to\infty}f(x)-mxh:=limx→∞f(x)−mx exists then y=mx+hy=mx+hy=mx+h is an oblique asymptote
Upper and lower Darboux sums
Let f:[a,b]→Rf:[a, b] \rightarrow \mathbb{R}f:[a,b]→R be a bounded function and σ=(xi)\sigma=\left(x_{i}\right)σ=(xi) a partition of [a,b][a, b][a,b] Then, the upper Darboux sum of fff with respect to σ\sigmaσ is Sˉσ=∑i=1nMi(xi−xi−1)\bar{S}_{\sigma}=\sum_{i=1}^{n} M_{i}\left(x_{i}-x_{i-1}\right)Sˉσ=∑i=1nMi(xi−xi−1) where Mi:=Sup⁡x∈[xi−1,xi]f(x)M_{i}:=\operatorname{Sup}_{x \in\left[x_{i-1}, x_{i}\right]} f(x) Mi:=Supx∈[xi−1,xi]f(x) . The lower Darboux sum of fff with respect to σ\sigma…
Boundedness of partitions of intervals
Let M and m be upper and lower bounds for f:[a,b]→Rf:[a, b] \rightarrow \mathbb{R}f:[a,b]→R . Then, for any partition σ\sigmaσ of [a,b],m(b−a)≤Sˉσ,S‾σ≤M(b−a)[a, b], m(b-a) \leq \bar{S}_{\sigma}, \underline{S}_{\sigma} \leq M(b-a)[a,b],m(b−a)≤Sˉσ,Sσ≤M(b−a) . In particular, the sets {Sˉσ∣σ\left\{\bar{S}_{\sigma} \mid \sigma\right.{Sˉσ∣σ is a partition of [a,b]}\left.[a, b]\right\}[a,b]} and {S‾σ∣σ\left\{\underline{S}_{\sigma} \mid \sigma\right.{Sσ∣σ is a partition of [a,b]}\left.[a, b]\righ…
Upper and lower Darboux integral
Let f:[a,b]→Rf:[a, b] \rightarrow \mathbb{R}f:[a,b]→R be a bounded function, then the upper Darboux integral of f on [a, b] is Sˉ:=Inf⁡{Sˉσ∣σ\bar{S}:=\operatorname{Inf}\left\{\bar{S}_{\sigma} \mid \sigma\right.Sˉ:=Inf{Sˉσ∣σ is a partition of [a,b]}\left.[a, b]\right\}[a,b]} and the lower Darboux integral of f( on [a, b]) is S‾:=Sup⁡{S‾σ∣σ\underline{S}:=\operatorname{Sup}\left\{\underline{S}_{\sigma} \mid \sigma\right.S:=Sup{Sσ∣σ is a partition of [a,b]}\left.[a, b]\right\}[a,b]}
relation between upper and lower darboux sums
Let f:[a,b]→Rf:[a, b] \rightarrow \mathbb{R}f:[a,b]→R be a bounded function. (1) If σ\sigmaσ is a partition of [a,b][a, b][a,b] and σ′\sigma^{\prime}σ′ is a refinement of σ\sigmaσ , then: S‾σ≤S‾σ′, and Sˉσ≥Sˉσ′\underline{S}_{\sigma} \leq \underline{S}_{\sigma^{\prime}}, \text { and } \bar{S}_{\sigma} \geq \bar{S}_{\sigma^{\prime}}Sσ≤Sσ′, and Sˉσ≥Sˉσ′ (2) If σ\sigmaσ is a partition of [a,b][a, b][a,b] then: S‾σ≤Sˉσ \underline{S}_{\sigma} \leq \bar{S}_{\sigma}Sσ≤Sˉσ If f:[a,b]→Rf:[a, b]…
integrability of f, darboux definition
Let f:[a,b]→Rf:[a, b] \rightarrow \mathbb{R}f:[a,b]→R be a bounded function. We say that fff is integrable, if Sˉ=S‾,\bar{S}=\underline{S},Sˉ=S, in which case this common value is called the integral of f between a and b, and it is denoted by ∫abf(x)dx\int_{a}^{b} f(x) d x∫abf(x)dx
continuity and integrability
If f:[a,b]→Rf:[a, b] \rightarrow \mathbb{R}f:[a,b]→R is continuous then it is integrable. More precisely, we know that the above assumptions imply uniform continuity of f.f .f. So, fix 0"> ε>0\varepsilon>0ε>0 . Let 0"> δ>0\delta>0δ>0 be the constant in the definition of uniform continuity associated to εb−a\frac{\varepsilon}{b-a}b−aε (that is, ∣x−y∣≤δ⇒|x-y| \leq \delta \Rightarrow∣x−y∣≤δ⇒ ∣f(x)−f(y)∣≤εb−a),\left.|f(x)-f(y)| \leq \frac{\varepsilon}{b-a}\right),∣f(x)−f(y)∣≤b−aε), an…
Antiderivative
Let f:[a,b]→Rf:[a, b] \rightarrow \mathbb{R}f:[a,b]→R be a continuous function. A function G:[a,b]→RG:[a, b] \rightarrow \mathbb{R}G:[a,b]→R is called an anti-derivative of fff if GGG is continuous on [a,b],[a, b],[a,b], it is differentiable on ]a,b[,] a, b[,]a,b[, and G ′(x)=f(x)G~~^{\prime}(x)=f(x)G ′(x)=f(x) for all x∈]a,b[\left.x \in\right] a, b[x∈]a,b[
Mean value theorem for integrals
If f:[a,b]→Rf:[a, b] \rightarrow \mathbb{R}f:[a,b]→R is continuous, then there is a c∈[a,b],c \in[a, b],c∈[a,b], such that ∫abf(x)dx=f(c)(b−a)\int_{a}^{b} f(x) d x=f(c)(b-a)∫abf(x)dx=f(c)(b−a)
Fundamental theorem of calculus I
Let f:[a,b]→Rf:[a, b] \rightarrow \mathbb{R}f:[a,b]→R be continuous. Then, F(x):=∫axf(t)dt F(x):=\int_{a}^{x} f(t) d tF(x):=∫axf(t)dt is an anti-derivative of $f$.
Fundamental theorem of calculus II
Let f:[a,b]→Rf:[a, b] \rightarrow \mathbb{R}f:[a,b]→R be continuous and let GGG be an anti-derivative of fff . Then ∫abf(x)dx=G(b)−G(a) \int_{a}^{b} f(x) d x=G(b)-G(a)∫abf(x)dx=G(b)−G(a)
Integration by Substitution
∫ϕ(α)ϕ(β)f(x)dx=∫αβf(ϕ(t))ϕ′(t)dt\int_{\phi(\alpha)}^{\phi(\beta)} f(x) d x=\int_{\alpha}^{\beta} f(\phi(t)) \phi^{\prime}(t) d t∫ϕ(α)ϕ(β)f(x)dx=∫αβf(ϕ(t))ϕ′(t)dt
improper integral
In the above case we define the improper integral of f on I as (1) if I=[a,b[I=[a, b[I=[a,b[ , then ∫ab−f(t)dt:=lim⁡x→b−(∫axf(t)dt)\int_{a}^{b-} f(t) d t:=\lim _{x \rightarrow b-}\left(\int_{a}^{x} f(t) d t\right)∫ab−f(t)dt:=limx→b−(∫axf(t)dt) (2) if I=]a,b],I=] a, b],I=]a,b], then ∫a+bf(t)dt:=lim⁡x→a+(∫xbf(t)dt)\int_{a^{+}}^{b} f(t) d t:=\lim _{x \rightarrow a+}\left(\int_{x}^{b} f(t) d t\right)∫a+bf(t)dt:=limx→a+(∫xbf(t)dt) (3) if I=]a,b[I=] a, b[I=]a,b[ , ∫a+b−f(t)dt:=∫a+cf(t)dt+∫cb−f(…