Single Variable Calculus

Attention

  • All the variables are real and all the functions are real valued functions.

  • We will be using points instead of elements since \(\mathbb{R}\) is a metric space with a distance function defined in terms of the absolute value function \(|\cdot|\).

  • The set \(U=\{f(x)\mathop{:}x\in S\}=f(S)\) is called the image of \(S\) under \(f\).

Metric Topology

Definitions

Open and Closed Balls and Intervals

Note

  • For any \(\epsilon > 0\), we can create an open \(\epsilon\)-ball around any point \(x\) as

    \[B_\epsilon(x)=\{y\mathop{:} |x-y|< \epsilon\}\]

  • Closed ball is defined similarly as

    \[\bar{B}_\epsilon(x)=\{y\mathop{:} |x-y|\leq \epsilon\}\]

  • Open interval: \((a,b)=\{x\mathop{:} a < x < b\}\)

  • Closed interval: \([a,b]=\{x\mathop{:} a \leq x \leq b\}\)

Limit Point and Closure

Sequence and Convergence

Accumulation Point

Limit Point

Note

  • Let \((x_n)_{n=1}^\infty\) be a sequence such that \(\forall x_n\in S\subset\mathbb{R}\).

  • The sequence is said to have a limit \(\lim\limits_{n\to\infty} x_n=L\in\mathbb{R}\) iff

    • \(\forall\epsilon > 0\)

    • \(\exists N_\epsilon\in\mathbb{N}^{+}\) (depends on how small of a \(\epsilon\) we’re given) such that

    • if we skip \(N_\epsilon\) number of terms in that sequence, the remaining values are guaranteed to be inside \(B_\epsilon(x)\).

      • Formally, \(n > N_\epsilon\implies |x_n-L|< \epsilon\)

  • A sequence with a limit point \(L\in\S\subset\mathbb{R}\) is said to be convergent in \(S\).

Important Theorems

Attention

  • Limit of a sequence is unique.

  • If a sequence is convergent, it is bounded.

  • Every limit point is an accumulation point. Converse doesn’t hold.

  • Every open ball around a limit point contains all but a finite number of terms in a convergent sequence.

See also

  • Null sequence

  • Sequence of nested intervals

Cauchy convergence

Note

The sequence is said to be Cauchy convergent iff

  • \(\forall\epsilon > 0\)

  • \(\exists N_\epsilon\in\mathbb{N}^{+}\) such that

  • if we skip \(N_\epsilon\) number of terms in that sequence, any two terms from the rest of it is within a \(\epsilon\)-ball around one another.

    • Formally, \(m, n> N_\delta\implies |x_m-x_n|< \epsilon\)

Attention

  • For a sequence to be Cauchy convergent, the limit value doesn’t need to be in \(S\).

  • Example: We can imagine a sequence in rationals

    \[1,1.4,1.41,1.414,1.4142,1.41421,1.414213,\cdots\]

  • This sequence is Cauchy convergent as it tends to \(\sqrt{2}\) but it’s not convergent in \(\mathbb{Q}\).

Monotonic Sequences

Attention

  • If a monotonic sequence is bounded, it is convergent.

  • Term-wise order relationship between two sequences is preserved at limit points.

  • Squeeze/sandwich theorem

Functional Limit and Continuity

Continuity

Let \(f:X\subset\mathbb{R}\mapsto Y\subset\mathbb{R}\).

Note

The function \(f:X\mapsto Y\) is said to be continuous at a point \(p\in X\) iff

  • \(\forall\epsilon > 0\)

  • we can create an open ball around \(p\) with some \(\delta_{\epsilon, p} > 0\) such that

    • (note: the size depends on \(\epsilon\) as well as \(p\) and can be arbitrarily small)

  • if we force \(x\) to be in \(B_{\delta_{\epsilon, p}}(p)\), then the image \(f(x)\) is guaranteed to be in \(B_\epsilon(f(p))\).

    • Formally, \(\forall x\in X, |p-x|< \delta_{\epsilon, p}\implies |f(p)-f(x)|< \epsilon\)

See also

  • If the function varies quite drastically, we’d only able to choose extremely small \(\delta_{\epsilon, p}\) to push the image inside \(B_\epsilon(f(p))\).

  • If we’re allowed to take larger \(\delta\), then the function is considered smoother.

Sequential Continuity

Tip

Under a continuous function \(f\), \(\lim\limits_{n\to\infty} x_n=x\in X\implies \lim\limits_{n\to\infty} f(x_n)=f(x)\in Y\)

Properties

Note

  • If \(f\) and \(g\) are continuous at \(x\), so is \(f\cdot g\).

  • If \(f\) and \(g\) are continuous at \(x\), so is \(f\circ g\).

Continuous Everywhere

Note

If the function is continuous \(\forall p\in X\), then it is said to be continuous everywhere.

Uniform Continuity

This is a stricter form of continuity.

Note

The function \(f:X\mapsto Y\) is said to be uniformly continuous in \(X\) iff

  • \(\forall\epsilon > 0\)

  • we can create an open ball around any \(p\) with some \(\exists\delta_\epsilon > 0\) such that

    • (note: a universal one as it doesn’t depend on \(p\) anymore, however can still be arbitrarily small)

  • if we force \(x\) to be in \(B_{\delta_\epsilon}(p)\), the image \(f(x)\) is guaranteed to be in \(B_\epsilon(f(p))\).

    • Formally, \(\forall p, x\in X, |p-x|< \delta_\epsilon\implies |f(p)-f(x)|< \epsilon\)

Tip

  • The same \(\delta\) works for every point \(p\in X\), hence the term uniform.

Lipschitz Continuity

This is an even stricter form of continuity.

Note

The function \(f:X\mapsto Y\) is said to be Lipschitz continuous in \(X\) with Lipschitz constant \(K\) iff

  • \(\exists K\geq 0\) such that \(\forall x,y\in X, \frac{|f(x)-f(x)|}{|x-y|}\leq K\)

See also

  • For the image to be in a \(\epsilon\)-ball around any \(p\), we can afford to be in a \(\epsilon/K\)-ball in the domain.

  • These functions are a lot smoother.

Differentiation

Let \(f:(a,b)\subset\mathbb{R}\mapsto \mathbb{R}\) be a continuous function at some \(x\in(a,b)\).

Differentiation as a rate of change

Note

The derivative of \(f\) at \(x\in(a,b)\) is defined to be (assuming that the limit exists),

\[f'(x)=\lim\limits_{h\to 0}\frac{f(x+h)-f(x)}{h}\]

Warning

We need the point \(x\) to be inside the open interval because we need to be able to create an open \(h\)-ball around it and we need the function to be well defined in that region.

Differentiation as a linear approximation

We can define the derivative as a linear approximation of the function at close proximity of \(x\).

Note

  • We consider the open-ball \(B_h(x)\), and assume that inside this, the function is approximately linear.

  • Therefore, we introduce a linear transform \(\alpha:\mathbb{R}\mapsto\mathbb{R}\) to replace our original function \(f:\mathbb{R}\mapsto\mathbb{R}\).

  • The change in value as we move from \(x\) to \(x+h\) is

    • \(f(x+h)-f(x)\) under the actual function.

    • \(\alpha(x+h)-\alpha(x)=\alpha h\) under the approximation.

  • The error in this approximation is

    \[\epsilon_x(h)=f(x+h)-f(x)-\alpha h\]

  • We assume that \(\lim\limits_{h\to 0}\frac{|\epsilon_x(h)|}{|h|}=0\) and define \(f'(x)=\alpha\).

Tip

If the derivative of a function exists at a point, then the function is continuous at that point.

Properties

Note

  • Sum Rule: \((f+g)'=f'+g'\)

  • Product Rule: \((f\cdot g)'=f\cdot g'+f'\cdot g\)

  • Chain Rule: \((f\circ g)'=(f'\circ g)\cdot g'\)

Important Theorems

Boundedness theorem

Note

  • Let \(f:[a,b]\mapsto\mathbb{R}\) is continuous \(\forall x\in[a,b]\). Then it is bounded.

  • More formally, here exists \(m, M\in\mathbb{R}\) such that \(m\leq f(x)\leq M\).

EVT: Extreme value theorem

Note

  • Let \(f:[a,b]\mapsto\mathbb{R}\) is continuous \(\forall x\in[a,b]\). Then the function achives a min and a max.

  • More formally, there exists \(c,d\in[a,b]\) such that \(f(c)\leq f(x)\leq f(d)\).

Bolzano’s theorem

Note

  • Let \(f:[a,b]\mapsto\mathbb{R}\) is continuous \(\forall x\in[a,b]\).

  • Also assume that \(f(a)\) and \(f(b)\) have opposite signs.

  • Then \(\exists c\in(a,b)\) such that \(f(c)=0\)

IVT: Intermediate value theorem

Note

  • Let \(f:[a,b]\mapsto\mathbb{R}\) is continuous \(\forall x\in[a,b]\).

  • Let \(a\leq p < q\leq b\) be two arbitrary points with \(f(p)\neq f(q)\).

  • Then \(f(x)\) takes every possible value in \((f(p), f(q))\) within the interval \((a,b)\).

MVT: Mean value theorem

Note

  • Let \(f:[a,b]\mapsto\mathbb{R}\) is continuous \(\forall x\in[a,b]\).

  • Then \(\exists c\in[a,b]\) such that \(f(c)\) acts as the mean value of the integral \(\int\limits_a^b f(x)\mathop{dx}\).

  • Formally, \(\int\limits_a^b f(x)\mathop{dx}=f(c)\cdot(b-a)\)

See also

  • This can also be stated using derivatives as \(\frac{F(b)-F(a)}{b-a}=f(c)\) or \(\frac{g(b)-g(a)}{b-a}=g'(c)\)

Rolle’s theorem

Note

  • Special case of MVT.

  • Assuming that all the MVT conditions are satisfied, if \(f(a)=f(b)\), then \(\exists c\in(a,b)\) such that \(f'(c)=0\).

Application: Local extremum

Critical Point

Note

  • Let the function be \(f:X\mapsto Y\) and let \(c\in X\).

  • \(c\) is called a relative (local) maximum iff

    \[\exists\epsilon>0,x\in B_\epsilon(c)\implies f(x)\leq f(c)\]

Note

  • Relative minimum is defined in the same way.

  • This is usually defined in terms of an open interval, i.e. \(c\in(a,b)\).

  • Maxima and minimum are jointly called an extremum.

First derivative test

For critical points

Attention

Let \(c\in(a,b)\) be a local extremum. Then \(f'(c)=0\).

Tip

  • The point \(c\in(a,b)\) is called a critical point.

  • First derivative test doesn’t tell us whether it’s a maximum or a minimum.

For monotonic functions

Attention

  • If \(\forall x\in (a,b), f'(x)> 0\), then \(f\) is strictly increasing in \([a,b]\).

  • If \(\forall x\in (a,b), f'(x)< 0\), then \(f\) is strictly decreasing in \([a,b]\).

  • If \(\forall x\in (a,b), f'(x)= 0\), then \(f\) is constant in \([a,b]\).

Second derivative test

For critical points

Tip

Think of the slope of tangent for a convex function as it reaches a minimum.

Attention

  • For a minimum \(c\in(a,b)\), the second derivative is a strictly increasing function in \([a,b]\), i.e. \(\forall x\in(a,b), f''(x)> 0\).

  • For a maximum \(c\in(a,b)\), the second derivative is a strictly decreasing function in \([a,b]\), i.e. \(\forall x\in(a,b), f''(x)< 0\).

Integration

Integration of step functions

Let \(f:[a,b]\subset\mathbb{R}\mapsto \mathbb{R}\) be a step-function defined on a partition \(P=\{x_0,\cdots,x_n\}\) such that within each open interval \((x_{k-1},x_k)\), the function takes a constant value \(s_k\).

Note

The integral of such function is defined as

\[\int\limits_a^b f(x)\mathop{dx}=\sum_{k=1}^n s_k\cdot(x_k-x_{k-1})\]

Properties

Note

  • If \(f(x)<g(x)\) for all \(x\in[a,b]\), then \(\int\limits_a^b f(x)\mathop{dx}<\int\limits_a^b g(x)\mathop{dx}\).

Integration of general function

Warning

  • We try to approximate the integral \(I=\int\limits_a^b f(x)\mathop{dx}\) by 2 step functions \(s\) and \(t\), one above and one below, that is

    \[s(x)\leq f(x)\leq t(x)\]

  • This is not possible if the function \(f\) is unbounded (such as \(f(x)=1/x\)).

Let \(f:[a,b]\subset\mathbb{R}\mapsto \mathbb{R}\) be any bounded function.

Note

  • Let \(s\) and \(t\) be arbitrary step functions such that \(s(x)\leq f(x)\leq t(x)\).

  • We define \(S=\left\{\int\limits_a^b s(x)\mathop{dx}\mathop{:}\forall s\leq f\right\}\) and \(T=\left\{\int\limits_a^b t(x)\mathop{dx}\mathop{:}\forall t\geq f\right\}\).

  • It is in general true that

    \[\int\limits_a^b s(x)\mathop{dx}\leq\sup_s S\leq I\leq\inf_t T\leq \int\limits_a^b t(x)\mathop{dx}\]

  • The integral \(I\) exists when \(\sup_s S=\inf_t T\) and takes that exact same value

    \[I=\int\limits_a^b f(x)\mathop{dx}=\sup_s S=\inf_t T\]

Attention

  • Let \(f:[a,b]\mapsto\mathbb{R}\) is continuous \(\forall x\in[a,b]\).

  • Then it is integrable (follows since it is bounded).

Properties

Note

  • \(\int\limits_a^b (f(x)+g(x))\mathop{dx}=\int\limits_a^b f(x)\mathop{dx}=+\int\limits_a^b g(x)\mathop{dx}\)

  • \(\int\limits_a^b c\cdot f(x)\mathop{dx}=c\cdot\int\limits_a^b f(x)\mathop{dx}\)

  • \(\int\limits_a^b f(x)\mathop{dx}=-\int\limits_b^a f(x)\mathop{dx}\)

  • \(\int\limits_a^c f(x)\mathop{dx}=\int\limits_a^b f(x)\mathop{dx}+\int\limits_b^c f(x)\mathop{dx}\)

  • \(\int\limits_a^a f(x)\mathop{dx}=0\)

Indefinite Integral

Note

  • For every \(a\leq x\leq b\), we can define a function of \(x\) which is obtained via the integral

    \[A(x)=\int\limits_a^x f(t)\mathop{dt}\]

  • This is known as an indefinite integral of \(f\).

  • We can define another indefinite integral with a different lower limit \(c\in[a,b]\) as

    \[C(x)=\int\limits_c^x f(t)\mathop{dt}\]

Attention

  • These two differ by only a constant as

    \[A(x)-C(x)=\int\limits_a^x f(t)\mathop{dt}-\int\limits_c^x f(t)\mathop{dt}=\int\limits_a^c f(t)\mathop{dt}=k\]

Fundamental theorem of calculus

Note

  • Let \(f:[a,b]\mapsto\mathbb{R}\) be a function that is integrable for every \([a,x]\). Let \(c\in[a,b]\).

  • Let \(F(x)\) be an indefinite integral of \(f\)

    \[F(x)=\int\limits_c^x f(t)\mathop{dt}\]

  • Then the derivative of \(F\) exists at all \(x\in(a,b)\) wherever \(f(x)\) is continuous and

    \[F'(x)=f(x)\]

Tip

  • \(F\) is called an antiderivative of \(f\).

  • Any other antiderivative differs only by a constant.

  • Leibniz Notation: Therefore, we can use the notation

    \[\int f(x)\mathop{dx}=F(x)+C\]

Attention

\[\int\limits_a^b f(x)\mathop{dx}=F(b)-F(a)\]

Integration Strategies

Integration by parts

Let \(u(x)\) and \(v(x)\) be two integrable functions. We want to find out the integral of the product, \(\int u(x)\cdot v(x) \mathop{dx}\).

Note

  • To derive this formula, it becomes easier if we consider \(w(x)=\int v(x) \mathop{dx}\) (such that \(w'(x)=v(x)\)) and consider \(g(x)=u(x)\cdot w(x)\).

  • Taking derivatives on both sides \(g'(x)=u'(x)\cdot w(x)+u(x)\cdot w'(x)\) which gives

    \[u(x)\cdot w'(x)=g'(x)-u'(x)\cdot w(x)\]

  • Taking integration on both sides and ignoring the constant

    \[\int u(x)\cdot w'(x)\mathop{dx}=\int g'(x)\mathop{dx}-\int u'(x)\cdot w(x)\mathop{dx}=u(x)\cdot w(x)-\int u'(x)\cdot w(x)\mathop{dx}\]

  • Replacing \(w(x)\)

    \[\int u(x)\cdot v(x)\mathop{dx}=u(x)\cdot \int v(x)\mathop{dx}-\int u'(x)\left(\int v(x)\mathop{dx}) \right)\mathop{dx}\]

Tip

  • ILATE: Dictates the order in which the functions should be chosen to be \(u\) or \(v\).

  • ILATE: Acronym for Inverse > Logarithmic > Algebraic > Trigonometric > Exponential. Choose left of the two as \(u\).

Integration Bee

Warning

Series

Series with Positive Terms

Comparison Tests

Ratio Test

Integral Test

Series with Mixed Terms

Absolute Convergence

Convergence of Alternating Series

Power Series

Root Test

Useful Resources

Important