Let \(f(x)\) be a real-valued function and let \(a\in\mathbb{R}\). We say that the limit of \(f\) as \(x\) approaches \(a\) is \(L\) if for every \(\epsilon>0\) there exist \(\delta>0\) such that \[|x-a|< \delta\implies |f(x)-L|<\epsilon.\] We write \(L = \lim_{x\to a}f(x)\).

The definition states that for any number \(\epsilon>0\) we pick, we can sketch two horizontal lines at \(L-\epsilon\) and \(L+\epsilon\). Then, it is possible to find another real number \(\delta>0\) that allows us to add in two vertical lines at \(a-\delta\) and \(a+\delta\). If we take any \(x\) close enough to \(a\), then the image of \(x\) under \(f\), i.e., \(f(x)\) will be within \(L-\epsilon\) and \(L+\epsilon\), i.e., it will be close enough to \(L\). In other words, by making \(x\) sufficiently close to \(\boldsymbol{a}\), you can set \(\boldsymbol{f(x)}\) as close as you want to \(\boldsymbol{L}\).

A sequence \(x_n\in\mathbb{R}\), \(n\in\mathbb{N}\), is said to converge to \(a\in\mathbb{R}\) if for all \(\epsilon>0\) there is \(N_\epsilon\in\mathbb{N}\) such that for all \(n>N_\epsilon\), \(x_n\) does not deviate from \(a\) by more than \(\epsilon\), i.e., if \[\forall \epsilon>0\hskip0.2cm \exists N_\epsilon\in\mathbb{N}: n>N_\epsilon\implies |x_n-a|< \epsilon.\]

We write \(a=\lim_{n\to\infty}x_n\).

1. The limit of \(f\) at \(a\) is \(L\) if and only if the limit from above (right) and below (left) of \(f\) at \(a\) are equal to \(L\), i.e., \[\lim_{x\to a^+}f(x) = \lim_{x\to a^-}f(x) = L.\]

2. Continuity. The function \(f\) is continuous at \(a\) if and only if \(\lim_{x\to a}f(x) = f(a).\)

3. The algebraic limit theorem. If \(f\) and \(g\) are real-valued functions, then taking the limit of an operation on \(f(x)\) and \(g(x)\) is under certain conditions compatible with the algebraic operations and exponentiation. The main condition needed for applying the following rules is that the limits on the right-hand sides of the equations exist.

   3.1. \(\lim_{x\to a} [f(x) \pm g(x)] = \lim_{x\to a} f(x) \pm \lim_{x\to a}g(x).\)

   3.2. \(\lim_{x\to a}f(x)\cdot g(x) = \lim_{x\to a}f(x)\cdot\lim_{x\to a}g(x).\)

   3.3. \(\lim_{x\to a}f(x)/g(x) = \lim_{x\to a}f(x)/\lim_{x\to a}g(x),\) provided \(\lim_{x\to a}g(x)\neq0.\)

   3.4. \(\lim_{x\to a}f(x)^{g(x)} = \lim_{x\to a}f(x)^{\lim_{x\to a}g(x)}\), if the base is positive, or zero while the exponent comes out positive (but finite).

4.L’Hôpital’s rule. Let \(f(x)\) and \(g(x)\) be two functions defined over an open interval \(I\) containing the desired limit point \(a\). Assume that

• \(\lim_{x\to a}f(x) = \lim_{x\to a}g(x) = 0\), or \(\lim_{x\to a}f(x) = \pm \lim_{x\to a}g(x) = \pm\infty\).

• \(f\) and \(g\) are differentiable over \(I\setminus\{a\}\).

• \(g'(x)\neq 0\) for all \(x\in I\setminus\{a\}\).

• \(\lim_{x\to a}f'(x)/g'(x)\) exists.

  Then \[\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}.\] Normally, the first condition is the most important one.

Similar properties can be derived for the limit of a sequence.

If a curve \(C\) has equation \(y=f(x)\) and we want to find the tangent line to \(C\) at the point \(P(a,f(a))\), then we consider a nearby point \(Q(x,f(x))\), where \(x\neq a\), and compute the slope of the secant line \(PQ\): \[m_{PQ} = \frac{f(x)-f(a)}{x-a}.\] Then we let \(Q\) approach \(P\) along the curve \(C\) by letting \(x\) approach \(a\). If \(m_{PQ}\) approaches a number \(m\), then we define the tangent \(t\) to be the line through \(P\) with slope \(m\).

Definition. The tangent line to the curve \(y=f(x)\) at the point \(P(a,f(a))\) is the line through \(P\) with slope \[m = \lim_{x\to a}\frac{f(x)-f(a)}{x-a},\] provided that this limit exists.

Definition. The derivative of a function \(\boldsymbol{f}\) at a number \(\boldsymbol{a}\), denoted by \(f'(a)\), is \[f'(a) = \lim_{h\to 0}\frac{f(a+h)-f(a)}{h},\] provided that this limit exists. If we write \(x=a+h\), then we have \(h=x-a\) and \(h\) approaches 0 if and only if \(x\) approaches \(a\). Therefore, an equivalent way of stating the definition of he derivative, as we saw in finding tangent lines, is \[f'(a) = \lim_{x\to a}\frac{f(x)-f(a)}{x-a}.\]

Definition. A function \(f\) is differentiable at a if \(f'(a)\) exists. It is differentiable on an open interval (a,b) if it differentiable at every number in the interval.

Theorem. If \(f\) is differentiable at \(a\), then \(f\) is continuous at \(a\).

• Differentiation is linear. For any functions \(f\) and \(g\) and any real numbers \(a\) and \(b\), the derivative of the function \(h(x)=af(x)+bg(x)\) with respect to \(x\) is \[h'(x)=af'(x)+bg'(x).\]

• The product rule. The derivative of the function \(h(x) = f(x) g(x)\) with respect to \(x\) is \[h'(x)=(fg)'(x)=f'(x)g(x)+f(x)g'(x).\]

• Chain rule. The derivative of the function \(h(x)=f(g(x))\) is \[h'(x)=f'(g(x))\cdot g'(x).\]

• The inverse function rule. If the function \(f\) has an inverse function \(g\), meaning that \(g(f(x))=x\) and \(f(g(y))=y\), then \[g'=\frac{1}{f'\circ g}.\]

• The polynomial or elementary power rule. If \(f(x) = x^r\), for any real number \(r\neq 0\), then \[f'(x)=rx^{r-1}.\]

Increasing and Decreasing Functions

An increasing function is a function where: if \(x_1 > x_2\), then \(f(x_1) > f(x_2)\) , so as \(x\) increases, \(f(x)\) increases. A decreasing function is a function which decreases as \(x\) increases. Of course, a function may be increasing in some places and decreasing in others. A point where a function changes from an increasing to a decreasing function or visa-versa is known as a turning point. A turning point is a type of stationary point (see below). We can use differentiation to determine if a function is increasing or decreasing:

A function is increasing if its derivative is always positive. A function is decreasing if its derivative is always negative.

Examples

1. \(y = -x\) has derivative -1 which is always negative and so \(-x\) is decreasing.

2. \(y = x^2\) has derivative \(2x\), which is negative when \(x<0\) and positive when \(x>0\). Hence \(x^2\) is decreasing for \(x<0\) and increasing for \(x>0\).

Stationary Points

Stationary points are points on a graph where the gradient is zero. The gradient is defined for a scalar-valued differentiable function \(f(x_1,\ldots,x_n)\) of several variables as the vector whose components are the partial derivatives. If \(n=1\), then the gradient is just the derivative of \(f\). There are three types of stationary points: maximum, minimum and points of inflection. The three are illustrated in Figure 5.

Figure 5. The three types of stationary points.

Maximum, Minimum or Point of Inflection?

At all the stationary points, the gradient is the same (\(=0\)) but it is often necessary to know whether you have found a maximum point, a minimum point or a point of inflection. Therefore the gradient at either side of the stationary point needs to be looked at (alternatively, we can use the second derivative, see below).

At maximum points, the gradient is positive just before the maximum, it is zero at the maximum and it is negative just after the maximum. At minimum points, the gradient is negative, zero then positive. Finally at points of inflexion, the gradient can be positive, zero, positive or negative, zero, negative.

The second derivative is what you get when you differentiate the derivative. The second derivative can be used as an easier way of determining the nature of stationary points (whether they are maximum points, minimum points or points of inflection).

A stationary point \(x_0\) on a curve occurs when \(f'(x_0)=0\). Once we have established where there is a stationary point, the type of stationary point (maximum, minimum or point of inflexion) can be determined using the second derivative as follows:

• If \(f''(x_0)>0\), then \(f(x_0)\) is a minimum point.

• If \(f''(x_0)<0\), then \(f(x_0)\) is a maximum point.

• If \(f''(x_0)=0\), then \(f(x_0)\) could be a maximum, minimum or point of inflexion.

There are many ways of formally defining an integral, not all of which are equivalent. The most commonly used definitions of integral are Riemann integrals and Lebesgue integrals. Here, we focus on Riemann integrals.

Definition. The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval. Let \([a, b]\) be a closed interval of the real line; then a tagged partition of \([a, b]\) is a finite sequence \[a=x_{0}\leq t_{1}\leq x_{1}\leq t_{2}\leq x_{2}\leq \cdots \leq x_{n-1}\leq t_{n}\leq x_{n}=b.\]

This partitions the interval \([a, b]\) into n sub-intervals \([x_{i−1}, x_i]\) indexed by \(i\), each of which is “tagged” with a distinguished point \(t_i \in [x_{i−1}, x_i]\). A Riemann sum of a function \(f\) with respect to such a tagged partition is defined as \[{\displaystyle \sum _{i=1}^{n}f(t_{i})\,\Delta _{i}.}\] Thus, each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width. Let \(\Delta_i = xi−x_{i−1}\) be the width of sub-interval \(i\); then the mesh of such a tagged partition is the width of the largest sub-interval formed by the partition, \(\max_{i=1,\ldots,n} \Delta_i\). The Riemann integral of a function \(f\) over the interval \([a, b]\) is equal to \(S\) if: for all \(\epsilon > 0\) there exists \(\delta > 0\) such that, for any tagged partition \([a, b]\) with mesh less than \(\delta\), we have \[{\displaystyle \left|S-\sum _{i=1}^{n}f(t_{i})\,\Delta _{i}\right|<\varepsilon .}\] We write \(S = \int _{a}^{b}f(x)\,dx.\)

An integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral \[\int _{a}^{b}f(x)\,dx\] can be interpreted informally as the signed area of the region in the plane that is bounded by the graph of \(f\), the x-axis and the vertical lines \(x = a\) and \(x = b\). The area above the \(x\)-axis adds to the total and that below the \(x\)-axis subtracts from the total (see Figure 6).

The fundamental theorem of calculus is a theorem that links the concept of differentiation with the concept of integration. There are two parts to the theorem. The first part deals with the derivative of an antiderivative, while the second part deals with the relationship between antiderivatives and definite integrals.

First part. Let \(f\) be a continuous real-valued function defined on a closed interval \([a, b]\). Let \(F\) be the function defined, for all \(x \in [a, b]\), by

\[F(x)=\int _{a}^{x}\!f(t)\,dt.\] Then \(F\) is uniformly continuous on \([a, b]\) and differentiable on the open interval \((a, b)\), and

\[F'(x)=f(x), \qquad\text{for all }x \in (a, b).\]

Corollary. If \(f\) is a real-valued continuous function on \([a,b]\) and \(F\) is an antiderivative of \(f\) in \([a,b]\) then

\[\int _{a}^{b}f(t)\,dt=F(b)-F(a).\] The corollary assumes continuity on the whole interval. This result is strengthened slightly in the following part of the theorem.

Second part (the Newton–Leibniz axiom). Let \(f\) be a real-valued function on a closed interval \([a,b]\) and \(F\) an antiderivative of \(f\) in \([a,b]\), i.e., \(F'(x)=f(x).\) If \(f\) is Riemann integrable on \([a,b]\) then

\[\int _{a}^{b}f(x)\,dx=F(b)-F(a).\] The second part is somewhat stronger than the corollary because it does not assume that \(f\) is continuous.

When an antiderivative \(F\) exists, then there are infinitely many antiderivatives for \(f\), obtained by adding an arbitrary constant to \(F\). Also, by the first part of the theorem, antiderivatives of \(f\) always exist when \(f\) is continuous.

The definite integral \[\int_a^bf(x)\,dx\] is a number and represents the area under the curve \(f(x)\) from \(x=a\) to \(x=b\). The indefinite integral (also known as antiderivative) \[\int f(x)\,dx\] is a function and answers the question What function when differentiated gives f(x)?

Example. The function \({\displaystyle F(x)={\tfrac {x^{3}}{3}}}\) is an antiderivative (indefinite integral) of \({\displaystyle f(x)=x^{2}}\) while \(\int_1^2 f(x)\, dx = F(2) - F(1) = {\tfrac {2^{3}}{3}} - {\tfrac {1^{3}}{3}} = {\tfrac {7}{3}}\) is the definite integral of \(f\) between 1 and 2.

Here is a list of some techniques that are frequently successful when seeking antiderivatives of functions.

• Substitution or u-substitution or change of variables.
• Trigonometric substitution.
• Integration by parts or partial integration.