Week 5: Preparation

Week 5: Preparation#

Reading Material#

We recommend that you read the textbook. Watching YouTube videos on the week’s topics can be useful, but it should not replace proper preparation for the week’s program and is not recommended as a standalone approach.

Read and study the following:

Key Concepts#

Note

Note that degree and order of polynomials are used synonymously. We use “degree”.

Orders of magnitude and Big O notation are not part of the curriculum, but it is useful to be familiar with “Big O” notation, as it appears, for example, in the SymPy demo.

After reading, you should be able to explain the following key concepts:

  • Tangent lines and tangent planes

  • Taylor polynomials in one variable

  • Taylor polynomials in \(n\) variables

  • Taylor polynomials of vector functions

  • Taylor’s theorem

  • Taylor’s limit formula \(f(x) = P_K(x) + \) a term with an \(\varepsilon\)-function.

  • Remainder term and remainder estimation

This week, we will explore these key concepts in great detail. We expect you to have familiarized yourself with these topics before lectures.

Preparatory Exercises#

I: Tangent Lines and Tangent Planes#

Question a: Tangent Line (One Variable)#

A function \(f : \mathbb{R} \to \mathbb{R}\) is given by:

\[\begin{equation*} f(x) = x^2. \end{equation*}\]

Find the equation for the tangent line at \(x_0 = 1\).

Hint

Calculate \(f(1)\) and \(f'(1)\), and then use the well-known formula for the slope of the line:

\[\begin{equation*} f'(1) = \frac{y - f(1)}{(x-1)}. \end{equation*}\]

Now isolate \(y\).

Answer

\(y = 1 + 2(x-1)\)

Question b: Tangent Plane (Two Variables)#

A function \(f : \mathbb{R}^2 \to \mathbb{R}\) is given by:

\[\begin{equation*} f(x,y) = x^2 + y^2. \end{equation*}\]

Find the equation of the tangent plane at the point \((1,1)\).

Hint

Calculate the gradient \(\nabla f(1,1)\), and use the formula

\[\begin{equation*} z = f(1,1) + \frac{\partial f}{\partial x}(1,1)(x-1) + \frac{\partial f}{\partial y}(1,1)(y-1). \end{equation*}\]

Answer

\(z = 2 + 2(x-1) + 2(y-1)\)

Intuitive explanation

A tangent line is the line that “touches” a curve at a single point without crossing it. Similarly, a tangent plane is a surface that “touches” a 3D surface at a single point. The tangent line and tangent plane are described by first-degree Taylor polynomials.

II: Taylor Polynomials in One Variable#

Question a: First-Order Approximation#

Find the first-degree Taylor polynomial of

\[\begin{equation*} f(x) = \sin(x) \end{equation*}\]

from the expansion point \(x_0 = 0\). This polynomial is also called the approximating polynomial of degree one.

Hint

Use \(f(0)\) and \(f'(0)\).

Answer

\(P_1(x) = 0 + 1\cdot x = x\)

Question b: Second-Order Approximation#

Find the second-degree Taylor polynomial of

\[\begin{equation*} f(x) = \cos(x) \end{equation*}\]

from the expansion point \(x_0 = 0\). This polynomial is also called the approximating polynomial of degree two.

Hint

Use \(f(0)\), \(f'(0)\) and \(f''(0)\).

Answer

\(P_2(x) = 1 + 0\cdot x + \frac{-1}{2}x^2 = 1 - \frac{1}{2}x^2\)

Intuitive explanation

Taylor polynomials provide a way to approximate a “complicated” function by a sum of simple polynomial terms. The first-degree polynomial gives a linear approximation, while the second-degree polynomial includes a quadratic term for better accuracy.

III: “Big O” Notation and Growth Rate#

Note

Here, we go slightly beyond the curriculum: “Big O” notation is used to describe how fast a function grows approximately. For example, if a function is \(O(n^2)\), it means that its growth rate is at most proportional to \(n^2\) as \(n\) approaches infinity.

For a given functional expression, it is usually sufficient to focus on the term that grows the fastest, as lower-order terms and constants do not affect asymptotic growth. For example, a function \(f : \mathbb{N} \to \mathbb{N}\) is given by

\[\begin{equation*} f(n) = 2^n + 100 n^5, \end{equation*}\]

of order \(O(2^n)\), since exponential growth, due to \(2^n\), causes a much faster increase than polynomial growth with the term \(n^5\) as \(n\) becomes large.

Intuitive explanation:
For large \(n\), the term \(2^n\) is dominating, and \(100n^5\) becomes relatively insignificant. Therefore, we only consider the fastest-growing term when determining asymptotic growth.

Now to the exercise. Let

\[\begin{equation*} f(n) = 4n^2 + 10n + 6. \end{equation*}\]

Show that \(f(n)\) is \(O(n^2)\). That is, find constants \(C > 0\) and \(n_0 \in \mathbb{N}\) such that for all \(n \geq n_0\), the following holds:

\[\begin{equation*} |4n^2 + 10n + 6| \leq C \cdot n^2. \end{equation*}\]

Hint

For \(n \geq 1\) note that:

\[\begin{equation*} 10n \leq 10n^2 \quad \text{and} \quad 6 \leq 6n^2. \end{equation*}\]

Use these inequalities to show that:

\[\begin{equation*} 4n^2 + 10n + 6 \leq 4n^2 + 10n^2 + 6n^2 = 20n^2. \end{equation*}\]

Answer

From the hint, it can be seen that one can, for example, choose \(C = 20\) and \(n_0 = 1\).

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