Maths 1 Week 8 Summary

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Understanding Tangents, Continuity, and Differentiability

Our journey began with the concept of finding the slope of a line. The slope is a measure of the steepness of a line and is calculated as the ratio of the vertical change to the horizontal change between two points on the line. Mathematically, the slope $m$ is given by:

$$m=\frac{y_2-y_1}{x_2-x_1}$$

When we moved from straight lines to curves, we became interested in finding the tangent to the curve at a given point. The tangent to a curve at a point is a straight line that just touches the curve at that point. The slope of the tangent line at a point on the curve gives us the slope of the curve at that point.

However, we soon realized that drawing a tangent at every point on a curve is not always possible. This issue became apparent when we dealt with step functions and modulus functions. These functions posed challenges because their graphs have points where they break or take sharp turns. At these points, the tangent does not exist.

To address this, the concept of continuity was introduced. Continuity means that a graph has no breaks, jumps, or sharp turns. The hypothesis was that if a graph is continuous, then a tangent will exist at every point on the graph. Conversely, if a tangent does not exist at a point, the graph is not continuous at that point.

However, the modulus function contradicted this hypothesis. The modulus function is continuous at all points, but at the sharp turn (at $x=0$), it is not possible to draw a unique tangent. This issue also arose with cusp and kink graphs. These graphs are continuous but do not have a unique tangent at certain points.

Observe the graph carefully. For a very minute second, the tangent line(red line) on modulus graph disappers and it disappera at place where it changes its direction, while the tangent on parabola never disapper.

When we say that a unique tangent does not exist at a point, it means that we can draw infinitely many tangents at that point, each with a different slope. For example, at the sharp turn of the modulus function, we can draw infinitely many tangents, resulting in infinitely many slopes. However, we are interested in a unique slope, so we say that the tangent does not exist at that point.

To prove that the tangent indeed does not exist for the modulus function, cusp graphs, and kink graphs, the concept of differentiability was introduced. Differentiability means that the left-hand derivative and the right-hand derivative at a point are the same. If a function is differentiable at a point, then a tangent will surely exist at that point.

Mathematically, a function $f(x)$ is differentiable at a point $x=a$ if the following limit exists:

$$\underset{h\to 0}{lim}\frac{f(a+h)-f(a)}{h}$$

This limit represents the slope of the tangent to the curve at $x=a$. If the left-hand derivative and the right-hand derivative at $x=a$ are equal, then the function is differentiable at that point, and a unique tangent exists.

In summary, while continuity ensures that a graph has no breaks or jumps, it does not guarantee the existence of a unique tangent at every point. Differentiability, on the other hand, ensures that a unique tangent exists at a point. Therefore, for a function to have a unique tangent at every point, it must be both continuous and differentiable.

Limits

The concept of a limit is fundamental in calculus and analysis. It describes the behavior of a function as its argument approaches a particular point. Formally, the limit of a function $f(x)$ as $x$ approaches $a$ is $L$, written as:

$$\underset{x\to a}{lim}f(x)=L$$

This means that as $x$ gets arbitrarily close to $a$, $f(x)$ gets arbitrarily close to $L$.

Limit of a Sequence

A sequence is an ordered list of numbers. The limit of a sequence $\{a_n\}$ as $n$ approaches infinity is $L$, written as:

$$\underset{n\to \mathrm{\infty }}{lim}{a}_n=L$$

This means that as $n$ becomes very large, the terms $a_n$ get arbitrarily close to $L$.

Limit of a Function

The limit of a function at a point is the value that the function approaches as the input approaches that point. For example, the limit of $f(x)$ as $x$ approaches $a$ is $L$ if:

$$\underset{x\to a}{lim}f(x)=L$$

Continuity of a Function

A function $f(x)$ is continuous at a point $x=a$ if the following three conditions are met:

• $f(a)$ is defined.
• $\underset{x\to a}{lim}f(x)$ exists.
• $\underset{x\to a}{lim}f(x)=f(a)$.

In other words, the function has no breaks, jumps, or holes at $x=a$.

Formula of Continuity

The formula for continuity at a point $x=a$ is:

$$\underset{x\to a}{lim}f(x)=f(a)$$

Sandwich Principle

The Sandwich Principle, also known as the Squeeze Theorem, states that if $f(x)\le g(x)\le h(x)$ for all $x$ in some interval around $a$ (except possibly at $a$), and:

$$\underset{x\to a}{lim}f(x)=\underset{x\to a}{lim}h(x)=L$$

then:

$$\underset{x\to a}{lim}g(x)=L$$

Floor Function

The floor function, denoted $\lfloor x\rfloor$, returns the greatest integer less than or equal to $x$. For example:

$$\lfloor 3.7\rfloor =3\text{and}\lfloor -2.3\rfloor =-3$$

Ceiling Function

The ceiling function, denoted $\lceil x\rceil$, returns the smallest integer greater than or equal to $x$. For example:

$$\lceil 3.7\rceil =4\text{and}\lceil -2.3\rceil =-2$$

Modulus Function

The modulus function, denoted $|x|$, returns the absolute value of $x$. For example:

$$|3|=3\text{and}|-3|=3$$

Concept of Differentiability

A function $f(x)$ is differentiable at a point $x=a$ if the derivative $f^{\prime }(a)$ exists. Differentiability implies continuity, but continuity does not necessarily imply differentiability.

Derivative

The derivative of a function $f(x)$ at a point $x=a$ is the slope of the tangent line to the function at that point. It is defined as:

$$f^{\prime }(a)=\underset{h\to 0}{lim}\frac{f(a+h)-f(a)}{h}$$

Tangent and Linear Approximation

The tangent line to the function $f(x)$ at $x=a$ is the line that best approximates the function near that point. The equation of the tangent line is:

$$y=f(a)+f^{\prime }(a)(x-a)$$

Linear approximation uses the tangent line to approximate the value of the function near $x=a$.

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