MATH1011 - Lecture 2

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MATH1011 - Lecture 1
Skeleton Slides

Be able to convert between parametrial equations into an equation with a relationship between $x$ and $y$ (and $z$) allowing you to sketch curves/shapes in 2D/3D space.

Functions of Two Variables

• A function of two variables is a rule that assigns to each $(x,y)\in D$ a real number $f(x,y)$
  • Converts a vector into a real number

Level Curves

• Gives visual information about a surface (read graphing functions of two variables on slides) in a 2D form

  

  • Level curves are traditionally found in maps in order to inform the reader on the shape of the terrain

• Imagine a horizontal plane $k$ cutting through the surface

  • $k$ is some real constant (in the range of $f$) that defines a curve in $D$ called a level curve of $f$
  • The level curve is created where the surface intersects the plane $k$

• When asked to sketch level curves, decide an integer constant to be the difference between each level of $k$

  • e.g. $k_0=0,k_1=1,...,k_n=n$

Level Surfaces

• Just as we can represent a 3D shape in 2D with level curves, we can represent a 4D shape, one that normally cannot be represented geometrically, in 3D, through level surfaces

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End of Chapter 1

Limits and Continuity: Slides

Limits of Scalar-Valued Functions

• For a refresher on limits of scalar-valued functions, watch lecture recording

One-Sided Limits

• For some functions, approaching $c$ from either side would produce a different limit
  • In this scenario, there is no true limit, however, there is a special notation to write a limit for either side
• We write:
  • $\underset{x\to c^{-}}{\lim }f(x)=L$ if $f(x)$ gets close to $L$ as $x$ gets close to $c$ with $x<c$
  • $\underset{x\to c^{+}}{\lim }f(x)=L$ if $f(x)$ gets close to $L$ as $x$ gets close to $c$ with $x>c$
• $\underset{x\to c}{\lim }f(x)=L$ only exists if the limits from both sides exist and are equal

Limit Laws

1. $\underset{x\to c}{\lim }(f(x)+g(x))=\underset{x\to c}{\lim }f(x)+\underset{x\to c}{\lim }g(x)$
2. For any constant $a\in \mathbb{R},$ $\underset{x\to c}{\lim }(a\cdot f(x))=a\cdot \underset{x\to c}{\lim }f(x)$
3. $\underset{x\to c}{\lim }(f(x)\cdot g(x))=(\underset{x\to c}{\lim }f(x))\cdot (\underset{x\to c}{\lim }g(x))$
4. If $\underset{x\to c}{\lim }g(x)\ne 0$, then $\underset{x\to c}{\lim }\frac{f(x)}{g(x)}=\frac{{\lim }_{x\to c}f(x)}{{\lim }_{x\to c}g(x)}$

Need to practice applying and using limit laws

Squeeze Theorem

Read about Squeeze Theorem on slides
Useful for proving the limit of an oscillating function (e.g. a trigonometric function)

• Worked Example:

  

Diverging to $\pm \infty$

• Whenever a function approaches $\pm \infty$ as $x\to a$, we consider that the limit does not exist
  • It can still be denoted as $\underset{x\to a}{\lim }f(x)=\infty$ or $\underset{x\to a}{\lim }f(x)=-\infty$

L’Hospital’s Rule

• Watch lecture recording or read slides

Limits of Vector-Valued Functions

• Limit of $r(t)$ exists if the limit of $x(t),y(x)$ and $z(t)$ exist
  • ($r(t)=(x(t),y(t),z(t))$)

Bolded variables represent vectors
e.g. $f(\mathbf{x})$ could mean $f(x,y)$

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MATH1011 - Lecture 3
Annotated Slides