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Tutor: Maysum Panju

- 3B Computational Mathematics
- Lots of tutoring experience
- Interests:
- Reading
- Pokémon
- Calculus

2010 Outreach Trip

Summary

Date Aug 20 – Sept 4

Location Cusco, Peru

# Students 22

Project Cost $16,000

Building Projects

Kindergarten Classroom provides free education

Sewing Workshop enables better job prospects

ELT Classroom enables better job prospects

More info @ studentsofferingsupport.ca/blog

Outline

- Introduction
- Sets, inequalities, absolute values
- Basic function concepts
- Exponentials and Logarithms
- Trigonometry
- Limits
- Second Degree Equations
- Questions

Writing Solutions

A good solution includes…

- An introductory statement: what you are given and what you have to show/find;
- A concluding statement: summarize the conclusion briefly;
- Justifications of the main steps: refer to definitions, rules, and known properties;
- Some sentences of guidance for the reader, e.g. how you are going to solve the problem.

Sets of Numbers

- Set builder notation:S = { x | x satisfies some property }
- Sets you should know:
- Intervals:

Absolute Values

- Absolute value: the “size” of a number

Absolute Value Equation

Solve for x:

What is a Function?

- Function: Turns objects from one set into objects in another set.
- For each x in X, assign some value y in Y.
- You can only assign one y per x.
- This implies the “Vertical Line Test”!

X

Y

f

Domain and Range

- Domain: What is the input (x) allowed to be?
- Range: What values (y) does the function hit?

Even and Odd Functions

- Even functions:
- Reflect along y-axis
- e.g. Polynomials with only even degree terms
- Odd functions:
- Reflect around the origin
- e.g. Polynomials with only odd degree terms

Increasing /Decreasing Functions

- Increasing functions: implies
- As x increases, so does f(x).
- Odd functions: implies
- As y increases, so does f(y).
- Note: these inequalities are “strict”.

Transformation of Functions

- Given a function We can transform it:
- Here,
- Compress f horizontally by k (reflect in y-axis if k < 0)
- Translate f to the right p units
- Stretch f vertically by a (reflect in x-axis if a < 0)
- Translate f upwards by q
- Horizontal transformations are “backwards” and appear inside the function f. These affect the domain.
- Vertical transformations are “normal” and appear outside the function f. These affect the range.

Function Compositions

- Given two functions f and g, sometimes we can “compose” them to make a new function.
- Given and The composition is given as

where

C

A

B

f

g

Composition of Functions

- You can only compose functions when the range of the innerfunction is within the domain of the outer function
- If the inner function hits a value that the outer function isn’t defined on, there’s a problem!

Graphing Reciprocals

- Given a function , graphing the reciprocal is easy:
- Find all points where These points remain fixed.
- Find all points where These become vertical asymptotes.
- Increasing sections become decreasing sections, and vice versa.Positive sections remain positive sections, and vice versa.Maxima become minima, and vice versa.
- Check any special points on a point-wise basis.

Example

Given , graph .

Inverse Functions

- An inverse function: Given a function f, define a new function f -1that “undoes” f.
- The inverse must be a function:
- Must pass the vertical line test!
- A function has a inverse if and only if
- It is “one-to-one” (passes the horizontal line test)
- It is strictly increasing or decreasing (if continuous)
- Do NOT confuse f -1with 1/f !!

Inverse Functions

- If we have a functionThen the inverse function satisfies
- Sometimes we must restrict the domain of f to ensure that the inverse is a function.
- In this case, the range of the inverse f -1is restricted.

Finding inverse functions

- If you have an equation , swap x with y and rearrange for y.
- If you have to choose between +/-, take the +. This corresponds to restricting the domain.
- If you have a graph, reflect the graph in the line y = x.
- Remember to restrict the domain to a 1-to-1 interval first (so it passes the horizontal line test)!
- Example: find the inverse ofon a suitable interval.

Exponent Laws

- Some common exponent laws:

- In general, the exponential operation is really powerful.
- Weak operations in exponents become stronger once you pull them out.
- Examples:
- Addition in the exponent becomes multiplication outside.
- Multiplication in the exponent becomes exponentiation outside.

Exponential Graphs

- Given , the shape of the graph depends entirely on the choice of a.

If a > 1 If 0 < a < 1

Logarithm Laws

- Think of logs as the inverse of exponentiations.
- In general, the logarithm operation is really weak.
- Strong operations in logarithms become weaker once you pull them out.
- Examples:
- Multiplication in the log becomes addition outside.
- Exponents in the log become multiplication outside.

Logarithm Graphs

- Given , the shape of the graph depends entirely on the choice of a.

If a > 1 If 0 < a < 1

Example problem

- Graph the function
- What is the domain?
- What is the range?
- Find the equation of the inverse on a suitable interval.

Review of Trigonometry

- Main ratios: sin, cos, tan (SOHCAHTOA)
- Reciprocals: csc, sec, cot

Unit Circle: r = 1

TrigGraphs

http://www.algebra-help.org/graphs-of-trigonometric-functions.html

Trig Graphs

http://www.eighty-twenty.org/

Trig Equations: Formulas to Know

- Compound Angles:

Example Problem

- Compute .
- Solution:

Limits

- In calculus, the main ideas involve working with very small numbers and very big numbers.
- Limits help us…
- Use extremely small values
- Reach really large values
- Predict the value of a function at a place it isn’t defined
- Relax the rules of domain restrictions
- Heuristically: the limit of a function at a point is the value the function “should” take at that point if it were nice and smooth.

Left and Right Limits

- The left limit of f(x) at the point x = a is the value the function approaches as x gets close to a from the left.
- The right limit of f(x) at the point x = a is the value the function approaches as x gets close to a from the right.

Limits

- If the left limit of f(x) AND the right limit of f(x) BOTH exist at the point x = a, and are EQUAL, then we say the limit of f(x) at a exists as well.
- In this case, the limit is equal to the limits from the left and right.

so

How to think about Limits

- When considering the limit of f(x) as x approaches a …

Think of the curve here and here

But NOT here

Computing a Limit

- Computing limits is an art.
- Adapt the technique for the problem!
- Given a graph, guess the limit by inspection.

No Limit Limit = L

Computing a Limit

- Given
- First thing to try: Substitute x = a in the equation. If it works, and the function is “continuous”, you’re done!
- Usually (in “interesting” problems), you get an indeterminate form:

Computing a Limit

- Some tricks to try:
- Factor and cancel out the problems.
- Multiply by A/A for some clever A to eliminate square roots (set up a difference of squares)
- Use trig identities to help you cancel things
- If absolute values are involved, use cases (show the limit does NOT exist by showing left limit is not the same as right limit)
- Change the variables
- Squeeze Law

Change of Variables

- To turn a limit to a more recognizable form, try a variable change.
- Example:
- If x approaches infinity, let h = 1/x. Then h approaches 0.
- Try to get to familiar limits you can apply.
- Practice:

Squeeze Law

- If the function f(x) is always between g(x) and h(x), and these two boundary functions approach a common limit L at x = a, then f(x) must also approach L at x = a.
- Common for limits involving [something that goes to 0] * sin( something )
- Example:

Continuity

- A function f(x) is continuous at x = a if it has a limit at that point, and the value of the limit is the same as the value of the function.
- The function is “smooth”; no need to “lift your pencil” when drawing
- A very nice property!
- Examples:
- Polynomials, trig functions, logarithms, exponentials, compositions of continuous functions… ON THEIR DOMAIN

Intermediate Value Theorem

- Continuous functions can’t skip values.
- “If I started down there, and ended up there, and moved continuously the whole way…”
- Usually use for proving existence of a root: Can’t go from negative to positive without passing 0.

Horizontal Asymptotes

- Does the curve flatten out as x gets very large (approaches infinity)?
- A curve MAY cross the horizontal asymptote, possibly MANY times.
- Usually to check this, divide out by the “strongest thing” in the limit.
- Example:

Vertical Asymptotes

- Places where the curve flies upwards or plummets downwards without bound
- Like a barrier the curve cannot pass
- Use limits to determine behaviour around asymptote: up or down?
- Check the sign of one-sided limits.
- Example:

Finding Slopes of Tangents

To find the slope of the tangent to f(x) at x = a:

Compute the limit

If this limit exists, this is the slope of the tangent to f(x) at x = a.

- Then f(x) is “differentiable” at a!

Second Degree Curves..

Second degree curve: An equation of the form

Conic Sections

Each second degree equation corresponds to slicing a double cone using a plane (or knife).

Graphing Ellipses

http://www.intmath.com/Plane-analytic-geometry/5_Ellipse.php

Complete the squares to get something of the form

Graphing Hyperbolas

http://people.richland.edu/james/lecture/m116/conics/translate.html

Complete the squares to get something of the form

Graphing Parabolas

http://people.richland.edu/james/lecture/m116/conics/translate.html

Complete the square to get something of the form

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