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MATH 127 MIDTERM 1. 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

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tutor maysum panju
Tutor: Maysum Panju
  • 3B Computational Mathematics
  • Lots of tutoring experience
  • Interests:
    • Reading
    • Pokémon
    • Calculus
2010 outreach trip
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
Outline
  • Introduction
  • Sets, inequalities, absolute values
  • Basic function concepts
  • Exponentials and Logarithms
  • Trigonometry
  • Limits
  • Second Degree Equations
  • Questions
writing solutions
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
Sets of Numbers
  • Set builder notation:S = { x | x satisfies some property }
  • Sets you should know:
  • Intervals:
absolute values
Absolute Values
  • Absolute value: the “size” of a number
what is a function
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 and Range
  • Domain: What is the input (x) allowed to be?
  • Range: What values (y) does the function hit?
even and odd functions
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 /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
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
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
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!
example1
Example

Domain of

Domain of

graphing reciprocals
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.
example2
Example

Given , graph .

inverse functions
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 functions1
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
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
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
Exponential Graphs
  • Given , the shape of the graph depends entirely on the choice of a.

If a > 1 If 0 < a < 1

logarithm laws
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
Logarithm Graphs
  • Given , the shape of the graph depends entirely on the choice of a.

If a > 1 If 0 < a < 1

example problem
Example problem
  • Graph the function
  • What is the domain?
  • What is the range?
  • Find the equation of the inverse on a suitable interval.
example solution
Example - Solution
  • Graph:
  • Domain:
  • Range:
  • Inverse on :

(0, 2)

review of trigonometry
Review of Trigonometry
  • Main ratios: sin, cos, tan (SOHCAHTOA)
  • Reciprocals: csc, sec, cot

Unit Circle: r = 1

trig graphs
TrigGraphs

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

trig graphs1
Trig Graphs

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

example problem1
Example Problem
  • Compute .
  • Solution:
inverse trig functions
Inverse Trig FUNCTIONS

arcsin

arccos

arctan

examples
Examples

Prove that .

Picture proof…

limits
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
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.
limits1
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
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 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 limit1
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 limit2
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
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
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
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
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
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
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:
tangent line slopes
Tangent Line Slopes

What’s the point of limits?

Finding the slope of a tangent!

finding slopes of tangents
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 curves1
Second Degree Curves..

Second degree curve: An equation of the form

conic sections
Conic Sections

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

graphing ellipses
Graphing Ellipses

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

Complete the squares to get something of the form

graphing hyperbolas
Graphing Hyperbolas

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

Complete the squares to get something of the form

graphing parabolas
Graphing Parabolas

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

Complete the square to get something of the form