MATH 127 MIDTERM 1

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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
• 3B Computational Mathematics
• Lots of tutoring experience
• Interests:
• 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

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
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!
Example

Domain of

Domain of

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.
Example - Solution
• Graph:
• Domain:
• Range:
• Inverse on :

(0, 2)

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/

Example Problem
• Compute .
• Solution:
Inverse Trig FUNCTIONS

arcsin

arccos

arctan

Examples

Prove that .

Picture proof…

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

• 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)
• 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:
Tangent Line Slopes

What’s the point of limits?

Finding the slope of a tangent!

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