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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


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 @

  • 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”!




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







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!

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.

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

trig graphs1
Trig Graphs

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





Prove that .

Picture proof…

  • 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.
  • 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.


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:
  • 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

Complete the squares to get something of the form

graphing hyperbolas
Graphing Hyperbolas

Complete the squares to get something of the form

graphing parabolas
Graphing Parabolas

Complete the square to get something of the form