MATH 127 MIDTERM 1

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• 3B Computational Mathematics

• Lots of tutoring experience

• Interests: – Reading– Pokémon– Calculus

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Outline

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

Writing Solutions

• 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.

A good solution includes…

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 Inequality

Absolute Value EquationSolve for x:

Introduction to Functions

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”!

fX Y

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

Even and Odd Polynomials

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,1. Compress f horizontally by k (reflect in y-axis if k < 0) 2. Translate f to the right p units3. Stretch f vertically by a (reflect in x-axis if a < 0) 4. 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.

Example

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

A CB gf

Composition of Functions

• You can only compose functions when the range of the inner function 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:1. Find all points where

These points remain fixed.2. Find all points where

These become vertical asymptotes. 3. Increasing sections become decreasing sections, and vice

versa.Positive sections remain positive sections, and vice versa.Maxima become minima, and vice versa.

4. 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 -1 that “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 -1 with 1/f !!

Inverse Functions

• If we have a function

Then 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 -1 is 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 of

on a suitable interval.

Exponents and Logarithms

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)

Trigonometry

Review of Trigonometry

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

Unit Circle: r = 1

Trig Graphs

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Trig Equations: Formulas to Know

• Compound Angles:

Example Problem

• Compute .• Solution:

Inverse Trig FUNCTIONS

arcsin

arccos

arctan

Examples

Prove that .

Picture proof…

Limits and Tangents

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:

Limits to Remember

Tangent Line SlopesWhat’s the point of limits?

Finding the slope of a tangent!

Finding Slopes of TangentsTo 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 Curves..

• Second degree curve: An equation of the form

Case ShapeA = B Circle

AB > 0 EllipseAB < 0 Hyperbola

A = B = 0 LineAB = 0 Parabola

Conic Sections

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

Graphing Ellipses

• Complete the squares to get something of the form

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Graphing Hyperbolas

• Complete the squares to get something of the form

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

Graphing Parabolas

• Complete the square to get something of the form

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

Questions and Practice Problems