Class 15: Test 1

[ List of Lectures | Math 1100 Index ]

[ List of Lectures | Math 1100 Index ]

Problem of the day

This problem is due on Friday, March 5:
POW 8

Assignments during second test period.

These are the problems you should work before March 5.

Section 2.2 page 205 problems 8n+1, for n=0,...,11;

Section 2.3 page 218 problems 1-6 and 6n+1, for n=1,...,8;

Section 2.4 page 232 problems 1-6 and 4n+1, for n=2,…,22;

These are the problems you should work before March 19.

Section 2.5 page 244 problems 1-6 and 6n+1, for n=1,...,14;

Review page 249 problems 4n+1, n=0,…,14;

Section 2.6 page 232 problems 1-6 and 4n+1, for n=2,…,22;

Section 3.1 page 263 problems 1-6 and 4n+1, for n=0,…,19;

These are the problems you should work before March 26.

Section 3.2 page 277 problems 4n+1, n=0,...,12;

Section 3.3 page 289 problems 4n+1, n=0,...,7;

Section 3.4 page 298 problems 4n+1, n=2,...,7

Brief review of the lecture

The single most important operation on functions is composition of functions. There are many ways to form new functions out of old ones, like adding them, multiplying them, or subtracting them. These are operations we can perform on all real numbers, so we can easily understand these operations. But composition of functions is an operation that we can only perform on functions. You'll see in the calculus course a formula called the Chain Rule. Its purpose is to enable you to find the derivative of a function which is build as a composition. In order to understand the use of this important rule, you must understand composition of functions very well. For that reason, we'll spend more than one full lecture discussing this operation.

Leftovers

  1. First we discussed the notion of slope and talked about the several forms of equations for lines, slope-intercept, point-slope, and the general form.
  2. Reflection across a line. Symmetry with respect to
    1. the x-axis
    2. the y-axis
    3. the line y=x
    4. the origin
  3. The mid-point formula was discussed, as well as a formula to find an arbitrary point between any other two points

  4. The idea of a relation was introduced. The following definition was given:

    A relation is a set of ordered pairs of real numbers.

    Several examples of relations were discussed and graphed in the class, such as

    S1={(x,y):x²+y²=1}

    S2={(x,y):y<2x-3}

    S3={(x,y):|x|+|y|<=2}

    S4={(x,y):xy>0}

  5. A property of relations called symmetry was introduced. These are
    1. x-axis symmetry
    2. y-axis symmetry
    3. (0,0) or origin symmetry
    4. xy-symmetry, or symmetry with respect to the line y=x.

    When is a relation a function?

    Given the graph of a relation, we may perform the Vertical Line Test: If every vertical line crosses the graph at most once, then the relation is a function.

    Given an equation, we should be able to solve it for y in terms of x and get only one y for each x.

    Given a set of ordered pairs, there cannot be two ordered pairs with the same first coordinate.

  6. More of the relations given in Lecture 14 were discussed and graphed today.

  7. Symmetry , a property introduced in Lecture 14, was discussed in more details. Definitions were given:

    A relation R is symmetric with respect to
    1. the x-axis if for each pair (x,y) in R, (x,-y) is also in R,
    2. the y-axis if for each pair (x,y) in R, (-x,y) is also in R,
    3. the origin if for each pair (x,y) in R, (-x,-y) is also in R,
    4. the xy-line if for each pair (x,y) in R, (y,x) is also in R.

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