Download From Midterm 2, up to the Final Exam. - Math KSU

FINAL EXAM PREPARATION
MATHEMATICS 320
INSTRUCTOR: MICHAEL B. SCOTT
Preparation for a successful final exam.
• You should be able to understand, apply, and explain all definitions, theorems, and concepts we have
covered in class. Refer to the list of topics for each of the midterms, the list below, and your lecture
notes.
• In addition to being able to understand and solve each of the problems given in the homework, group
exercises, projects, and the set of final exam review problems, you should also be able to explain and
justify your solutions to the problems.
Topics list (Sections 7.3,7.4,12.1-12.4,13.1-13.4). Note: Refer to your lecture notes for solutions to any
examples given below.
Section 7.3
Ratios.
• Definition. A ratio is an ordered pair of numbers, written a : b, with b 6= 0.
Note:
a
a:b⇔ .
b
Comment. You should think of ratios as an application of fractions that allow us to compare the
relative sizes of two quantities.
• Models
– Part-to-Part. Example: The ratio of undergraduate students to graduate students at K-State.
– Part-to-Whole. Example: The ratio of graduate students to all students at K-State.
– Whole-to-Part. Example: The ratio of all students to graduate students at K-State.
• Ratio Equality:
a : b = c : d if and only if ad = bc.
Proportions.
• Definition. A Proportion is a statement the two given ratios are equal.
Example. Stan is trying to find the height of his apartment building. Stan, who is 6ft tall, measures
the length of his shadow to be 5ft. Then he measures the shadow of his apartment building to be
35ft. How tall is his apartment building?
Section 7.4
• Idea: You should think of a percent as another representation of a fraction. A percent is designed
to analyze a part-to-whole model where the whole is always equal to 100 parts. Therefore, 1 whole
is equivalent to 100%.
• Convert from one representation to another. We now have three representations of fractions: fractions, decimals, and percents. Just as we had to learn to convert between decimals and fractions, we
now have to be able to convert between decimals, fractions, and percents.
• Solving Problems. Use a proportion to solve percent problems.
part
percent
=
.
whole
100
Example: A 4200-pound automobile contains 357 pounds of rubber. What percent of the car’s total
weight is rubber? Answer: 85%
Section 12.1
• Definitions. (Note: Not all definitions are the same in every book. For example, in some books a
square is not a rectangle.)
– Line: Shortest set of points between two distinct points. A line segment is a line of finite length.
Two line segments with equal length are said to be congruent.
Date: Spring 2003.
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– Angle: An angle is the common point of two lines segments called a vertex. Two angles of equal
measure are said to be congruent.
∗ Right Angle: A 90◦ angle.
∗ Acute Angle: Less than 90◦ .
∗ Obtuse: Greater than 90◦ , but less than 180◦ .
∗ Straight Angle: A 180◦ angle (think straight line).
– Triangle: A triangle consists of three line segments joining three non-collinear points. Each of
the three segments is called a side, and each of the three non-collinear points is a vertex.
Types of Triangles:
∗ Acute: All angles less than 90◦ .
∗ Right: Contains a right angle.
∗ Obtuse: Contains one obtuse angle.
∗ Equilateral: All sides are equal.
∗ Equiangular: All interior angles are the same.
∗ isosceles: Contains two sides that are equal.
∗ scalene: All sides have different length.
– Quadrilaterals: Consists of four line segments joining for non-collinear points.
Types of Quadrilaterals:
∗ Trapezoid: Has exactly one pair of parallel sides. A trapezoid is said to be isosceles if its
non-parallel sides are congruent.
∗ Parallelogram: Has two pairs of parallel sides.
∗ Rhombus: A quadrilateral with all equal sides.
∗ Rectangle: A parallelogram with four right angles.
∗ Square: A parallelogram with four right angles and all equal sides.
Section 12.2
Symmetry
– Reflection Symmetry.(flip about an axis) Examples: scalene, isosceles, and equilateral triangles.
Also, a parallelogram.
– Rotation Symmetry. (get same figure by rotating around less than one full turn) Examples:
Equilateral triangle, isosceles triangle, square, circle, and parallelogram.
Regular Polygons
– simple closed curve: curve traced with same starting and end points without crossing or retracing
any part of curve.
– Polygon: Simple closed curve made up of line segments.
– regular polygon or n-gon: Polygon where all sides are congruent.
– Convex: A shape is convex if any line segment joining any two points in the shape stays inside
the shape.
Section 12.3
Points, Lines, and Planes. Give Examples.
Parallel Lines: Two lines that do not intersect.
Transversal: A line that intersects two other lines.
Angles
– Vertical angles.
– Supplementary Angles: Two angles that sum to 180◦ are said to be supplementary.
– Complementary angles: Two angles that sum to 180◦ are said to be complementary.
– Corresponding angles.
Corresponding angles are equal if and only if when a transversal crosses two parallel lines.
– alternate interior angles.
Alternate interior angles are equal if the transversal crosses two parallel lines.
The interior angles of a triangle sum to 180◦ .
Section 12.4
Two requisite facts:
– The sum of the interior angles of a triangle is 180◦ .
– The sum of a angles around a point is one rotation or 360◦ .
Notation. m∠A means the measure of angle A usually in degrees.
Central and Vertex angle measures of a an n-gon?
• Alternative computation on table 12.3 page 587.
Section 13.1
• The measurement process.
(1) Select attribute to measure. (length, area, volume, weight, temperature, time, etc.)
(2) Select an appropriate unit of measurement.
– nonstandard units. Examples: hands, steps, heartbeats for time, etc.
– standard units. Example: English system, metric system.
(3) Determine the quantity of units to measure the attribute.
• Ideal System of Units
– Portability
– Simple ratios among units of same type.
– Different types of units are interrelated. Example: 1 dm3 = 1 L = 1 kg of water.
• Length, Area, and Volume.
How many square inches are in a square foot?
How many cubic inches are in a cubic foot?
• Dimensional Analysis. (Converting between units.)
Example: Using the conversion ratio of 2.54 cm to 1 inch, determine how many kilometers are in
one mile.
Section 13.2
• Perimeter: The sum of the lengths of the sides of a polygon.
• Perimeters of Common Figures
– Parallelogram: 2a + 2b
– Kite: 2a + 2b
• Circumference: The perimeter of a circle given by 2πr.
Example: Find the circumference of a circle with diameter 6.
• Area. Measured in square units.
– Area of a square: s2 or b · h
– Area of a rectangle: l · w or b · h
– Area of a Triangle: 21 b · h
– Parallelogram. b · h Prove.
• Area of a circle: A = πr2 .
• Pythagorean Theorem. a2 + b2 = c2 .
Section 13.3
• Surface Area: The total area of an objects exterior surfaces.
• Lateral Area: (side) The surface area not including the area of the bases.
• Basic formula for the surface area: Surface Area = Lateral Area + Area of base(s).
• Surface area of a rectangular solid with length l, width w, and height h.
SA = LA + 2 · Area of Base = 2 · (l · h + w · h) + 2 · l · w
• Surface area of a Right Circular Cylinder with radius r and height h.
SA = LA + 2 · Area of Base = (2πr)h + 2πr2
• Surface area of a Sphere with radius r.
SA = 4πr2
Section 13.4
• Principle for finding volume: Volume = Area of Base × height.
Examples: Cylinder, and Rectangular solids.
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• The volume of a sphere, πr3 .
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Department of Mathematics, Kansas State University
E-mail address: [email protected]