Book cover for Applied Calculus for the MLSS A Brief Approach

Applied Calculus for the MLSS A Brief Approach

Soo T. Tan

ISBN #9781285464640

10th Edition

3,245 Questions

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46,959 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter begins with a review of essential algebraic concepts, introducing the Cartesian coordinate system to represent points with ordered pairs. It explains how to calculate the distance between two points and introduces the concept of the slope of a line, which is foundational in calculus. Additionally, the chapter illustrates the use of these algebraic techniques to solve real-world problems, such as calculating the funds required to purchase a minimum number of shares in a company.

Learning Objectives

1

Review and apply basic algebra concepts as a foundation for more advanced topics.

2

Understand and use the Cartesian coordinate system to represent points using ordered pairs.

3

Compute the distance between two points using the algebraic distance formula.

4

Determine the slope of straight lines and recognize its importance in calculus.

5

Apply algebraic methods to real-world problems, such as calculating the cost for purchasing shares.

Key Concepts

CONCEPT

DEFINITION

Algebra Review

A refresher on basic algebraic principles including operations, equations, and manipulation of expressions.

Cartesian Coordinate System

A system that represents points in a plane using ordered pairs of real numbers (x, y).

Ordered Pair

A pair of numbers (x, y) used to denote a specific location in the Cartesian plane.

Distance Formula

A formula derived from the Pythagorean theorem used to calculate the distance between two points: d = √[(x2 - x1)² + (y2 - y1)²].

Slope

A measure of the steepness of a straight line, defined as the ratio of the change in y to the change in x between two points.

Real-World Financial Calculation

The application of algebra to compute costs, such as determining the total money needed to purchase a given number of shares.

Example Problems

Example 1

Show the interval on a number line. $$(3,6)$$

Example 2

Show the interval on a number line. $$(-2,5]$$

Example 3

Show the interval on a number line. $$[-1,4)$$

Example 4

Show the interval on a number line. $$\left[-\frac{6}{5},-\frac{1}{2}\right]$$

Example 5

Show the interval on a number line. $$(0, \infty)$$
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Step-by-Step Explanations

QUESTION

How do you calculate the distance between two points, (x1, y1) and (x2, y2), in the Cartesian plane?

STEP-BY-STEP ANSWER:

Step 1: Subtract the x-coordinate of the first point from the x-coordinate of the second point to find the horizontal difference (Δx = x2 - x1).
Step 2: Subtract the y-coordinate of the first point from the y-coordinate of the second point to find the vertical difference (Δy = y2 - y1).
Step 3: Square both differences: (Δx)² and (Δy)².
Step 4: Add the squared differences: (Δx)² + (Δy)².
Step 5: Take the square root of the sum to obtain the distance: d = √[(Δx)² + (Δy)²].
Final Answer: This formula gives the algebraic distance between the two points.

Distance Between Two Points

QUESTION

How can you calculate the total amount of money needed to purchase at least 100,000 shares of a company?

STEP-BY-STEP ANSWER:

Step 1: Identify the number of shares to be purchased (at least 100,000 shares).
Step 2: Determine the price per share from market data or provided example (e.g., from Example 11 on page 21).
Step 3: Multiply the number of shares by the price per share: Total Cost = Number of Shares × Price per Share.
Step 4: Review the calculation to ensure that it meets the minimum share requirement.
Final Answer: The result is the total cost required to acquire at least 100,000 shares.

Share Purchase Cost Calculation

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Common Mistakes

  • Misapplying the distance formula by not correctly subtracting corresponding coordinates.
  • Ignoring the sign of the differences when calculating ?x and ?y, which can lead to errors in squaring.
  • Confusing the concepts of slope and distance.
  • Overlooking real-world factors in financial calculations, such as market fluctuations, when applying algebraic methods.