Business Calculus Demystified

By Rhonda Huettenmueller

Take the FEAR OUT of Business Calculus

Business Calculus Demystified clarifies the concepts and processes of calculus and demonstrates their applications to the workplace. Best-selling math author Rhonda Huettenmueller uses the same combination of winning step-by-step teaching techniques and real-world business and mathematical examples that have succeeded with tens of thousands of college students, regardless of their math experience or affinity for the subject.

With Business Calculus Demystified, you learn at your own pace. You get explanations that make differentiation and integration -- the main concepts of calculus -- understandable and interesting. This unique self-teaching guide reinforces learning, builds your confidence and skill, and continuously demonstrates your mastery of topics with a wealth of practice problems and detailed solutions throughout, multiple-choice quizzes at the end of each chapter, and a "final exam" that tests your total understanding of business calculus.

Learn business calculus for the real world! This self-teaching course conquers confusion with clarity and ease. Get ready to:

• Get a solid foundation right from the start with a review of algebra
• Master one idea per section -- develop complete, comfortable understanding of a topic before proceeding to the next
• Find a well-explained definition of the derivative and its properties; instantaneous rates of change; the power, product, quotient, and chain rules; and layering different formulas
• Learn methods for maximizing revenue and profit... minimizing cost... and solving other optimizing problems
• See how to use calculus to sketch graphs
• Understand implicit differentiation, rational functions, exponents, and logarithm functions -- learn how to use log properties to simplify differentiation
• Painlessly learn integration formulas and techniques and applications of the integral
• Take a "final exam" and grade it yourself!

Who says business calculus has to be boring? Business Calculus Demystified is a lively and entertaining way to master this essential math subject!

• Language: English
• Publisher: McGraw-Hill Education
• Release date: Jan 6, 2006
• ISBN: 9780071483438

Book preview

CHAPTER 1

The Slope of a Line and the Average Rate of Change

Calculus is the study of the rate of change. We use the slope of a line to describe the rate of change of a function. Instead of thinking of the slope of a line as a simple rise over run, we need to think of it as a number that measures how one variable changes compared to a change in the other variable. The numerator of the slope describes the change in y, and the denominator describes the change in x. For example, a slope of says that as x increases by 5, y increases by 3. A slope of says that as x increases by 5, y decreases by 3.

EXAMPLES

Interpret the slope.

As x increases by 3, y increases by 2.

As x increases by 9, y decreases by 20.

As x increases by 8, y increases by 1.

As x increases by 1, y decreases by 0.03. If we view −0.03 as instead, we see that as x increases by 100, y decreases by 3.

As x increases by 1, y increases by 1.

• y = 7 (This is the same as y = 0x + 7.)

The slope of this line is 0. If we think of 0 as , then we see that as x increases by 1, y does not increase nor does it decrease. In other words, the y-value does not change. In fact, x can change by any amount and y does not change.

• The daily cost of producing x units of a product is given by the equation y = 3.52x + 490.

The cost is y, and x is the number of units produced. The slope of tells us that as x increases by 1, y increases by 3.52. In other words, each unit costs $3.52 to produce.

• The property tax for a property valued at x dollars is .

As the value of property increases by $100, the tax increases by $0.5981.

• The demand function for a product is given by , where y units are demanded when x is the price per unit.

As the demand increases by 5 units, the price decreases by $4. (We could also interpret this slope to mean that as the price decreases by $4, demand increases by 5 units.)

• The monthly salary for an office manager is given by the equation y = 3800.

The slope of the line for this equation is 0, which means that no matter what happens to x, the y-value is always $3800. No matter how much (or how little) the manager works, her monthly salary stays the same.

PRACTICE

Interpret the slope.

1.

2. y = −2x + 1

3. y = −x

4. y = 10

5. The sales tax on purchases costing x dollars is y = 0.08x.

6. The pressure on a certain object submerged in the ocean is approximated by y = 170x + 6000, where x is the depth of the object, in feet, and y is the pressure, in pounds.

7. The nonfarm average weekly pay from 1997 to 2002 can be approximated by the equation y = 15.09x − 29708. (This equation is based on data from The Statistical Abstract of the United States, 123rd edition.)

SOLUTIONS

1. As x increases by 3, y increases by 7.

2. As x increases by 1, y decreases by 2.

3. As x increases by 1, y decreases by 1.

4. No matter how x changes, y does not change.

5. As the amount spent on purchases increases by $1, the sales tax increases by $0.08.

6. As the depth increases by 1 foot, the pressure on the object increases by 170 lbs.

7. The average weekly nonfarm wage increased by $15.09 each year from 1997 to 2002.

The rate of change for most functions is not the same for all x-values as it is with linear functions. For example, if a cup of hot coffee sits on a table for ten minutes, it will cool down faster in the third minute than in the eighth minute. So, the rate of temperature change varies for different periods of time. For most of the functions in this book, the y-values will increase or decrease at different rates for different values of x. In fact, for some values of x, the y-values can increase and for other values of x, the y-values can decrease. We will look at the average rate of change of a function between two x-values. The average rate of change of the function between two values of x is the slope of the line containing the two points on the graph of the function.

EXAMPLES

• Find the average rate of change for f(x) = x² − x − 2 between (1, −2) and (3, 4) and between (−2, 4) and (0, −2).

The average rate of change between (1, −2) and (3, 4) is the slope of the line between these two points.

Between x = 1 and x = 3, the average increase of the function is 3 as x increases by 1. See Figure 1.1.

Fig. 1.1.

The rate of change between (−2, 4) and (0, −2) is

Between x = −2 and x = 0, the average decrease of the function is 3 as x increases by 1. See Figure 1.2.

Fig. 1.2.

• Find the average rate of change for between x = 3 and x = 8.

Once we have computed the y-values for x = 3 and x = 8, we will put the points into the slope formula.

The points are (3, 2) and (8, 3). The average rate of change is

Between x = 3 and x = 8, the function increases by 1 on average as x increases by 5.

• Find the average rate of change for between x = −2 and x = 2. See Figure 1.3.

Fig. 1.3.

Because the slope of the line is 0, the average rate of change of the function is zero between x = −2 and x = 2. The function obviously changes in value but the changes negate each other.

• Find the average rate of change for between x = −3 and x = 0 and between x = 6 and x = 12.

We will first find the y-values for x = −3 and x = 0.

Between x = −3 and x = 0, the function increases, on average, by 2 as x increases by 3. The average rate of change is the same between x = 6 and x = 12.

The average rate of change for a linear function is the same between any two points on its graph.

PRACTICE

Find the average rate of change.

1. f(x) = x³ + x² − 4 between the points (−1, −4) and (2, 8).

2. f(x) = x⁴ − 4x², between x = −3 and x = 0.

3. between x = 0 and x = 2.

4. See Figure 1.4.

Fig. 1.4.

SOLUTIONS

1.

Between x = −1 and x = 2, the function increases by 4, on average, as x increases by 1.

2.

y1 = f(x1) = f(−3) = (−3)⁴ − 4(−3)² = 45

y2 = f(x2) = f(0) = 0⁴ − 4(0)² = 0

Between x = 0 and x = −3, the function decreases, on average, by 15 as x increases by 1.

3.

Between x = 0 and x = 2, the function increases, on average, by 1 as x increases by