Schaum's Outline of Mathematical Methods for Business, Economics and Finance, Second Edition

By Luis Moises Pena-Levano

The most useful tool for reviewing mathematical methods for business and economics classes—now with more content

Schaum's Outline of Mathematical Methods for Business, Economics and Finance, Second Edition is the go-to study guide for students enrolled in business and economics courses that require a variety of mathematical skills. No mathematical proficiency beyond the high school level is assumed, enabling students to progress at their own rate and adapt the book to their own needs.

With an outline format that facilitates quick and easy review, this guide helps you understand basic concepts and get the extra practice you need to excel in business and economics courses. Schaum's Outline of Mathematical Methods for Business, Economics and Finance, Second Edition supports the bestselling textbooks and is ideal study aid for classes such as Calculus for Business, Applied Calculus, Calculus for Social Sciences and Calculus for Economics. Chapters include Equations and Graphs, Functions, Systems of Equations, Linear (or Matrix) Algebra, Linear Programming, Differential Calculus, Exponential and Logarithmic Functions, Integral Calculus, Calculus of Multivariable Functions, and more.

Features

• NEW in this edition: Additional problems at the end of each chapter
• NEW in this edition: An additional chapter on sequences and series
• NEW in this edition: Three computer applications of Linear Programming in Excel
• More than 1,000 fully solved problems
• Outline format to provide a concise guide for study
• Clear, concise explanations covers all course fundamentals
• Supplements the major bestselling textbooks in economics courses
• Appropriate for the following courses: Calculus for Business, Applied Calculus, Calculus for Social Sciences, Calculus for Economics

• Language: English
• Publisher: McGraw-Hill Education
• Release date: Nov 15, 2021
• ISBN: 9781264266883

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Preface

Students of undergraduate business, finance, and economics, as well as candidates for the MBA and MA degrees in economics today need a variety of mathematical skills to successfully complete their degree requirements and compete effectively in their chosen careers. Unfortunately, the requisite mathematical competence is not the subject of a single course in mathematics such as Calculus I or Linear Algebra I, and many students, pressed with the demands from business and economics courses, do not have space in their schedules for a series of math courses. Mathematical Methods for Business, Economics, and Finance, Second Edition is designed to cull the mathematical tools, topics, and techniques essential for success in business and economics today. It is suitable for a one- or two-semester course in business mathematics, depending on the previous background of the students. It can also be used profitably in an introductory calculus or linear algebra course by professors and students interested in business connections and applications.

The theory-and-solved-problem format of each chapter provides concise explanations illustrated by examples, plus numerous problems with fully worked-out solutions. No mathematical proficiency beyond the high-school level is assumed. The learning-by-doing pedagogy will enable students to progress at their own rate and adapt the book to their own needs.

Mathematical Methods for Business, Economics, and Finance, Second Edition can be used by itself or as a supplement to other texts for undergraduate and graduate students in business and economics. It is largely self-contained. Starting with a basic review of high-school algebra in Chapter 1, the book consistently explains all the concepts and techniques needed for the material in subsequent chapters.

This edition introduces one new chapter on sequences and series, and two sections on computing linear programming and series using Microsoft Excel. The end of book also presents a new section of problems called Additional Practice Problems, which provides a template that can be used to test students’ knowledge on each chapter, with solutions in the subsequent pages (called ‘Solution to Practice Problems’).

This book contains over 1,200 problems, all of them solved in considerable detail. To derive the most from the book, students should strive as soon as possible to work independently of the solutions. This can be done by solving problems on individual sheets of paper with the book closed. If difficulties arise, the solution can then be checked in the book.

For best results, students should never be satisfied with passive knowledge—the capacity merely to follow or comprehend the various steps presented in the book. Mastery of the subject and doing well on exams require active knowledge—the ability to solve any problem, in any order, without the aid of the book.

Experience has proved that students of very different backgrounds and abilities can be successful in handling the subject matter of this text when the material is presented in the current format.

I wish to express my gratitude to father Alberto, my mother Betty, my sister Mirella, and my brother Willy, for raising me to be the man I am today. I also want to thank Ernesto, Mario, Daniele, Shaheer, Jared, Jose, Víctor, Hugo, Edward, Irvin, and Andres, my 11 best friends who are part of my family and have encouraged me and helped me improve my skills and abilities over these last 15 years. I would also like to dedicate this book to my goddaughters Briggitte, Ashley, and Valentina, my nephew Fitzgerald, and my godson Adrian, who I envision to be the future of the upcoming generations.

LUIS MOISÉS PEÑA LÉVANO

Contents

Chapter 1     REVIEW

1.1       Exponents

1.2       Polynomials

1.3       Factoring

1.4       Fractions

1.5       Radicals

1.6       Order of Mathematical Operations

1.7       Use of a Pocket Calcular

Chapter 2     EQUATIONS AND GRAPHS

2.1       Equations

2.2       Cartesian Coordinate System

2.3       Linear Equations and Graphs

2.4       Slopes

2.5       Intercepts

2.6       The Slope-Intercept Form

2.7       Determining the Equation of a Straight-Line

2.8       Applications of Linear Equations in Business and Economics

Chapter 3     FUNCTIONS

3.1       Concepts and Definitions

3.2       Graphing Functions

3.3       The Algebra of Functions

3.4       Applications of Linear Functions for Business and Economics

3.5       Solving Quardratic Equations

3.6       Facilitating Nonlinear Graphing

3.7       Applications of Nonlinear Functions in Business and Economics

Chapter 4     SYSTEM OF EQUATIONS

4.1       Introduction

4.2       Graphical Solutions

4.3       Supply-and-Demand Analysis

4.4       Break-Even Analysis

4.5       Elimination and Substitution Methods

4.6       Income Determination Models

4.7       IS-LM Analysis

4.8       Economic and Mathematical Modeling (Optional)

4.9       Implicit Functions and Inverse Functions (Optional)

Chapter 5     LINEAR (OR MATRIX) ALGEBRA

5.1       Introduction

5.2       Definitions and Terms

5.3       Addition and Subtraction of Matrices

5.4       Scalar Multiplication

5.5       Vector Multiplication

5.6       Multiplication of Matrices

5.7       Matrix Expression of a System of Linear Equations

5.8       Augmented Matrix

5.9       Row Operations

5.10     Gaussian Method of Solving Linear Equations

Chapter 6     SOLVING LINEAR EQUATIONS WITH MATRIX ALGEBRA

6.1       Determinants and Linear Independence

6.2       Third-Order Determinants

6.3       Cramer’s Rule for Solving Linear Equations

6.4       Inverse Matrices

6.5       Gaussian Method of Finding an Inverse Matrix

6.6       Solving Linear Equations with an Inverse Matrix

6.7       Business and Economic Applications

6.8       Special Determinants

Chapter 7     LINEAR PROGRAMMING: USING GRAPHS

7.1       Use of Graphs

7.2       Maximization Using Graphs

7.3       The Extreme-Point Theorem

7.4       Minimization Using Graphs

7.5       Slack and Surplus Variables

7.6       The Basis Theorem

Chapter 8     LINEAR PROGRAMMING: THE SIMPLEX ALGORITHM AND THE DUAL

8.1       The Simplex Algorithm

8.2       Maximization

8.3       Marginal Value or Shadow Pricing

8.4       Minimization

8.5       The Dual

8.6       Rules of Transformation to Obtain the Dual

8.7       The Dual Theorems

8.8       Shadow Prices in the Dual

8.9       Integer Programming .

8.10     Zero-One Programming

Chapter 9     DIFFERENTIAL CALCULUS: THE DERIVATIVE AND THE RULES OF DIFFERENTIATION

9.1       Limits

9.2       Continuity

9.3       The Slope of a Curvilinear Function

9.4       The Derivative

9.5       Differentiability and Continuity

9.6       Derivative Notation

9.7       Rules of Differentiation

9.8       Higher-Order Derivatives

9.9       Implicit Functions

Chapter 10   DIFFERENTIAL CALCULUS: USES OF THE DERIVATIVE

10.1     Increasing and Decreasing Functions

10.2     Concavity and Convexity

10.3     Relative Extrema

10.4     Inflection Points

10.5     Curve Sketching

10.6     Optimization of Functions

10.7     The Successive-Derivative Test

10.8     Marginal Concepts in Economics

10.9     Optimizing Economic Functions for Business

10.10   Relationships Among Total, Marginal, and Average Functions

Chapter 11   EXPONENTIAL AND LOGARITHMIC FUNCTIONS

11.1     Exponential Functions

11.2     Logarithmic Functions

11.3     Properties of Exponents and Logarithms

11.4     Natural Exponential and Logarithmic Functions

11.5     Solving Natural Exponential and Logarithmic Functions

11.6     Logarithmic Transformation of Nonlinear Functions

11.7     Derivatives of Natural Exponential and Logarithmic Functions

11.8     Interest Compounding

11.9     Estimating Growth Rates from Data Points

Chapter 12   INTEGRAL CALCULUS

12.1     Integration

12.2     Rules for Indefinite Integrals

12.3     Area Under a Curve

12.4     The Definite Integral

12.5     The Fundamental Theorem of Calculus

12.6     Properties of Definite Integrals

12.7     Area Between Curves

12.8     Integration by Substitution

12.9     Integration by Parts

12.10   Present Value of a Cash Flow

12.11   Consumers’ and Producers’ Surplus

Chapter 13   CALCULUS OF MULTIVARIABLE FUNCTIONS

13.1     Functions of Several Independent Variables

13.2     Partial Derivatives

13.3     Rules of Partial Differentiation

13.4     Second-Order Partial Derivatives

13.5     Optimization of Multivariable Functions

13.6     Constrained Optimization with Lagrange Multipliers

13.7     Income Determination Multipliers

13.8     Optimizing Multivariable Functions in Business and Economics

13.9     Constrained Optimization of Multivariable Economic Functions

13.10   Constrained Optimization of Cobb-Douglas Production Functions

13.11   Implicit and Inverse Function Rules (Optional)

Chapter 14   SEQUENCES AND SERIES

14.1     Sequences

14.2     Representation of Elements

14.3     Series and Summations

14.4     Property of Summations

14.5     Special Formulas of Summations

14.6     Economics Applications: Mean and Variance

14.7     Infinite Series

14.8     Finance Applications: Net Present Value

EXCEL PRACTICE A

EXCEL PRACTICE B

ADDITIONAL PRACTICE PROBLEMS

ADDITIONAL PRACTICE PROBLEMS: SOLUTIONS

INDEX

Chapter 1

Review

1.1       EXPONENTS

Given a positive integer n, xn signifies that x is multiplied by itself n number of times. Here x is referred to as the base; n is called an exponent. By convention an exponent of one is not expressed: x(1) = x, 8(1) = 8. By definition any nonzero number or variable raised to the zero power is equal to 1: x⁰ = 1, 3⁰ = 1, and 0⁰ is undefined. Assuming a and b are positive integers and x and y are real numbers for which the following exist, the rules of exponents are presented below, illustrated in Examples 1 and 2, and treated in Problems 1.1, 1.24, 1.26, and 1.27.

1.   xa · xb = xa+b

2.   

3.   (xa)b = xab

4.   (xy)a = xa ya

5.   

6.   

7.   

8.   

9.   

EXAMPLE 1.   In multiplication, exponents of the same variable are added; in division, exponents of the same variable are subtracted; when raised to a power, the exponents of a variable are multiplied, as indicated by the rules above and shown in the examples below followed by illustrations.

Since and from Rule 1 exponents of a common base are added in multiplication, the exponent of , when added to itself, must equal 1. With , the exponent of must equal . Thus,

See Problems 1.1, 1.24, 1.26, and 1.27.

EXAMPLE 2.   From Rule 2, it can easily be seen why any variable or nonzero number raised to the zero power equals one. For example, x³/x³ = x³–³ = x⁰ = 1; 8⁵/8⁵ = 8⁵−⁵ = 8⁰ = 1.

1.2       POLYNOMIALS

Given an expression such as 9x⁵, x is called a variable because it can assume any number of different values, and 9 is referred to as the coefficient of x. Expressions consisting simply of a real number or of a coefficient times one or more variables raised to the power of a positive integer are called monomials. Monomials can be added or subtracted to form polynomials. Each monomial constituting a polynomial is called a term. Terms that have the same variables and respective exponents are called like terms. The degree of a monomial is the sum of the exponents of its variables. The degree of a polynomial is the degree of its highest term. Rules for adding, subtracting, multiplying, and dividing polynomials are explained below, illustrated in Examples 3 to 5, and treated in Problems 1.3 and 1.4.

1.2.1   Addition and Subtraction of Polynomials

Like terms in polynomials can be added or subtracted by adding or subtracting their coefficients. Unlike terms cannot be so added or subtracted.

EXAMPLE 3.

(a)   6x³ + 15x³ = 21x³

(b)   18xy – 7xy = 11xy

(c)   (4x³ + 13x² − 7x) + (11x³ − 8x² − 9x) = 15x³ + 5x² − 16x

(d)   (22x – 19y) + (7x + 6z) = 29x − 19y + 6z

See also Problem 1.3.

1.2.2   Multiplication and Division of Terms

Like and unlike terms can be multiplied or divided by multiplying or dividing both the coefficients and variables.

EXAMPLE 4.

(a)   20x⁴ · 7y⁶ = 140x⁶y⁶

(b)   6x²y³ · 8x⁴y⁶ = 48x⁶y⁹

(c)   12x³y² · 5y⁴z⁵ = 60x³y⁶z⁵

(d)   3x³y²z⁵ · 15x⁴y³z⁴ = 45x⁷y⁵z⁹

(e)   

(f)   

1.2.3   Multiplication of Polynomials

To multiply two polynomials, multiply each term in the first polynomial by each term in the second polynomial and then add their products together.

EXAMPLE 5.

See also Problem 1.4.

1.3       FACTORING

Factoring reverses the process of polynomial multiplication in order to express a given polynomial as a product of simpler polynomials called factors. A monomial such as the number 14 is easily factored by expressing it as a product of its integer factors 1 · 14, 2 · 7, (−1) · (−14), or (−2) · (−7). A binomial such as 5x⁴ − 45x³ is easily factored by dividing or factoring out the greatest common factor, here 5x³, to obtain 5x³(x − 9). Factoring a trinomial such as mx² + nx + p, however, generally requires the following rules:

1.   Given (mx² + nx + p), the factors are (ax + c)(bx + d), where (1) ab = m; (2) cd = p; and (3) ad + be = n.

2.   Given (mx² + nxy + py²), the factors are (ax + cy)(bx + dy), where (1) ab = m; (2) cd = p; and (3) ad + bc = n, exactly as above. For proof of these rules, see Problems 1.28 and 1.29.

EXAMPLE 6.   To factor (x² + 11x + 24), where in terms of Rule 1 (above) m = 1, n = 11, and p = 24, we seek integer factors such that:

1)   a · b = 1. Integer factors: 1 · 1, (−1) · (−1). For simplicity we shall consider only positive sets of integer factors here and in step 2.

2)   c · d = 24. Integer factors: 1 · 24, 2 · 12, 3 · 8, 4 · 6, 6 · 4, 8 · 3, 12 · 2, 24 · 1.

3)   ad + bc = 11. With a = b = 1, c + d must equal 11.

Adding the different combinations of factors from step 2, we have 1 +24 = 25, 2+ 12 = 14, 3 + 8 = 11,4 + 6 = 10, 6 + 4 = 10, 8 + 3 = 11, 12 + 2 = 14, and 21 + 1 = 25. Since only 3 + 8 and 8 + 3 = 11 in step 3, 3 and 8 are the only candidates for c and d from step 2 which, when used with a = b = 1 from step 1, fulfill all the above requirements, and the order does not matter. Hence

See Problems 1.5 to 1.13. For derivation of the rules, see Problems 1.28 and 1.29.

1.4       FRACTIONS

Fractions, or rational numbers, consist of polynomials in both numerator and denominator, assuming always that the denominator does not equal zero. Reducing a fraction to lowest terms involves the cancellation of all common factors from both the numerator and the denominator. Raising a fraction to higher terms means multiplying the numerator and denominator by the same nonzero polynomial. Assuming that A, B, C, and D are polynomials and C and D ≠ 0, fractions are governed by the following rules:

1.   

2.   

3.   

4.   

5.   

The properties of fractions are illustrated in Example 7 and treated in Problems 1.14 to 1.21.

EXAMPLE 7.

(a)   Multiplying or dividing both the numerator and the denominator of a fraction by the same nonzero number or polynomial leaves the value of the fraction unchanged.

Rule 1 provides the basis for reducing a fraction to its lowest terms as well as for raising a fraction to higher terms.

(b)   To multiply fractions, simply multiply the numerators and the denominators separately. The product of the numerators then forms the numerator of the product and the product of the denominators forms the denominator of the product.

(c)   To divide fractions, simply invert the divisor and multiply.

(d)   Fractions can be added or subtracted only if they have exactly the same denominator, called a common denominator. If a common denominator is present, simply add or subtract the numerators and set the result over the common denominator. Remember always to subtract all the terms within a given set of parentheses.

(e)   To add or subtract fractions with different denominators, a common denominator must first be found. Multiplication of one denominator by the other will always produce a common denominator. Each fraction can then be restated in terms of the common denominator using Rule 1 and the numerators added as in (d).

(f)   Similarly,

The least common denominator (LCD) of two or more fractions is the polynomial of lowest degree and smallest coefficient that is exactly divisible by the denominators of the original fractions. Use of the LCD helps simplify the final sum or difference. See Problems 1.19 to 1.21. Fractions are reviewed in Problems 1.14 to 1.21.

1.5       RADICALS

If bn = a, where b > 0, then by taking the nth root of both sides of the equation, where is a radical (sign), a is the radicand, and n is the index. For square roots, the index 2 is not expressed. Thus, . From Rules 7 and 8 in Section 1.1, we should also be aware that and

Assuming x and y are real nonnegative numbers and m and n are positive integers such that and exist, the rules of radicals are given below. For proof of Rule 1, see Problem 1.30.

1.   

2.   

3.   

4.   

EXAMPLE 8.   The laws of radicals are used to simplify the following expressions. Note that for even-numbered roots, positive and negative answers are equally valid.

See also Problems 1.22, 1.23, and Problems 1.25 to 1.27.

1.6       ORDER OF MATHEMATICAL OPERATIONS

Given an expression involving multiple mathematical operations, computations within parentheses are performed first. If there are parentheses within parentheses, computations on the innermost set take precedence. Within parentheses, all constants and variables are first raised to the powers of their respective exponents. Multiplication and division are then performed before addition and subtraction. In carrying out operations of the same priority, the procedure is from left to right. In sum,

1.   Start within parentheses, beginning with the innermost.

2.   Raise all terms to their respective exponents.

3.   Multiply and divide before adding and subtracting.

4.   For similar priorities, move from left to right.

EXAMPLE 9.   The following steps are performed to solve

1.   5² = 25

2.   25 · 6 = 150

3.   

4.   15 – 8 = 7

Thus

1.7       USE OF A POCKET CALCULATOR

Pocket calculators are helpful for checking one’s ordinary calculations and performing arduous or otherwise time-consuming computations. Rules for the different mathematical operations are set forth and illustrated below, including some rules which will not be used or needed until later in the text.

1.7.1   Addition of Two Numbers

To add two numbers, enter the first number, press the key, and enter the second number. Then press the key to find the total.

EXAMPLE 10.

(a)   To find 139 + 216, enter 139, press the key, enter 216, and press the key to find 139 + 216 = 355.

(b)   To find 1025 + 38.75, enter 1025, press the key, then enter 38.75, and hit the key to find 1025 + 38.75 = 1063.75. Practice this and subsequent examples using simple numbers to which you already know the answers to see if you are doing the procedure correctly.

1.7.2   Addition of More Than Two Numbers

To add more than two numbers, simply follow each entry of a number by pressing the key until all the numbers have been entered. Then press the key to find the total. Pressing the key at any time after a number will give the subtotal at that point.

EXAMPLE 11.   To find 139 + 216 + 187, enter 139, press the key, enter 216, press the key again, enter 187, and hit the key to find 139 + 216 + 187 = 542. Hitting the key after 216 would reveal the subtotal of 139 + 216 is 355, as in the example above.

1.7.3   Subtraction

To find the difference A − B, enter A, press the key, and enter B. Then press the key to find the remainder. Multiple subtractions can be done as multiple additions in 1.7.2 above, with the key substituted for the key.

EXAMPLE 12.

(a)   To find 315–708, enter 315, press the key, then enter 708 followed by the key to find 315–708 = −393.

(b)   To find 528 – 79.62, enter 528, hit the key, then enter 79.62 followed by the key to find 528 – 79.62 = 448.38.

1.7.4   Multiplication

To multiply two numbers, enter the first number, press the key, enter the second number, and press the key to find the product. Serial multiplications can be done in the same way as multiple additions in 1.7.2. with the key substituted for the key.

EXAMPLE 13.

(a)   To find 486 · 27, enter 486, press the key, then enter 27, and hit the key to learn 486 · 27 = 13,122.

(b)   To find 149 · −35, enter 149, press the key, then enter 35 followed by the key to make it negative, and hit the key to learn that 149 · −35 = −5215.

Note: Be aware of the distinction between the key and the key. The key initiates the process of subtraction; the key simply changes the value of the previous entry from positive to negative or negative to positive.

1.7.5   Division

Dividing A by B is accomplished by entering A, pressing the key, then entering B and pressing the key.

EXAMPLE 14.

(a)   To find 6715 ÷ 79, enter 6715, hit the key, then enter 79 followed by the key. The display will show 85, indicating that 6715 ÷ 79 = 85.

(b)   To find −297.36 ÷ 72.128, enter 297.36 followed by the key to make it negative, then press the key, enter 72.128, and hit the key to find −297.36 ÷ 72.128 = −4.1226708.

1.7.6   Raising to a Power

To raise a number to a power, enter the number, hit the key, then enter the exponent and press the key.

EXAMPLE 15.

(a)   To find 8⁵, enter 8, press the key, then enter 5 followed by the key to learn that 8⁵ = 32,768. Continue to practice these and subsequent exercises by using simple numbers for which you already know the answers.

(b)   To find 36⁰.²⁵, enter 36, hit the key, then enter 0.25, and press the key to see 36⁰.²⁵ = 2.4494897.

(c)   To find 2−3, enter 2, hit the key, then press 3 followed by the key to make it negative, and hit the key to discover 2−3 = 0.125.

See also Problem 1.24.

1.7.7   Finding a Square Root

To find the square root of a number, enter the number, then press the key to find the square root immediately without having to press the key. Note that on many calculators the key is the inverse (shift, or second function) of the key, and to activate the key, one must first press the ( or ) key followed by the key.

EXAMPLE 16.   To find , enter 529, then press the key to see immediately that ±23 is the square root of 529.

If the key is the inverse, shift, or second function of the key, enter 529, then press the ( or key) followed by the key to activate the key, and you will see immediately that without having to press the key.

1.7.8   Finding the nth Root

To find the nth root of a number, enter the number, press the key, then enter the value of the root n and hit the key to find the root. If the is the inverse, shift, or second function of the key, enter the number, press the ( , or ) key followed by the key, then enter the value of the root n and hit the key to find the answer.

EXAMPLE 17.

(a)   To find , enter 17,576, press the key followed by the key, enter 3, then hit the key to learn .

(b)   To find , enter 32,768, hit the key followed by the key, then enter 5 and hit the key to learn .

(c)   From Rule 8 in Section 1.1, . To use this latter form, simply enter 32,768, press the key, enter 0.2, and hit the key to find 32,768⁰.² = 8.

To make use of similar conversions, recall that

, and so forth. See Problems 1.25 to 1.27.

1.7.9   Logarithms

To find the value of the common logarithm log10x, enter the value of x and simply press the key. The answer will appear without the need to press the key.

EXAMPLE 18.

(a)   To find the value of log 24, enter 24 and hit the key. The screen will immediately display 1.3802112, indicating that log 24 = 1.3802112.

(b)   To find log 175, enter 175 and hit the key. You will see 2.243038, which is the value of log 175.

1.7.10   Natural Logarithms

To find the value of the natural logarithm ln x, enter the value of x and press the key. The answer will appear immediately without the need to press the key.

EXAMPLE 19.

(a)   To find ln 20, enter 20 and hit the key to see 2.9957323 = ln 20.

(b)   For ln 0.75, enter 0.75 and hit the key. You will find ln 0.75 = −0.2876821.

1.7.11   Exponential Functions

To find the value of an exponential function y = ax, enter the value of a and press the key, then enter the value of x and hit the key, similar to what was done in Section 1.7.6.

EXAMPLE 20.

(a)   Given y = 1.5³.², enter 1.5, press the key, then enter 3.2, and hit the key to get 1.5³.² = 3.6600922.

(b)   For y = 256−1.25, enter 256, hit the key, then enter 1.25 followed immediately by the key to make it negative, and then press the key to learn 256−1.25 = 0.0009766.

1.7.12   Natural Exponential Functions

To find the value of a natural exponential function y = ex, enter the value of