[PDF] Differentiation and its Uses in Business Problems

Apply the formula of differentiation to solve business problems Introduction Calculus Multiple Choice Questions (√ the most appropriate answer) 1 Find dy

Problems Given At the Math 151 - Calculus I and Math 150 - Calculus I With 2 4 Tangent Lines and Implicit Differentiation 3 Applications of Differentiation 31 sorted by topic and most of them are accompanied with hints or solutions 16 Habits of Mind (1 page summary): http://www chsvt org/wdp/Habits of Mind pdf

[PDF] Exercises and Problems in Calculus - Portland State University

1 août 2013 · 11 3 Problems 72 11 4 Answers to Odd-Numbered Exercises 74 Chapter 12 APPLICATIONS OF THE DERIVATIVE 75 12 1 Background

[PDF] Problem Set Solutions: Applications of Differentiation - MIT

SOLUTIONS TO 18 01 EXERCISES 2 Applications of Differentiation 2A Approximation 2A-1 d √ a + bx = b f(x) ≈ √ a + b x by formula dx 2 √ a + bx ⇒ 2

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14 mar 2013 · log x x Solution: It is natural to want to use L'Hopital's rule Taking the derivatives , we would find it equals limx→0 1/x 1 , which is undefined

[PDF] DIFFERENTIAL CALCULUS QUESTIONS FOR - UBC Math

This book is organized into four sections: Questions, Hints, Answers, and Solutions As you are working problems, resist the 3 Applications of derivatives 53

[PDF] Problem Set Solutions: Differentiation - MIT OpenCourseWare

problem using the function s(t) = 16t2, representing the distance down measured from the top apply the chain rule, the rule to be treated below in E4 ) The chain rule answer is the same as the one using implicit differentiation because

[PDF] Word Problems - University of Kentucky

Chapter7 pdf word problems that one usually encounters in a first Calculus course: to give up on hoping to mechanically apply a set of steps Take the derivative and find the critical points (11 ) Give your answer in feet per minute 81

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7 mar 2019 · Now applying the Sandwich Theorem and using the fact that lim x c SfИxНS 0 we obtain Solution: First we will make a mathematical model of the problem We will Solution: a For x ~ 0 we compute the derivative using the rules of di erentiation: fЬ http://sertoz bilkent edu tr/depo/limit pdf 7 Determine

[PDF] Differentiation and its Uses in Business Problems - Bangladesh

Apply the formula of differentiation to solve business problems Introduction Calculus Multiple Choice Questions (√ the most appropriate answer) 1 Find dy

Differentiation and its Uses in

Business Problems

The objectives of this unit is to equip the learners with differentiation and to realize its importance in the field of business. The unit surveys derivative of a function, derivative of a multivariate functions, optimization of lagrangian multipliers and Cobb-Douglas production function etc. Ample examples have been given in the lesson to demonstrate the applications of differentiation in practical business contexts. The recognition of 8

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Unit-8 Page-164 differentiation in decision making is extremely important in the filed of business.

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Business Mathematics Page-165 Lesson-1: Differentiation After studying this lesson, you should be able to:

Explain the nature of differentiation; State the nature of the derivative of a function; State some standard formula for differentiation; Apply the formula of differentiation to solve business problems.

Introduction

Calculus is the most important ramification of mathematics. The present and potential managers of the contemporary world make extensive uses of this mathematical technique for making pregnant decisions. Calculus is inevitably indispensable to measure the degree of changes relating to different managerial issues. Calculus makes it possible for the enthusiastic and ambitious executives to determine the relationship of different variables on sound footings. Calculus in concerned with dynamic situations, such as how fast production levels are increasing, or how rapidly interest is accruing. The term calculus is primarily related to arithmetic or probability concept. Mathematics resolved calculus into two parts - differential calculus and integral calculus. Calculus mainly deals with the rate of changes in a dependent variable with respect to the corresponding change in independent variables. Differential calculus is concerned with the average rate of changes, whereas Integral calculus, by its very nature, considers the total rate of changes in variables.

Differentiation

Differentiation is one of the most important operations in calculus. Its theory solely depends on the concepts of limit and continuity of functions. This operation assumes a small change in the value of dependent variable for small change in the value of independent variable. In fact, the techniques of differentiation of a function deal with the rate at which the dependent variable changes with respect to the independent variable. This rate of change is measured by a quantity known as derivative or differential co-efficient of the function. Differentiation is the process of finding out the derivatives of a continuous function i.e., it is the process of finding the differential co- efficient of a function.

Derivative of a Function

The derivative of a function is its instantaneous rate of change. Derivative is the small changes in the dependent variable with respect to a very small change in independent variable.

Let y = f (x), derivative i.e.

dx dy means rate of change in variable y with respect to change in variable x.

Differential calculus

is concerned with the average rate of changes.

The techniques of

differentiation of a function deal with the rate at which the dependent variable changes with respect to the independent variable.

School of Business

Unit-8 Page-166 The derivative has many applications, and is extremely useful in optimization- that is, in making quantities as large (for example profit) or as small (for example, average cost) as possible.

Some Standard Formula for Differentiation Following are the some standard formula of derivatives by means of which we can easily find the derivatives of algebraic, logarithmic and

exponential functions. These are : 1. dx dc= 0, where C is a constant. 2. dx d dx dxn = [xn] = n. xn-1 3. ( )( )[ ]xfdx daxfa dx d=. 4. nn nxnxdxd 11 .1+---=))

5. ( )xxx

ee dx d dx de== 6. ( )[]( )( )[ ]xgdx dee dx dxgxg.=

7. If y = f(u) and U = g(x) then

dx du du dy dx dy×= 8. ()aaadx d exxlog.= 9. dx xgd dx xfd dx xgxfd±=± 10. ( )( )xxdx dx dx d e1lnlog==

11. If Y = [f(x)]n then,

dx dy= n [f(x)]n-1 .()[] dx xfd 12. dx dloga x = 1 x loga e 13. dx xfdxg dx xgdxf dx xgxfd+=.

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Business Mathematics Page-167 14.

( )[ ]2xgdxxgdxfdxxfdxg dxxgxfd- 15. ( ) ( )( )[ ]exgdx daa dx d axgxglog..=

16. If U = f(x, y),

-+=dxyxfydxxf dxdu,.and ()()? -+=dyyxfdyyxf dydu

17. If y = e

ax, then its first derivative is equal to dx deax = eax

Second derivative is equal to

22
dxedas = a2eax

Third derivative is equal to

3 3 dx edax= a3eax and the nth derivative is denoted by naxn dxed= aneax

Derivative of Trigonometrically Functions

18. d dx (Sin x) = Cos x. d dx (Cos x) = - Sinx 19. d dx (tan x) = Sec2x. d dx (Cot x) = - Cosec2x 20. d dx (Sec x) = Sec x. tan x; d dx(Cosec x) = - Cosec x. Cot x 21.
21
21
11 11 xxCosdxd xxSindxd 22.
()()21 21

11;11taxxCotdxd

xxndxd 23.
11 sec; 1.1 21
21
xxxCodxd xxxSecdxd Sin

2x + Cos2x = 1; tan x = Sin x

Cos x

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Unit-8 Page-168 Sec

2x - tan2x = 1; Cot x = Cos x

Sin x When x and y are separately expressed as the functions of a third variable in the equation of a curve is known as parameter. In such cases we can find dydx without first eliminating the parameter as follows:

Thus, if x =Q (t), y =

ψ (t)

Then, dy dt = dydx . dx dt dtdxdtdy dx dy=? Let us illustrate these different derivatives by the following examples.

Example - 1:

If y = f(x) = a; find

dydx

Solution:

dy dx = d(a) dx =0, since a is a constant, i.e.,"a" has got no relationship with variable x.

Example - 2:

Differentiate the following functions, with respect to x, (i) y = x , (ii) y = 8x-5 (iii) y= 3x3 - 6x2+2x - 8

Solution:

We know that

x = x 1 2 dy dx = d dx )) 21
x= 121
2 1-x = 2

121x- = 1

2 x . Hence dydx = 1 2x (ii) dy dx = d dx (8x-5) = 8 d dx (x-5) = 8(-5) x-6 = - 40x-6 = - 40 x 6 .

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Business Mathematics Page-169 Therefore,

dy dx = - 40 x 6 (iii) dy dx = d dx (3x3 - 6x2+2x - 8) d dx (3x3) - d dx (6x2) + d dx (2x) - d dx (8) = 3.3x

3-1- 2.6.x2-1+ 2-0 = 9x2-12x+2

Thus, dy dx = 9x2 - 12x + 2.

Example -3:

Differentiate e

x(log x) . (2x2+3) with respect to x.

Solution:

Let y = e

x.(log x) . (2x2 + 3) dy dx =ex (log x)d dx (2x2+3) +ex(2x2+3). d dx (logx) + (log x). (2x2 +3).d dx (ex) =e x(logx)(4x)+ex(2x2+3)1 x + logx(2x2+3)ex = e x[4x.logx + 2x2+3 x + (2x2 +3) logx] So, dy dx = ex[4x.logx + 2x2+3 x + (2x2 +3) logx]

Example-4:

If y =

2+3log x

x 2+5 , find dydx

Solution:

y = 2+3log x x 2+5 dx dy= 2222 )5()5()log32()log32()5( xx dxdxxdxdx = (x2+5) (3 x )-(2+3logx)(2x) (x

2+5)2 = 15

x -x -6xlogx (x 2+5)2 .Thus, dydx = 15 x -x -6xlogx (x 2+5)2

School of Business

Unit-8 Page-170 Example-5: Find the differential co-efficient of 7x5x

2e++with respect to x.

Solution:

Let y= e

x2+5x+y dy dx =yx5x

2e++. d

dx (x2 +5x +y) = ex2 + 5x + y. (2x + 5) dy dx = (2x + 5). ex2 + 5x + y

Example-6:

Find dy dx , if y = log4x+3

Solution:

Let y = log

4x+3 log(4x+3) 1

2 = 1

2 log (4x +3) dy dx = d dx [ 1

2 log (4x +3)] = 1

2 . 1

4x+3 . d

dx (4x +3) = 1 2 . 1

4x+3 . 4 = 2

4x+3 ? dydx = 2 4x+3

Example-7:

Find the first, second and third derivatives when y = x.e x2

Solution:

y = x.e x2

First derivative,

dy dx = x d dx ex2 +ex2. d dx (x) = x.ex2. 2x + ex2.1 = ex2 (2x2 +1)

Second derivative,

d2y dx 2 = ex2.2x(2x2+1) +ex2.4x = ex2. 4x3 +

2xe.2x + ex2. 4x = ex2. (4x3 + 2x + 4x)

= e x2. (4x3 + 6x)

Bangladesh Open University

Business Mathematics Page-171 Third derivative,

d3yquotesdbs_dbs14.pdfusesText_20

[PDF] [PDF] Differentiation and its Uses in Business Problems - Bangladesh

Differentiation and its Uses in

Business Problems

School of Business

Blank Page

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Introduction

Differentiation

Derivative of a Function

Let y = f (x), derivative i.e.

Differential calculus

The techniques of

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5. ( )xxx

7. If y = f(u) and U = g(x) then

11. If Y = [f(x)]n then,

Bangladesh Open University

Business Mathematics Page-167 14.

16. If U = f(x, y),

17. If y = e

Second derivative is equal to

Third derivative is equal to

Derivative of Trigonometrically Functions

11;11taxxCotdxd

2x + Cos2x = 1; tan x = Sin x

School of Business

Unit-8 Page-168 Sec

2x - tan2x = 1; Cot x = Cos x

Thus, if x =Q (t), y =

ψ (t)

Example - 1:

If y = f(x) = a; find

Solution:

Example - 2:

Solution:

We know that

121x- = 1

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Business Mathematics Page-169 Therefore,

3-1- 2.6.x2-1+ 2-0 = 9x2-12x+2

Example -3:

Differentiate e

Solution:

Let y = e

Example-4:

If y =

2+3log x

Solution:

2+5)2 = 15

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2e++with respect to x.

Solution:

Let y= e

2e++. d

Example-6:

Solution:

Let y = log

2 = 1

2 log (4x +3)] = 1

4x+3 . d

4x+3 . 4 = 2

Example-7:

Solution:

First derivative,

Second derivative,

2xe.2x + ex2. 4x = ex2. (4x3 + 2x + 4x)

Bangladesh Open University

Business Mathematics Page-171 Third derivative,