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DIFFERENTIAL CALCULUS
BATCH-6:
1. VANATHI.S
2. VARSHINIE.A.L
3. SRIRAM.R
4.THIRUNAVUKARASU.S
5. VISHAL.S
6. SURANJANA.N.S
>>> ENGINEERING MATHEMATICS
TABLE OF CONTENTS
1. INTRODUCTION
2. BAKING ANALOGY
3. DIFFERENTIAL CALCULUS
4. DERIVATIVE RULES
5. PRODUCT RULE
6. QUOTIENT RULE
7. DERIVATIVE OF TRIGNOMETRIC FUCTIONS
8. THE SANDWICH/SQUEEZE THEOREM
9. CONCLUSIONS
INTRODUCTION
>> Differential calculus is a procedure for finding the
exact derivative directly from the formula of the
function without having to use graphical methods.
>> It is a method that deals with the rate of change of
one quantity with respect to another.
BAKING ANALOGY
>> In this we will focus on the formulas and rules for
both differentiation, the method by which we calculate
the derivative of a function.
>> Before we dive into formulas and rules for
differentiation , let’s look at some notations for
differentiation.
>> we can write f(X) as d/dx f(x),f’(x),df(x) and Df(x)
We read these as d by dx of f of x, f’ prime of f of x, df of
x and cap Df of x.
DIFFERENTIAL CALCULUS
>> Differential calculus is the area of calculus dealing
with cutting something into smaller pieces in order to
analyze how it changes.
>> The primary operation of differential calculus is the
derivative. The derivative of a function given the
infinitesimal change of the function with respect to
one of it’s variable.
>> The derivative represents the slope of function at
every point it is defined.
DERIVATIVE RULES
LOGARITHMIC
FUNCTIONS (d/dx):
1. e^x = e^x
2. a^x = a^x ln(a)
3. Ln(x) = 1/x
4. Log a ^x = 1/xln(a)
TRIGONOMETRIC
FUNCTIONS (d/dx):
1. Sin(x)= cos(x)
2. Cos(x)= -sin(x)
3. Tan(x)= sec^2(x)
4. Cosec(x)= - cosec(x) cot(x)
5. Sec(x)= sec(x) tan(x)
6. Cot(x)= -cosec^2(x)
Real life applications of differential
calculus:
>> Calculation of profit or loss with respect to
business using graphs.
>> Calculation of rate of change of temperature.
>> To derive many physical equations.
>> Calculation of speed or distance covered such as
miles per hour , km/hour.
PRODUCT RULE
>> The derivative of the product of two differentiable
functions is equal to the addition of the first multiplied
by the derivative of the second and the second
function multiplied by the derivative of the first
function.
APPLICATION:
1. The product rule is used in calculus, when you are
asked to take derivative of the function.
2. It makes calculation clean and easier to solve.
3. It is used to differentiate product of two or more
functions.
DERIVATIVE PRODUCT RULE
If u and v are differentiable at x, then so is their product uv
and
d/dx(u.v) = u (dv/dx) +v (du/dx)
Example: Q) Find the derivative of y=(x^2 +1)(x^3+3)
Answer: d/dx(x^2+1)(x^3+3)=(x^2+1)(3x^2)+ (2x)(x^3+3)
=3x^4+3x^2+2x^4+6x
=5x^4+3x^2+6x
The particular product can be differentiated as well by
multiplying out the original expression for y and
differentiating the resulting polynomial.
Y=(x^2+1)(x^3+3)=x^5+x^3+3x^2+3
dy/dx=5x^4+3x^2+6x
This is in agreement with our first calculation.
QUOTIENT RULE
>> A quotient rule is similar to product rule. A quotient
rule is stated as the ratio of the quantity of the
denominator times the derivative of the numerator
function minus the numerator times the derivative of
the denominator function to the square of the
denominator function.
APPLICATION:
1. It is used for finding the derivative of a quotient of
functions.
2. It is used for extend the power rule to functions with
negative exponents.
3. To combine differentiation rule to find the derivative
of a polynomial or rational function.
DERIVATIVE QUOTIENT RULE
If u and v are differentiable at x and if v(x) is not equal
to 0 then the quotient u/v is differentiable at x and
d/dx(u/v)= v (du/dx) – u (dv/dx)/ v^2
Example: Q)Find the derivative of y=(t^2-1)/(t^3+1)
Answer: u=t^2-1 v=t^3+1
dy/dt=(t^3+1).2t- (t^2-1).3t^2/(t^3+1)^2
=2t^4+2t-3t^4+3t^2/(t^3+1)^2
=-t^4+3t^2+2t/(t^3+1)^2
SQUEEZE THEOREM
>> In calculus the squeeze theorem is a theorem
regarding the limit of a function that is trapped
between two other function.
>> The squeeze theorem is used in calculus and
mathematical analysis typically to confirm the limit of
a function via comparison with other function whose
limits are known.
>> If the right hand limits and left hand limits do not
equal eachother we cannot utilize squeeze theorem.
If f(x)<g(x)<h(x) when x is near a
If limxa f(x)=limxa h(x)=L then limxa g(x)=L.
WHY IS IT CALLED SANDWICH
THEOREM?
>> The squeeze theorem is also called as sandwich or
pinching theorem. It is a way to find the limit of one
function if we know the limits of two functions it is
“sandwiched” between.
APPLICATION:
It is used for calculating the limit of a given
trigonometric funtions.
EXAMPLE OF SANDWICH THEOREM
Q) Using sandwich theorem show that:
limx0 x^2 sin (1/x)=0
ANSWER:
Let -1<sin(1/x)<1
Multiply by x^2
-x^2<x^2 sin 1/x <x^2
Lim x0 (–x^2)<lim x0 x^2 sin (1/x)< lim x0 x^2
Lim x->0 (-x^2)=-0=0
Lim x x^2=0=0
Lim x0 (-x^2)= lim x (x^2)
Lim x0 x^2 sin (1/x)=0
THANK YOU!

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maths diff.calculus ppt.pptx

  • 1. DIFFERENTIAL CALCULUS BATCH-6: 1. VANATHI.S 2. VARSHINIE.A.L 3. SRIRAM.R 4.THIRUNAVUKARASU.S 5. VISHAL.S 6. SURANJANA.N.S >>> ENGINEERING MATHEMATICS
  • 2. TABLE OF CONTENTS 1. INTRODUCTION 2. BAKING ANALOGY 3. DIFFERENTIAL CALCULUS 4. DERIVATIVE RULES 5. PRODUCT RULE 6. QUOTIENT RULE 7. DERIVATIVE OF TRIGNOMETRIC FUCTIONS 8. THE SANDWICH/SQUEEZE THEOREM 9. CONCLUSIONS
  • 3. INTRODUCTION >> Differential calculus is a procedure for finding the exact derivative directly from the formula of the function without having to use graphical methods. >> It is a method that deals with the rate of change of one quantity with respect to another.
  • 4. BAKING ANALOGY >> In this we will focus on the formulas and rules for both differentiation, the method by which we calculate the derivative of a function. >> Before we dive into formulas and rules for differentiation , let’s look at some notations for differentiation. >> we can write f(X) as d/dx f(x),f’(x),df(x) and Df(x) We read these as d by dx of f of x, f’ prime of f of x, df of x and cap Df of x.
  • 5. DIFFERENTIAL CALCULUS >> Differential calculus is the area of calculus dealing with cutting something into smaller pieces in order to analyze how it changes. >> The primary operation of differential calculus is the derivative. The derivative of a function given the infinitesimal change of the function with respect to one of it’s variable. >> The derivative represents the slope of function at every point it is defined.
  • 6. DERIVATIVE RULES LOGARITHMIC FUNCTIONS (d/dx): 1. e^x = e^x 2. a^x = a^x ln(a) 3. Ln(x) = 1/x 4. Log a ^x = 1/xln(a) TRIGONOMETRIC FUNCTIONS (d/dx): 1. Sin(x)= cos(x) 2. Cos(x)= -sin(x) 3. Tan(x)= sec^2(x) 4. Cosec(x)= - cosec(x) cot(x) 5. Sec(x)= sec(x) tan(x) 6. Cot(x)= -cosec^2(x)
  • 7. Real life applications of differential calculus: >> Calculation of profit or loss with respect to business using graphs. >> Calculation of rate of change of temperature. >> To derive many physical equations. >> Calculation of speed or distance covered such as miles per hour , km/hour.
  • 8. PRODUCT RULE >> The derivative of the product of two differentiable functions is equal to the addition of the first multiplied by the derivative of the second and the second function multiplied by the derivative of the first function. APPLICATION: 1. The product rule is used in calculus, when you are asked to take derivative of the function. 2. It makes calculation clean and easier to solve. 3. It is used to differentiate product of two or more functions.
  • 9. DERIVATIVE PRODUCT RULE If u and v are differentiable at x, then so is their product uv and d/dx(u.v) = u (dv/dx) +v (du/dx) Example: Q) Find the derivative of y=(x^2 +1)(x^3+3) Answer: d/dx(x^2+1)(x^3+3)=(x^2+1)(3x^2)+ (2x)(x^3+3) =3x^4+3x^2+2x^4+6x =5x^4+3x^2+6x The particular product can be differentiated as well by multiplying out the original expression for y and differentiating the resulting polynomial. Y=(x^2+1)(x^3+3)=x^5+x^3+3x^2+3 dy/dx=5x^4+3x^2+6x This is in agreement with our first calculation.
  • 10. QUOTIENT RULE >> A quotient rule is similar to product rule. A quotient rule is stated as the ratio of the quantity of the denominator times the derivative of the numerator function minus the numerator times the derivative of the denominator function to the square of the denominator function. APPLICATION: 1. It is used for finding the derivative of a quotient of functions. 2. It is used for extend the power rule to functions with negative exponents. 3. To combine differentiation rule to find the derivative of a polynomial or rational function.
  • 11. DERIVATIVE QUOTIENT RULE If u and v are differentiable at x and if v(x) is not equal to 0 then the quotient u/v is differentiable at x and d/dx(u/v)= v (du/dx) – u (dv/dx)/ v^2 Example: Q)Find the derivative of y=(t^2-1)/(t^3+1) Answer: u=t^2-1 v=t^3+1 dy/dt=(t^3+1).2t- (t^2-1).3t^2/(t^3+1)^2 =2t^4+2t-3t^4+3t^2/(t^3+1)^2 =-t^4+3t^2+2t/(t^3+1)^2
  • 12. SQUEEZE THEOREM >> In calculus the squeeze theorem is a theorem regarding the limit of a function that is trapped between two other function. >> The squeeze theorem is used in calculus and mathematical analysis typically to confirm the limit of a function via comparison with other function whose limits are known. >> If the right hand limits and left hand limits do not equal eachother we cannot utilize squeeze theorem. If f(x)<g(x)<h(x) when x is near a If limxa f(x)=limxa h(x)=L then limxa g(x)=L.
  • 13. WHY IS IT CALLED SANDWICH THEOREM? >> The squeeze theorem is also called as sandwich or pinching theorem. It is a way to find the limit of one function if we know the limits of two functions it is “sandwiched” between. APPLICATION: It is used for calculating the limit of a given trigonometric funtions.
  • 14. EXAMPLE OF SANDWICH THEOREM Q) Using sandwich theorem show that: limx0 x^2 sin (1/x)=0 ANSWER: Let -1<sin(1/x)<1 Multiply by x^2 -x^2<x^2 sin 1/x <x^2 Lim x0 (–x^2)<lim x0 x^2 sin (1/x)< lim x0 x^2 Lim x->0 (-x^2)=-0=0 Lim x x^2=0=0 Lim x0 (-x^2)= lim x (x^2) Lim x0 x^2 sin (1/x)=0