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St. JOSEPH’S COLLEGE OF ENGINEERING St. JOSEPH’S INSTITUTE OF TECHNOLOGY MA8151 - ENGINEERING MATHEMATICS I UNIT I - DIFFERENTIAL CALCULUS FORMULA SHEET

Functions: Let X and Y be two non-empty sets. If there is a rule ‘f’ which associates to every element x  X , a unique element y  Y , then such a rule ‘f ’ is known as a function or mapping from the set X to the set Y. If x  X is associated to an element of y  Y , then y is called image of f and x is called the pre-image of y under f . We write it as y  f  x  . The set X is called the domain of f and Y is called the co-doma in of f . The set of all images under f is called the range of f and is denoted by f  X  .

Even and Odd functions: If a function f satisfies f  x  f  x for every number x in its domain, then f is called an even function. If f satisfies f  x    f  x  for every number x in its domain, then f is called an odd function.

Increasing and Decreasing Functions: A function f is called an increasing function in its domain D if f  x1   f  x2  , whenever x1  x2 ,

x1, x2  D . A function f is called an decreasing function in its domain D if f  x1   f  x2  , whenever x1  x2 ,

x1, x2  D . Note: Increasing and decreasing functions are also called Monotonic functions.

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Limit of a function: We say that f  x  approaches l  l  R as x approaches a if given  0 there exists   0 such that

f  x  l  , whenever x  a   . In this case we write lim f  x   l x a

One sided limit: A function f  x  is said to tend to l as x tends to a from right if to each  0 , there exists   0 such that x   a, a    , f  x  l  and we write lim f  x   l x a

A function f  x  is said to tend to l as x tends to a from left if to each  0 , there exists   0 such that x   a   , a  , f x   l  and we write lim f  x   l . x a

Limit Laws: Suppose that c is a constant and the limits lim f  x  and lim g  x  exist. Then x a

x a

(i)

lim  f  x   g  x  lim f  x  lim g  x

(ii)

lim  cf  x  clim f  x

x a

x a

x a

x a

x a

(iii) lim  f  x  g  x  lim f x lim g x xa x a x a

f  x  f  x  lim  x a (iv) lim   x a g  x g  x   lim x a



if lim g  x   0 x a

Continuity: A function f  x is continuous at a number a if lim f  x   f  a  x a

(i.e.) (i) f  a  is defined (ii) (iii)

lim f  x  exists x a

lim f  x   f a  x a

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A function f  x is continuous from the right at a number a if lim f  x   f  a  and f  x is x a

continuous from the left at a if lim f  x   f  a  x a

A function f  x is continuous on an interval if it is continuous every number in the interval.

Note: If f  x and g  x  are continuous at ‘a ’ and c is constant, then the following functions are also continuous at ‘a’: (i) f  x  g  x

(ii) cf  x

(iii)

f  x g  x

(iv)

f x  g  x

, g x   0

Note: The following types of functions are continuous at every number in their domain: (i) Polynomials (ii) Rational functions (iii) Root functions (iv) Trigonometric functions (v) Inverse trigonometric functions (vi) Exponential functions (vii) Logarithmic functions.

The Intermediate value theorem: Suppose that f  x  is continuous on the closed interval a, b  and let N be any number between

f  a and f  b , where f  a   f b  , then there exists a number c in  a, b such that f  c  N . Tangent line:





The tangent line to the curve y  f  x  at the point P a, f a  is the line through P with slope

m  lim x a

f  x   f a  provided that this limit exists. x a

Derivatives:

' The derivative of a function f  x  at a number a denoted by f '  a  is f  a  lim h 0

f  a  h  f  a h

if

this limit exists.

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Differentiation Rules:

d c   0 dx

(i)

Derivative of a constant:

(ii)

The Power rule: If n is any real number, then

(iii)

The constant multiple rule:

d n x   nx n1  dx

If c is a constant and f  x is a differentiable function, then (iv)

d d f x  cf  x    c dx dx

The sum and difference rule: If f  x and g  x  are both differentiable, then

(v)

Derivative of the natural exponential function:

(vi)

The product rule:

d d d  f  x  g  x  f  x  g  x dx dx dx

d x e   ex  dx

If f  x and g  x  are both differentiable, then

d d d f  x g  x    f  x . g  x  f  x  g x   dx dx dx (vii)

The quotient rule: If f  x and g  x  are both differentiable, then

d d g  x   f  x   f  x   g  x  d  f  x  dx dx   2 dx  g  x   g  x  (viii)

Derivative of trigonometric functions:

d sin x   cos x dx d sec x   sec x.tan x dx

d  cos x    sin x dx

d  tan x   sec 2 x dx

d  cosec x    cos ec x.cot x dx

d  cot x    cosec2 x dx

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(ix)

Derivatives of inverse trigonometric functions:

(x)

d 1 sin x  1    dx 1  x2

d 1 cos 1 x     dx 1 x 2

d 1 tan 1 x    1 x2 dx

d 1 sec 1 x    dx x x2  1

d 1 cos ec 1x     dx x x2  1

d 1 cot 1 x     1  x2 dx

Derivative of Logarithmic functions:

1 d loge x   , dx x (xi)

d  logb x   1 dx x log b

Derivative of Hyperbolic functions:

d d sinh x   cosh x   cosh x   sinh x dx dx d  cosech x    cos ec h x.coth x dx d  coth x    cosech 2 x dx

d tanh x   sech2 x  dx d sech x    sechx.tanh x dx

Maxima and Minima of functions of one variable:

Absolute maximum and absolute minimum: Let c be a number in the domain D of a function f . Then f  c is called the (i)

absolute maximum value of f on D if f  c  f  x for all x in D

(ii)

absolute minimum value of f on D if f  c  f  x for all x in D.

Note: An absolute maximum or minimum is sometimes called a globa l maximum or minimum. The maximum and minimum values of f are called extreme values of f .

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Critical Number: A critical number of a function f is a number c in the doma in of f such that either f '  c   0 or

f '  c  does not exist. The Extreme value theorem: If f is continuous on a closed interva l  a, b , then f attains an absolute maximum value f  c and an absolute minimum value f  d  at some number c and d in a, b  Fermat’s theorem: If f has a local maximum or minimum at c, and if f '  c  exists, then f '  c   0 .

The closed interval method: To find the absolute maximum and minimum va lues of a continuous function f on a closed interval a, b  : (i)

Find the values of f at the critical numbers of f in  a, b

(ii)

Find the values of f at the end points of the interval

(iii)

The largest of the values from steps (i) and (ii) is the absolute maximum va lue, and the smallest of these values is the absolute minimum value.

Increasing and Decreasing test: (i)

If f '  x   0 on an interval, then f is increasing on that interval

(ii)

If f '  x   0 on an interval, then f is decreasing on that interval

Local maximum and local minimum: Let c be a number in the domain D of a function f . Then f  c is called the (i)

local maximum value of f if f  c   f  x  , when x is near c

(ii)

local minimum value of f if f  c  f  x , when x is near c.

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The First Derivative test: Suppose that c is a critical number of a continuous function f . (i)

If f '  x  changes from positive to negative at c, then f  x has a local maximum at c

(ii)

' If f  x  changes from negative to positive at c, then f  x has a local minimum at c.

(iii)

If f '  x  has no sign change at c, then f  x  has no local maximum or minimum at c.

Concave upward and concave downward: If the graph of f lies above all of its tangents on an interval I, then it is called concave upward on I. If the graph of f lies below all of its tangents on I, it is called concave downward on I.

Inflection point: A point P on a curve y  f  x  is called an inflection point if f is continuous there and the curve changes from concave upward to concave downward or from concave downward to concave upward at P.

Concavity test: (i)

If f ''  x   0 for all x in I, then the graph of f is concave upward on I

(ii)

If f ''  x   0 for all x in I, then the graph of f is concave downward on I

The second derivative test: Suppose f ''  x  is continuous near c, (i)

' '' If f  c   0 , and f  c   0, then f has local minimum at c

(ii)

If f '  c   0, and f ''  c   0 , then f has local maximum at c.

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ENGINEERING MATHEMATICS – I (MA8151) UNIT V - DIFFERENTIAL EQUATIONS - NOTES 1.

1

Solve the initial value problem y ′′ + y = 0, y (0) = 2, y ′(0) = 3

( D2 + 1) y = 0

Solution: Given

The Auxiliary equation (A.E) is m 2+ 1 = 0 ⇒ m = ± i = 0 ± i = α ± iβ

C.F . = eα x ( A cos β x + B sin β x) = A cos x + B sin x P .I . = 0 General Solution y (x ) = A cos x + B sin x y′( x) = − A sin x + B cos x y (0) = 2 ⇒ y (0) = A cos 0 + B sin 0 ⇒2 = A y′(0) = − A sin 0 + B cos 0 ⇒3 =B Hence y (x ) = 2cosx + 3sin x 2.

(

)

Solve D 3 + D2 + 4 D + 4 y = 0

The A.E. is m3 + m2 + 4m + 4 = 0 m2 ( m + 1) + 4 ( m +1) = 0

Solution:

( m2 + 4 )(m + 1) = 0 m 2 = −4, m = −1

m = ±2i , m = −1

m1 = 1, m2 = 2i m3 = − 2i The roots are real and complex. ∴ C.F . = Ae −x + ( B cos 2 x + C sin 2 x ) R .H .S . = 0, ∴P.I . = 0

∵

∴ y = Ae − x + (B cos 2 x + C sin 2 x )

y = C .F . + P .I .

3.

Solve

d2 y dx

2

−5

dy − 6y = 0 dx

(

)

Solution: Given D 2 − 5D − 6 y = 0

The Auxiliary equation (A.E) is m2 − 5 m− 6 = 0 (m − 6)(m + 1) = 0 m = −1, 6

The roots are real and distinct.

Complementary function is (C.F) = Ae m1x + Be m 2x = Ae −x + Be 6x , Since R .H .S = 0 ∴ P .I . = 0 y = C .F . + P .I . Page No: 1 For More Visit : www.Learnengineering.in

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4.

Solve 4 y′′ + 12 y′ + 9 y = 0

(

)

Solution: Given 4 D2 +12 D +9 y = 0

The Auxiliary equation (A.E) is 4m2 + 12m + 9 = 0 (2 m + 3) 2 = 0 ⇒ m =

Complementary function is (C.F) = ( A + Bx)e Since R .H .S = 0 ∴ P .I . = 0

mx

−3 2

The roots are real and equal.

 −3   x = ( A + Bx)e  2  ,

y = C .F . + P .I . ∴ The general solution is y

5.

Solve

d2 y dx 2

−6

 −3  x   = ( A + Bx )e 2 

dy + 13 y = 0 dx

(

)

2 Solution: Given D − 6D + 13 y = 0<...