# Gtu Maths - 1(3110014) Papers, Syllabus, Study Material And Example For Examination

## Gtu Maths - 1(3110014) Papers, Syllabus, Study Material And Example For Examination

GTU Maths-1 (3110014) :

### Syllabus

#### Chapter 1 : Infinite Sequences and Series

Weightage Of Chapter : 20 %

• Introduction of Convergence, Divergence of Sequences and Infinite Series
• The nth term test for Divergence, Integral Test
• Comparison Test, Ratio Test, Root Test
• Alternating Series, Absolute convergence, Conditional convergence
• Power Series & Radius of convergence
• Taylor’s series
• Maclaurin series

#### Chapter 2 : Curve Sketching

Weightage Of Chapter : 10 %

• Concavity
• Polar co-ordinates, Relation between Polar and Cartesian Co-ordinates Graphs in Polar co-ordinates

#### Chapter 3 :Indeterminate Forms

Weightage Of Chapter : 10 %

Indeterminate forms

• 0/0 Form
• ∞/∞ Form
• 0 × ∞ Form
• ∞ − ∞ Form
• ∞0 Form
• 1∞ Form

Improper Integral

• Improper integrals of Type- I and Type – II

#### Chapter 4 : Applications of Integration

Weightage Of Chapter : 10 %

• Volume by slicing
• Volume by cylindrical shell.

#### Chapter 5 : Partial Derivatives

Weightage Of Chapter : 30 %

• Function of 2-variables, graphs, level curves
• Limit, continuity of function of several variables
• Tangent plane, Normal line
• Linear approximation, Total differential
• Chain rule, implicit differentiation
• Euler’s theorem for homogeneous function
• Maximum and minimum values by second derivative test
• Lagrange multipliers
• Taylor’s formula for two variables

#### Chapter 6 : Multiple Integrals

Weightage Of Chapter : 20 %

• Double integrals over general region And in polar co-ordinates
• Triple Integrals, Triple integrals in cylindrical coordinates ans spherical co-ordinates
• Change of Order of Integration
• Jacobian of several variables

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As Per New Syllabus Of Maths 1 (3110014)

#### 1. Indeterminate Forms and L’Hospital’s Rule

Weightage Of Chapter : 2 %

• Indeterminate Forms and L’Hospital’s Rule

#### 2. Improper Integrals

Weightage Of Chapter : 6 %

• Convergence and divergence of the integrals.
• Improper Integrals.
• Beta and Gamma functions and their properties.

#### 3. Curves & Integrals

Weightage Of Chapter : 7 %

• Applications of definite integral.
• Volume using cross-sections.
• Length of plane curves, Areas of Surfaces of Revolution.

#### 4. Convergence and divergence of sequences

Weightage Of Chapter : 20 %

• Convergence and divergence of sequences
• The Sandwich Theorem for Sequences.
• The Continuous Function Theorem for Sequences.
• Bounded Monotonic Sequences, Convergence and divergence of an infinite series, geometric series, telescoping series, nth term test for divergent series.
• Combining series, Harmonic Series,Integral test, The p – series.
• The Limit Comparison test. Ratio test. Raabe’s Test. Root test
• Alternating series test, Absolute and Conditional convergence
• Power series and Radius of convergence of a power series

Taylor Series :

• Taylor Series For One Variable
• Taylor Series For Two Variables
• Maclaurin series

#### 5. Fourier Series

Weightage Of Chapter : 10 %

• Fourier Series of 2PI periodic functions.
• Dirichlet’s conditions for representation by a Fourier series.
• Orthogonality of the trigonometric system.
• Fourier Series of a function of period 2L.. Fourier Series of even and odd functions, Half range expansions.

#### 6. Functions of several variables

Weightage Of Chapter : 20 %

• Functions of several variables
• Limits and continuity, Test for non existence of a limit.
• Partial differentiation.
• Mixed derivative theorem.
• differentiability.
• Chain rule, Implicit differentiation
• tangent plane and normal line
• total differentiation.
• Maxima And Minima
• Method of Lagrange Multipliers

#### 7. Functions of several variables

Weightage Of Chapter : 20 %

• Multiple integral.
• Double integral over Rectangles and general regions, double integrals as volumes
• Change of order of integration.
• double integration in polar coordinates
• Area by double integration
• Triple integrals in rectangular, cylindrical and spherical coordinates
• jacobian for Multiple Substitution

#### 8. Elementary row operations in Matrix

Weightage Of Chapter : 15 %

• Elementary row operations in Matrix, Row echelon and Reduced row echelon forms.
• Rank by echelon forms.
• Inverse by Gauss-Jordan method.
• Solution of system of linear equations by Gauss elimination and Gauss-Jordan methods.
• Eigenvalues and eigenvectors. Cayley-Hamilton theorem, Diagonal iration of a matrix.

### Reference And Text Books Of Maths 1

#### Thomas' Calculus

• Author : Maurice D. Weir, Joel Hass
• Publisher : Pearson

#### Calculus

• Author : Howard Anton. Id Bivens. Stephens Davis
• Publisher : Wiley

#### Calculus: Early Transcendentals with Coursemate

• Author : James Stewart
• Publisher : Cengage

#### Elementary Linear Algebra. Applications version

• Author : Anton and Rorres
• Publisher : Wiley India Edition

May 2019
May 2018
Dec 2017
May 2017
Jan 2017

May 2019
Jan 2019

### Most Imp Topics Of Maths 1

#### 1) Maclaurin Series with definition and examples

Maclaurin series is like a another form of taylor series.for learn maclaurin's series you have to about taylor series .

Taylor series defined as,

In above equation when we put a=0 then it is known as maclaurin series.

In other words we can say we find series at point zero (0) of Taylor series expansion or you can also say we find sum of terms of point at 0.

#### Definition Maclaurin Series :

Maclaurin series A special case arises when we take the Taylor series at the point 0. When we do this, we get the Maclaurin series. The Maclaurin series is the Taylor series at the point 0. The formula for the Maclaurin series then is this:

Above equation has value at point zero hence we find series which placed on point zero.

#### Some Useful Maclaurin series

All Series Given below is special case hence you can use directly.

therefore series are like :

#### Natural logarithm :

The natural logarithm (with base e) has Maclaurin series,

#### Binomial series :

The binomial series is the power series,

All series above given are special series therefore you can use directly if you have this kind of function in your equation.

#### Example :

Find Maclaurin Series Of f(x) = sinx

#### Solution :

To find the Maclaurin series for this function, we start the same way. We find the various derivatives of this function and then evaluate them at the point 0. We get these for our derivatives:

Derivative At the point 0,

f(x) = sin x : f(0) = 0

f'(x) = cos x : f'(0) = 1

f(x) = -sin x : f(0) = 0

f'(x) = -cos x : f'(0) = -1

f(x) = sin x : f(0) = 0

Firstly you have to find derivatives after that put x =0 in derivatives .

After all this  put equation and put value hence you get the series which is maclaurin series.

### Taylor Series Expansion With Examples

#### Definition :

Taylor series is a series expansion of a function about a point. A one-dimensional Taylor series is an expansion of a real function f(x) about a point x=a.

Distant economies change dramatically. Far away populations move their centers. Sometimes, it’s better to focus on what is happening locally. The same may be said for complicated mathematical expressions.

By looking at the localized behavior of a function, we can often gain amazing insights.

This is where the Taylor’s series is really useful. Simply put, the Taylor’s series is a representation of a function that can help us do mathematics.

Uses of the Taylor’s series include analytic derivations and approximations of functions.

If we have a function which we can differentiate, then we can express that function as a Taylor’s series.

The Taylor’s series for any polynomial is the polynomial itself.

The Maclaurin series for 1/1 − x is the geometric series

so the Taylor’s series for 1/x at a = 1 is

By integrating the above Maclaurin series, we find the Maclaurin series for log(1 − x), where log denotes the natural logarithm.

and the corresponding Taylor’s series for log x at a = 1 is

The Taylor’s series for the exponential function ex at a = 0 is

If You Want To Know About Or Clear Your Fundamental Of  Taylor’s Series Then Here Totally Explanation Is Available With Example And Also Their Solutions.

#### Solution :

In this example we have,

If we compare (x-a) = (x-1)   then a = 1

Then according to equation find f’ , f’’ and f’’’

Put this values in the Taylor’s series equation.

Put a = 1 :

So, After Simplification ,

So, the taylor series is :

You Can Also Read :

### Chain Rule And Implicit Functions

#### Example:

if u=x2+y2, x=at2, y=2at, find du/dt.

#### Solution:

here,u is a composite function of t.

du/dt = du/dx * dx/dt + du/dy * dy/dt
= 2x * 2at + 2y * 2a
=4atx + 4ay
=4a [t * at2 + 2at]
=4a2t (t2 + 2)

#### Example:

if u=x2+y2, x=a cost, y=b sint, find du/dt.

#### Solution:

here,u is a composite function of t.

du/dt = du/dx * dx/dt + du/dy *dy/dt
=2x (-asint) + 2y (bcost)
=-2a2 cost sint + 2b2 sint cost
=-a2 sin2t + b2sin2t
=sin2t (b2 – a2)

### Convergence,Divergence Of Sequences And Infinite Series

Convergence,Divergence And Infinite series is explain the neural mechanism of recollection.

#### Sequences :

If for each n ∈ N, a number an is assigned, then the numbers a1,a2,……an is said to form a sequence.

That is a sequence is a function from the set N to the set S of numbers
a1,a2,……an .

Here an is called the nth term of the sequence. Thus if f : N -> S is a sequence then we can write

f(n) = an

Symbolically, the sequence is denoted by {an}n=1 -> ∞  or simply {an} or ( an ) or < an >. The number of terms of a sequences is always infinite as n ∈ N.

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#### Convergence Of a Sequence :

To Understand the concept of convergence let us consider the sequence {1⁄3n}={1⁄3 , 1⁄32 , 1⁄33 ,….}. If we take the value of n sufficiently large, the value of 1⁄3n comes close and close to 0. That is as n -> ∞ we have an -> 0. Suppose if we want to find the value of n for which the distance between an = 1⁄3n and 0 will be less than 0.00001.

That is

| an – 0| < 0.00001

| 1⁄3n | < 0.00001

3n > 1000 > 36

n > 6

If we take n > 6 than the distance between 1⁄3n and 0 becomes less than 0.00001.

Hence we say that the sequence {1⁄3n} converges to 0, and 0 is the limit of the sequence and we write

1⁄3n  = 0.

On the other hand if we observe the series {2n – 1} = {1,3,5….} then for large value of n, an = 2n-1 increasing and increasing. It never attempts to tend to any fixed value. we say that this sequence is not convergent. Thus we led to the following definitions.

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#### Convergence Of A Sequence :

Definition :

Convergence is a sequence {an} is said to be converge to a real number l if given ε > 0 Ǝ a positive integer m such that | an – l | < ε,

we can write as,

n → ∞ an = l .

#### Divergence Of a Sequence :

Definition :

Divergence is a sequence {an} is diverge to ∞ if given any real number k > 0 Ǝ a positive integer m such that an > k for all n ≥ m, and we can write as,

lim n → ∞ an = ∞

Solution :

### Maxima And Minima By Second Derivative Test.

#### Definition :

Maxima and Minima from Calculus. One of the great powers of calculus is in the determination of the maximum or minimum value of a function.

Consider a function u=f(x,y) of two independent variables x,y whose domain is a certain region R in the xy-plane.

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We say that f(x,y) has a maximum at the point (x0,y0) of its domain R if f(x0,y0) ≥ f(x,y) for all (x,y) in R. such a maximum corresponds to a highest point of the surface S.

We say that f(x,y) has a strict maximum at the point (x0,y0) of its domain R. if actually, f(x0,y0) > f(x,y) for all (x,y) in R that are different from (x0,y0). Thus the greatest value is reach only at a single point.

Similarly, We say that f(x,y) has a minimum at the point (x1,y1) of R if f(x1,y1) < f(x,y) for all (x,y) in R. such a minimum corresponds to a lowest point of the surface S.

Again f(x,y) has a has a strict minimum if f(x1,y1) < f(x,y) for all (x,y) ≠ (x1,y1) in R.

For Example, consider the function u=x2+y2 defined in the closed disc given by x2+y2 < 1.

The function represents the surface of paraboloid lying below the plane u=1. Here the maximum of occur at all the points of the boundary of the circle x2+y2 = 1.

You can also see different plane equations like Normal Line And Tangent Plane.

#### Working Rule To Find The Extremum Value :

Consider the function u = f(x,y).

Obtain the first and second order derivatives, such as

p = fx

q = fy

r = fxx

s = fxy

t = fyy

1)Take fx = 0, fy = 0 and solve them simultaneously to obtain stationary points. LET (x0,y0) , (x1,y1) , …. be the stationary points.

2)Consider the stationary point (x0,y0).

obtain the values of r , s and t at (x0,y0) .

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(a) If rt – s2 > 0 then the extremum values exist.

-> r < 0 the value is maximum.

-> r > 0 the value is minimum.

(b) If rt – s2 < 0 the there is no extremum value exist. thus there is neither maximum nor minimum at (x0,y0) is called a saddle point.

(c) If rt – s2 = 0 then we can’t say about extremum value and further investigation is required.

(3) Follow the same procedure for other stationary points.

### Improper Integral Of Type-1 And Type-2

#### Improper Integral :

The definite integral is said to be improper (or singular) integral if one or both limits of integration are infinite and/or if the integrand is unbounded on the interval

If the definite integral is not an improper integral then it is said to be a proper (or regular) interval.

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Type – 1

1 . Integral With One end Point Is Infinity
1.1 When Upper Limit Is Infinity

Now If f is continuous on an interval [ a , ∞ ) then an improper integral is defined as

If this limit exist, we say that I is convergent; if not, it s divergent

Here If f is nonnegative and continuous then above integral has an important geometrical interpretation. For each value of t>a the definite integral represents the area under the curve y=f(x) over an interval [a,t].

when t -> ∞ this area will approach to the area under the curve y = f(x) over the entire interval [a, +∞ ).

Thus represents the area under the curve y=f(x) over the interval [a, ∞).

### improper integral

#### 1.2 When Lower Limit Is Infinity

Now if f is continuous on the interval ( -∞ , b) then the improper integral is defined as If the limit exists, we say that I is convergent: if not, it is divergent.

If f is nonnegative and continuous on (-∞, b ] then this integral, geometrically, represents the area under the curve y=f(x) over the interval (-∞ , b].

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#### Solution : #### 1.3 When both the limits of integration are Infinity

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that is if is given

then we write,

= I1 + I2

I is convergent if both I1 and I2 are convergent.

I is divergent I1 or I2 is divergent.

The improper integral defined in expression (1) can sometimes, defined by the value of the improper integral by the limit.

Called Cauchy Principal Value of the integral. Usually, This limiting procedure is designated by the special notation.

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### Type-2

#### 2. Integral when Integrated Is Unbounded

1) If f(x) is continuous on [a,b) such that

f(x) -> + ∞ as x -> b from left then we can write.

2) If f(x) is continuous on (a,b] such that

f(x) -> + ∞ as x -> a from left then we can write.

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3) if f(x) is continuous on [a,b] and not bounded at the point c ∈ (a,b) then we can write.

In each of the above three cases we say that the integral converges if limit exists; otherwise it diverges.

### Comparison test, The ratio & Cauchy’s Root test

Comparison Test,Integral comparison test, Ratio test pdf,Roof test examples problems,Ratio test practical problems,Root test vs Ratio test,Ratio test calculator,Can the value of a series be determined using the root test or the ratio test,Alternating series test,Page navigation

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This Post Is About For Check Either Given Series is Convergent or Divergent And Also Check Three Test For It.

• Comparison test
• Ratio test
• Cauchy’s Root test

In Addition You will Get Example With Solution in Brief Of These All Three tests.

If You Want To Know About Comparison test,Ratio test And Cauchy’s Root test Then You Are on Right Place.

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Read Till To The End And You Get Brief Explanation Of Comparison test,Ratio test and Root test.

#### Comparison Test :

Up To now we were finding sn the partial sum of the series, to test the convergence of the series. But unfortunately , it is not always possible to find the partial sum sn for every series and hence we cannot apply the definitions of convergence or divergence directly. Thus it becomes necessary to formulate other tests for series with all terms positive. Using these tests we can discuss the convergence of several standard series.

The following theorems gives different tests of comparison.

Limit comparison test : a method of testing for the convergence of an infinite series.

Direct comparison test : a way of deducing the convergence or divergence of an infinite series or an improper integral

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### Ratio Test :

In this section we shall discuss one more test for the convergence of the series known as Ratio test, which does not require the knowledge of any auxiliary series.

Conclusion : After Read This Page As A Result We Know About Test For Check Either Given Series Convergent or Divergent.We Check With Comparison test,Ratio test And Root test.

So, In this article You Can Learn Most IMP Topics Of the maths -1 with subject code 2110014 and 2110015 both.

You Can Get Maths-1 Syllabus , Question Papers, Study Material, Chapter Weightage And Example With Solutions.

I hope you may like it this post and this post may will help you.

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#### 1 comment:

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