ArulmiguMeenakshi Amman College of Engineering
Vadamavandal-604410.
Important Question and Answers
MA25C01-Applied Calculus
Prepared by
V.Umapathy,
Prof. /Mech.
S.N Question Answer
o
Syllabus
S.No. Description
1 Differential Calculus: Functions, graph of functions, New functions from old
functions, Limit of a function, Continuity, Limits at infinity, Derivative as a function,
Maxima and Minima of functions of single variable, Mean value theorem, Effect of
derivatives on the shape of a graph.
Activities: Visualization of the functions, Maxima and Minima of a function using
open-source software, Solving of Competitive Examination questions (Ex. GATE).
2 Functions of Several Variables: Partial derivatives, Chain rule, Total derivative,
Maxima and minima of functions of two variables, Method of Lagrange’s Multipliers,
Application problems in engineering.
Activities: Partial Derivatives with two or three variables, Maxima and Minima of a
function using open-source software, Solving of Competitive Examination questions
(Ex. GATE).
3 Integral Calculus: Fundamental theorem of Calculus, Indefinite integrals and the Net
Change Theorem, Improper integrals, Arc Length, Area of Region, Area of surface of
revolution.
Activities: Definite and Indefinite Integrals, Determination of Area, Solving of
Competitive Examination questions (Ex. GATE).
4 Multiple Integrals: Iterated integrals and Fubini’s theorem, Evaluation of double
integrals, change of order of integration, change of variables between Cartesian and
polar co-ordinates, evaluation of triple integrals-change of variables between Cartesian
and cylindrical and spherical co-ordinates.
Activities: Double integrals and triple integrals using open-source software, Solving
of Competitive Examination questions (Ex. GATE).
Unit-I
Differential Calculus
,Functions
- A function is a relation between a set of inputs (called the domain) and a set of possible outputs
(called the range).
- It assigns to each element in the domain exactly one element in the range.
- Functions can be represented algebraically, graphically, or numerically.
Graph of Functions
- The graph of a function f(x) is the set of all points (x, f(x)) in the Cartesian plane.
- The graph of a function can provide valuable information about the function's behavior, such as
its domain, range, and any patterns or symmetries.
New Functions from Old Functions
- Functions can be combined using arithmetic operations (addition, subtraction, multiplication,
and division) to create new functions.
- Functions can also be composed with each other to create new functions.
- Transformations such as shifts, stretches, and reflections can be applied to functions to create
new functions.
Limit of a Function
- The limit of a function f(x) as x approaches a is denoted by lim x→a f(x) and represents the
value that f(x) approaches as x gets arbitrarily close to a.
- Limits can be evaluated using various techniques, such as substitution, factoring, and L'Hopital's
Rule.
Continuity
- A function f(x) is continuous at a point x = a if lim x→a f(x) = f(a).
- A function is continuous on an interval if it is continuous at every point in the interval.
- Continuous functions have certain properties, such as being able to be drawn without lifting the
pencil from the paper.
Limits at Infinity
- The limit of a function f(x) as x approaches infinity is denoted by lim x→∞ f(x) and represents
the value that f(x) approaches as x gets arbitrarily large.
- Limits at infinity can be evaluated using various techniques, such as dividing by the highest
power of x or using L'Hopital's Rule.
,Derivative as a Function
- The derivative of a function f(x) is denoted by f'(x) and represents the rate of change of f(x) with
respect to x.
- The derivative can be interpreted as the slope of the tangent line to the graph of f(x) at a point.
Maxima and Minima of Functions of Single Variable
- A function f(x) has a local maximum at x = a if f(a) ≥ f(x) for all x in some interval containing a.
- A function f(x) has a local minimum at x = a if f(a) ≤ f(x) for all x in some interval containing a.
- The first derivative test and the second derivative test can be used to find local maxima and
minima.
Mean Value Theorem
- The Mean Value Theorem states that if a function f(x) is continuous on [a, b] and differentiable
on (a, b), then there exists a point c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).
- The Mean Value Theorem has various applications, such as proving the existence of maxima
and minima.
Effect of Derivatives on the Shape of a Graph
- The first derivative of a function can be used to determine the intervals on which the function is
increasing or decreasing.
- The second derivative of a function can be used to determine the concavity of the graph of the
function.
- The first and second derivatives can be used together to sketch the graph of a function.
Important Formulas
1. Functions:
- f(x) = y, where x is the input and y is the output
- Domain: set of all possible inputs
- Range: set of all possible outputs
2. Graph of Functions:
- Linear: f(x) = mx + c
, - Quadratic: f(x) = ax2 + bx + c
- Polynomial: f(x) = a_n xn + a_(n-1) x(n-1) + ... + a_1 x + a_0
- Trigonometric: f(x) = sin(x), f(x) = cos(x), etc.
3. New Functions from Old Functions:
- Addition: (f + g)(x) = f(x) + g(x)
- Subtraction: (f - g)(x) = f(x) - g(x)
- Multiplication: (fg)(x) = f(x)g(x)
- Division: (f/g)(x) = f(x)/g(x)
- Composition: (f ∘ g)(x) = f(g(x))
4. Limit of a Function:
- lim x→a f(x) = L
- lim x→a- f(x) = L (left-hand limit)
- lim x→a+ f(x) = L (right-hand limit)
5. Continuity:
- f(x) is continuous at x = a if lim x→a f(x) = f(a)
- f(x) is continuous on [a, b] if it is continuous at every point in [a, b]
6. Limits at Infinity:
- lim x→∞ f(x) = L (horizontal asymptote)
- lim x→-∞ f(x) = L (horizontal asymptote)
7. Derivative as a Function:
- f'(x) = dy/dx
- f'(a) = lim h→0 [f(a + h) - f(a)]/h
8. Maxima and Minima of Functions of Single Variable:
- Local maximum: f(a) ≥ f(x) for all x in some neighborhood of a
- Local minimum: f(a) ≤ f(x) for all x in some neighborhood of a
- Global maximum or minimum: above conditions hold for all x in the domain
9. Mean Value Theorem:
Vadamavandal-604410.
Important Question and Answers
MA25C01-Applied Calculus
Prepared by
V.Umapathy,
Prof. /Mech.
S.N Question Answer
o
Syllabus
S.No. Description
1 Differential Calculus: Functions, graph of functions, New functions from old
functions, Limit of a function, Continuity, Limits at infinity, Derivative as a function,
Maxima and Minima of functions of single variable, Mean value theorem, Effect of
derivatives on the shape of a graph.
Activities: Visualization of the functions, Maxima and Minima of a function using
open-source software, Solving of Competitive Examination questions (Ex. GATE).
2 Functions of Several Variables: Partial derivatives, Chain rule, Total derivative,
Maxima and minima of functions of two variables, Method of Lagrange’s Multipliers,
Application problems in engineering.
Activities: Partial Derivatives with two or three variables, Maxima and Minima of a
function using open-source software, Solving of Competitive Examination questions
(Ex. GATE).
3 Integral Calculus: Fundamental theorem of Calculus, Indefinite integrals and the Net
Change Theorem, Improper integrals, Arc Length, Area of Region, Area of surface of
revolution.
Activities: Definite and Indefinite Integrals, Determination of Area, Solving of
Competitive Examination questions (Ex. GATE).
4 Multiple Integrals: Iterated integrals and Fubini’s theorem, Evaluation of double
integrals, change of order of integration, change of variables between Cartesian and
polar co-ordinates, evaluation of triple integrals-change of variables between Cartesian
and cylindrical and spherical co-ordinates.
Activities: Double integrals and triple integrals using open-source software, Solving
of Competitive Examination questions (Ex. GATE).
Unit-I
Differential Calculus
,Functions
- A function is a relation between a set of inputs (called the domain) and a set of possible outputs
(called the range).
- It assigns to each element in the domain exactly one element in the range.
- Functions can be represented algebraically, graphically, or numerically.
Graph of Functions
- The graph of a function f(x) is the set of all points (x, f(x)) in the Cartesian plane.
- The graph of a function can provide valuable information about the function's behavior, such as
its domain, range, and any patterns or symmetries.
New Functions from Old Functions
- Functions can be combined using arithmetic operations (addition, subtraction, multiplication,
and division) to create new functions.
- Functions can also be composed with each other to create new functions.
- Transformations such as shifts, stretches, and reflections can be applied to functions to create
new functions.
Limit of a Function
- The limit of a function f(x) as x approaches a is denoted by lim x→a f(x) and represents the
value that f(x) approaches as x gets arbitrarily close to a.
- Limits can be evaluated using various techniques, such as substitution, factoring, and L'Hopital's
Rule.
Continuity
- A function f(x) is continuous at a point x = a if lim x→a f(x) = f(a).
- A function is continuous on an interval if it is continuous at every point in the interval.
- Continuous functions have certain properties, such as being able to be drawn without lifting the
pencil from the paper.
Limits at Infinity
- The limit of a function f(x) as x approaches infinity is denoted by lim x→∞ f(x) and represents
the value that f(x) approaches as x gets arbitrarily large.
- Limits at infinity can be evaluated using various techniques, such as dividing by the highest
power of x or using L'Hopital's Rule.
,Derivative as a Function
- The derivative of a function f(x) is denoted by f'(x) and represents the rate of change of f(x) with
respect to x.
- The derivative can be interpreted as the slope of the tangent line to the graph of f(x) at a point.
Maxima and Minima of Functions of Single Variable
- A function f(x) has a local maximum at x = a if f(a) ≥ f(x) for all x in some interval containing a.
- A function f(x) has a local minimum at x = a if f(a) ≤ f(x) for all x in some interval containing a.
- The first derivative test and the second derivative test can be used to find local maxima and
minima.
Mean Value Theorem
- The Mean Value Theorem states that if a function f(x) is continuous on [a, b] and differentiable
on (a, b), then there exists a point c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).
- The Mean Value Theorem has various applications, such as proving the existence of maxima
and minima.
Effect of Derivatives on the Shape of a Graph
- The first derivative of a function can be used to determine the intervals on which the function is
increasing or decreasing.
- The second derivative of a function can be used to determine the concavity of the graph of the
function.
- The first and second derivatives can be used together to sketch the graph of a function.
Important Formulas
1. Functions:
- f(x) = y, where x is the input and y is the output
- Domain: set of all possible inputs
- Range: set of all possible outputs
2. Graph of Functions:
- Linear: f(x) = mx + c
, - Quadratic: f(x) = ax2 + bx + c
- Polynomial: f(x) = a_n xn + a_(n-1) x(n-1) + ... + a_1 x + a_0
- Trigonometric: f(x) = sin(x), f(x) = cos(x), etc.
3. New Functions from Old Functions:
- Addition: (f + g)(x) = f(x) + g(x)
- Subtraction: (f - g)(x) = f(x) - g(x)
- Multiplication: (fg)(x) = f(x)g(x)
- Division: (f/g)(x) = f(x)/g(x)
- Composition: (f ∘ g)(x) = f(g(x))
4. Limit of a Function:
- lim x→a f(x) = L
- lim x→a- f(x) = L (left-hand limit)
- lim x→a+ f(x) = L (right-hand limit)
5. Continuity:
- f(x) is continuous at x = a if lim x→a f(x) = f(a)
- f(x) is continuous on [a, b] if it is continuous at every point in [a, b]
6. Limits at Infinity:
- lim x→∞ f(x) = L (horizontal asymptote)
- lim x→-∞ f(x) = L (horizontal asymptote)
7. Derivative as a Function:
- f'(x) = dy/dx
- f'(a) = lim h→0 [f(a + h) - f(a)]/h
8. Maxima and Minima of Functions of Single Variable:
- Local maximum: f(a) ≥ f(x) for all x in some neighborhood of a
- Local minimum: f(a) ≤ f(x) for all x in some neighborhood of a
- Global maximum or minimum: above conditions hold for all x in the domain
9. Mean Value Theorem:
