Engneering Mathematics

1. Fundamentals to Calculus

 

Definition and History of Calculus

Calculus is the mathematical study of continuous change. It provides a framework for understanding how quantities change over time or space and is divided into two main branches:

1. Differential Calculus – Deals with rates of change and slopes of curves.
2. Integral Calculus – Focuses on accumulation of quantities and areas under curves.

Historical Background:

• The foundations of calculus were independently developed in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz.
• Newton used calculus to describe motion and gravity in his work Principia Mathematica.
• Leibniz introduced the modern notation for calculus, including the integral (∫) and derivative (d/dx) symbols.
• Over time, mathematicians such as Euler, Cauchy, and Riemann refined the concepts, making calculus more rigorous and applicable to various scientific fields.

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Key Concepts of Calculus

1. Limits

   • A limit describes the value that a function approaches as the input (x) gets closer to a particular point.
   • It is essential in defining derivatives and integrals.
   • Example:$\lim_{{x \to 2}} (3x + 1) = 7$x→2lim​(3x+1)=7 This means that as$x$x approaches 2, the function$3x + 1$3x+1 approaches 7.

2. Continuity

   • A function is continuous at a point if its graph has no breaks, holes, or jumps.
   • Mathematically, a function$f(x)$f(x) is continuous at$x = a$x=a if:$\lim_{{x \to a}} f(x) = f(a)$x→alim​f(x)=f(a)
   • Example: The function$f(x) = x^2$f(x)=x2 is continuous everywhere, but$f(x) = \frac{1}{x}$f(x)=x1​ is discontinuous at$x = 0$x=0.

3. Functions

   • A function is a relationship between an input (x) and an output where each input has exactly one output.
   • Types of functions commonly used in calculus:
     • Polynomial functions (e.g.,$f(x) = x^3 - 2x + 5$f(x)=x3−2x+5)
     • Exponential functions (e.g.,$f(x) = e^x$f(x)=ex)
     • Trigonometric functions (e.g.,$f(x) = \sin x$f(x)=sinx)
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2. Differential Calculus

1. Derivatives and Their Physical Meaning

A derivative represents the rate at which a function changes with respect to a variable. Mathematically, if

$y = f(x)$

y=f(x), the derivative

$f'(x)$

f′(x) or

$\frac{dy}{dx}$

dxdy​ measures how

$y$

y changes as

$x$

x changes.

Physical Interpretations:

• Slope of a Curve: The derivative gives the slope of the tangent line at any point on a function’s graph.
• Velocity in Physics: If$s(t)$s(t) represents position at time$t$t, then$v(t) = s'(t)$v(t)=s′(t) is the velocity (rate of change of position).
• Rate of Growth: In economics and biology, derivatives measure growth rates, such as population increase or profit maximization.

Example:
If

$f(x) = x^2$

f(x)=x2, then the derivative is

$f'(x) = 2x$

f′(x)=2x. This means at

$x = 3$

x=3, the slope of the curve is

$2(3) = 6$

2(3)=6.

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2. Rules of Differentiation

To compute derivatives efficiently, calculus provides several differentiation rules:

a) Power Rule

If

$f(x) = x^n$

f(x)=xn, then

$f'(x) = n x^{n-1}$

f′(x)=nxn−1

Example:

$f(x) = x^3 \Rightarrow f'(x) = 3x^2$

f(x)=x3⇒f′(x)=3x2

b) Product Rule

If

$f(x) = u(x) \cdot v(x)$

f(x)=u(x)⋅v(x), then

$f'(x) = u'v + uv'$

f′(x)=u′v+uv′

Example:

$f(x) = x^2 \sin x$

f(x)=x2sinx, then

$f'(x) = 2x \sin x + x^2 \cos x$

f′(x)=2xsinx+x2cosx

c) Quotient Rule

If

$f(x) = \frac{u(x)}{v(x)}$

f(x)=v(x)u(x)​, then

$f'(x) = \frac{u'v - uv'}{v^2}$

f′(x)=v2u′v−uv′​

Example:

$f(x) = \frac{x^2}{x+1}$

f(x)=x+1x2​, then

$f'(x) = \frac{(2x)(x+1) - x^2(1)}{(x+1)^2}$

f′(x)=(x+1)2(2x)(x+1)−x2(1)​

d) Chain Rule

If

$f(x) = g(h(x))$

f(x)=g(h(x)), then

$f'(x) = g'(h(x)) \cdot h'(x)$

f′(x)=g′(h(x))⋅h′(x)

Example:

$f(x) = \sin(x^2)$

f(x)=sin(x2), then

$f'(x) = \cos(x^2) \cdot 2x$

f′(x)=cos(x2)⋅2x

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3. Applications of Derivatives

a) Tangents and Normals

• The tangent line to a function at$x = a$x=a has a slope given by$f'(a)$f′(a).
• The normal line is perpendicular to the tangent and has slope$-\frac{1}{f'(a)}$−f′(a)1​.
• Used in engineering and physics to find slopes of curves.

b) Optimization (Maxima and Minima)

• Derivatives help determine where a function reaches its highest (maximum) or lowest (minimum) points.
• Critical Points: Solve$f'(x) = 0$f′(x)=0 to find these points.
• Second Derivative Test:
  • If$f''(x) > 0$f′′(x)>0, the function has a local minimum.
  • If$f''(x) < 0$f′′(x)<0, the function has a local maximum.

Example: A company’s revenue function is

$R(x) = -2x^2 + 40x$

R(x)=−2x2+40x. To maximize revenue, solve

$R'(x) = 0$

R′(x)=0:

$R'(x) = -4x + 40 = 0 \Rightarrow x = 10$

R′(x)=−4x+40=0⇒x=10

3. Integral Calculus

Definition of Integration

Integration is the mathematical process of finding the accumulation of a quantity. It is the inverse operation of differentiation. The integral of a function represents the total accumulation over an interval, such as area under a curve or total distance traveled.

There are two main types of integrals:

• Indefinite Integral: Represents a family of functions and includes a constant of integration$C$C.
• Definite Integral: Evaluates the net accumulation over a specific interval$[a, b]$[a,b].

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2. Indefinite and Definite Integrals

a) Indefinite Integral

An indefinite integral finds a general function whose derivative is the given function. It is written as:

$\int f(x) \,dx = F(x) + C$

∫f(x)dx=F(x)+C

where

$C$

C is the constant of integration.

Example:

$\int 2x \,dx = x^2 + C$

∫2xdx=x2+C

b) Definite Integral

A definite integral calculates the total accumulation over an interval

$[a, b]$

[a,b] and is written as:

$\int_{a}^{b} f(x) \,dx = F(b) - F(a)$

∫ab​f(x)dx=F(b)−F(a)

where

$F(x)$

F(x) is the antiderivative of

$f(x)$

f(x).

Example:

$\int_{1}^{3} (x^2) \,dx = \left[ \frac{x^3}{3} \right]_1^3$

∫13​(x2)dx=[3x3​]13​

$= \frac{3^3}{3} - \frac{1^3}{3} = \frac{27}{3} - \frac{1}{3} = \frac{26}{3}$

=333​−313​=327​−31​=326​

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3. Techniques of Integration

a) Substitution Method

Used when an integral contains a composite function. We set

$u$

u as an inner function to simplify integration.

Example:

$\int x e^{x^2} \,dx$

∫xex2dx

Let

$u = x^2$

u=x2, so

$du = 2x \,dx$

du=2xdx.
Rewriting the integral:

$\frac{1}{2} \int e^u \,du = \frac{1}{2} e^u + C = \frac{1}{2} e^{x^2} + C$

21​∫eudu=21​eu+C=21​ex2+C

b) Integration by Parts

Used for integrating the product of two functions, based on:

$\int u \,dv = uv - \int v \,du$

∫udv=uv−∫vdu

Example:

$\int x \ln x \,dx$

∫xlnxdx

Let

$u = \ln x$

u=lnx, so

$du = \frac{1}{x}dx$

du=x1​dx, and let

$dv = xdx$

dv=xdx, so

$v = \frac{x^2}{2}$

v=2x2​.
Applying the formula:

$\int x \ln x \,dx = \frac{x^2}{2} \ln x - \int \frac{x^2}{2} \cdot \frac{1}{x} dx$

∫xlnxdx=2x2​lnx−∫2x2​⋅x1​dx

$= \frac{x^2}{2} \ln x - \int \frac{x}{2} dx$

=2x2​lnx−∫2x​dx

$= \frac{x^2}{2} \ln x - \frac{x^2}{4} + C$

=2x2​lnx−4x2​+C

4. Application of Integration

a) Area Under a Curve

The area between a curve

$y = f(x)$

y=f(x) and the x-axis from

$x = a$

x=a to

$x = b$

x=b is given by:

$A = \int_{a}^{b} f(x) dx$

A=∫ab​f(x)dx

b) Volume of Solids of Revolution

Using the disk method, the volume of a solid formed by rotating

$f(x)$

f(x) about the x-axis is:

$V = \pi \int_{a}^{b} [f(x)]^2 dx$

V=π∫ab​[f(x)]2dx

c) Work Done by a Force

The work done by a variable force

$F(x)$

F(x) moving an object from

$x = a$

x=a to

$x = b$

x=b is:

$W = \int_{a}^{b} F(x) dx$

W=∫ab​F(x)dx

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Multivariable Calculus (For Advanced Learners)

1. Partial Derivatives

When a function has multiple variables, its derivative with respect to one variable (while keeping others constant) is a partial derivative.

For

$f(x, y)$

f(x,y), the partial derivatives are:

$\frac{\partial f}{\partial x}, \quad \frac{\partial f}{\partial y}$

∂x∂f​,∂y∂f​

Example: If

$f(x, y) = x^2y + 3y^2$

f(x,y)=x2y+3y2, then:

$\frac{\partial f}{\partial x} = 2xy, \quad \frac{\partial f}{\partial y} = x^2 + 6y$

∂x∂f​=2xy,∂y∂f​=x2+6y

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2. Multiple Integrals

Extends definite integrals to functions of two or more variables.

a) Double Integrals

Used to compute areas and volumes:

$\int \int_{R} f(x, y) dA$

∫∫R​f(x,y)dA

b) Triple Integrals

Used for three-dimensional volume calculations:

$\int \int \int_{V} f(x, y, z) dV$

∫∫∫V​f(x,y,z)dV

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3. Vector Calculus

Deals with vector fields and their properties.

a) Gradient

For

$f(x, y, z)$

f(x,y,z), the gradient is:

$\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$

∇f=(∂x∂f​,∂y∂f​,∂z∂f​)

b) Divergence

Measures the rate at which a vector field spreads out:

$\nabla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}$

∇⋅F=∂x∂F1​​+∂y∂F2​​+∂z∂F3​​

c) Curl

Measures rotation of a vector field:

$\nabla \times \mathbf{F}$

∇×F