# Calculus I : Pokhara University Notes & Solutions

Calculus I Notes & Solutions

## Subject Contents

## Overview

## Syllabus

Unit I: Limit Continuity and Derivatives (5 hrs)

1.1 Introduction

1.2 Limit, continuity and differentiability

1.3 Higher order derivatives by Leibnitz method.

Unit II: Applications of Derivatives (8 hrs)

2.1 Mean value theorems: Rolle’s theorem, Lagrange’s

Theorem (Geometrical interpretation and verification) and

applications

2.2 Higher order mean value theorem: Taylor’s Series,

Maclaurin’s Series expansion of function.

2.3 Asymptotes to Cartesian curves up to four degrees.

2.4 Curve tracing in Cartesian form and parametric form

2.5 Curvature

Unit III: Integral Calculus (6 hrs)

3.1 Introduction

3.2 Review on Indefinite Integral and fundamental theorem of integral calculus.

3.3 Definite integral and its properties

3.4 Improper Integrals; comparison test.

3.5 Reduction formula, Beta Gamma function

Unit IV: Application of Integral (6 hrs)

4.1 Application of integrals for finding area beneath a curve

and between two curves and arc length

4.2 Surface and volume of solid of revolution in the plane for

Cartesian and parametric curves.

Unit V: Partial Differentiation (3 hrs)

5.1 Introduction

5.2 Partial Derivatives

5.3 Homogeneous function and Euler’s theorem for the

function of two and three variables

5.4 Total Derivatives and Differentiation of Implicit

functions.

Unit VI: Application of Partial Differentiation (4 hrs)

6.1 Extrema of functions of two and three variables.

6.2 Lagrange’s method of undetermined Multipliers (up to 2

multipliers)

Unit VII: First Order Ordinary Differential Equations

(6 hrs)

7.1 Review of separable, homogeneous and exact differential

equation with engineering applications

7.2 Linear, Bernoulli equation and Riccati’s equation with

engineering application.

7.3 Mathematical modeling of engineering problems using

first order equation.

Unit VIII: Second Order Ordinary Differential Equations

(7 hrs)

8.1 Second order Homogeneous ODE with constant and

variable coefficients, Euler-Cauchy equation.

8.2 Existence and uniqueness of solutions, Wronskian and

general solutions for solving ODE.

8.3 Non-homogeneous second order ODE and Solution by

undetermined coefficients and variation of parameters and

engineering application

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