MATH10250 Introduction to Calculus for Engineers

Academic Year 2022/2023

This is a mathematics module designed for engineering students. It provides an introduction to differential and integral calculus of functions of one variable, and to differential equations. Course Outline:1. Review: Functions. .2. Limits: Notion of a limit, statements of basic limit theorems.3. Differentiation: Notion of derivative, product and quotient rules, derivatives of polynomial functions, review of trigonometry, derivatives of trigonometric functions and their inverses, chain rule, implicit and logarithmic differentiation, higher order derivatives. 4. Hyperbolic functions and their inverses 5. Applications of differentiation: maxima and minima, second order derivative test. 6.Indefinite and definite integrals, the fundamental theorem of calculus, integration by parts, integration by substitution and the method of partial fractions. 7. Applications of integration: area under the curve, volume of a solid, length of a graph, surface area. 8. Series. Geometric series, Ratio Test, Harmonic series. Power series. MacLaurin and Taylor series of a function of one variable. 9. Differential equations: first order linear equations with constant coefficients.

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Curricular information is subject to change

Learning Outcomes:

On completion of this module students should be able to:1. Calculate limits.2. Calculate derivatives via implicit and logarithmic differentiation. 3. Display knowledge of the properties of polynomial and rational functions, trigonometric functions and their inverses, exponential, logarithmic and hyperbolic functions. 4. Use calculus to find local extrema of a function of one variable and apply these methods to optimisation problems. 5. Calculate definite and indefinite integrals. 6. Determine the Interval the Convergence of a power series. Obtain the expansion for MacLaurin and Taylor series of a function of a single variable. 6. Solve first order linear differential equations .

Indicative Module Content:

Functions and Limit
Definition of a function; bijective and inverse functions
Geometric criteria for bijective and inverse functions
Limit of a function. Computing the limit by multiplying with the rational conjugate
Continuous functions

Differentiation
Definition. Mechanical and geometrical interpretation of derivati
Basic rules of differentiation (sum, difference, product, quotient, chain rule)
Logarithmic differentiation
Implicit differentiation; the eq. of the tangent line to a curve
Trigonometric functions $\sin x$ and $\cos x$
]The inverse trig. functions $\sin^{-1} x$ and $\cos^{-1} x$ and their derivatives


Hyperbolic functions
The hyperbolic functions
Inverse hyperbolic functions their graphs and derivatives
Various identities for inverse hyperbolic functions

Optimisation
Critical points of functions of one variable
Maximum, minimum and point of inflection
Convexity and concavity
The Second Derivative Test for functions of one variable.

Integral Calculus-Methods of integration
Integration by parts, LIATE rule
Integration by substitution
Method of partial fractions
mproper integrals

Numerical Approximation of integrals
Midpoint rule, Trapezoidal rule and Simpson's rule

Applications of Integral Calculus} (Formulae provided on question sheet)
Total area
Area of the region between two graphs
Volume of a solid of rotation
Length of a Graph
Area of a Surface:

Differential equations
First order differential equations with separable variables
Applications from Physics and Chemistry
Linear first order differential equations . General solution using Integrating factor

Sequences and series
Geometric series
Harmonic series test
Ratio test
Alternating series test and Comparison series test
Power series
Interval of convergence, radius of convergence for a power series
Taylor series
Maclaurin series

Applications of Differential Equations
Applications of Differential Equations using Separation of Variables
Applications of Differential Equations using Integrating Factor

Student Effort Hours: 
Student Effort Type Hours
Lectures

30

Tutorial

12

Autonomous Student Learning

52

Total

94

Approaches to Teaching and Learning:
Lectures, tutorials, enquiry and problem-based learning.. 
Requirements, Exclusions and Recommendations
Learning Recommendations:

At least H4 in the Leaving Certificate Exam (or equivalent)


Module Requisites and Incompatibles
Incompatibles:
ACM00010 - Intro to Mechanics, ACM00020 - Applied Maths: Methods & Appli, ACM10010 - Mathematical ModellingI, ACM10020 - Mathematical Modelling II, ACM10030 - Mechanics & Special Relativity, ACM10070 - Math Modelling in the Sciences, ACM10080 - Intro to Applied & Comp Math, ECON10030 - Intro Quantitative Economics, MATH00010 - Introduction to Mathematics, MATH10030 - Maths for Business, MATH10050 - Linear Algebra & Geom, MATH10060 - Diff & Integral Calc, MATH10070 - Introduction to Calculus, MATH10080 - Calculus & Statistics, MATH10120 - Linear Algebra Apps to Econ, MATH10140 - Advanced Calculus (E&F), MATH10230 - Mathematics for Agriculture I , MATH10240 - Mathematics for Agriculture II, MATH10280 - Linear Algebra in the Phys.Sci, MATH10290 - Linear Algebra for Science, MATH10300 - Calculus in the Math. Sciences, MATH10310 - Calculus for Science, MATH10330 - Calculus in the Phy Sciences, MATH10420 - Intro Calculus Engineers(NUin), MATH20130 - Fund. Actuarial Mathematics I, MST00050 - Mathematics: An introduction, MST10010 - Calculus I, MST10020 - Calculus II, MST10030 - Linear Algebra I, STAT10010 - Research Methods, STAT10050 - Practical Statistics, STAT10060 - Statistical Modelling


 
Assessment Strategy  
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Continuous Assessment: Web Work Throughout the Trimester n/a Alternative linear conversion grade scale 40% No

10

Examination: Midterm Examination Unspecified No Alternative linear conversion grade scale 40% No

17

Class Test: Proficiency tests Week 1 n/a Alternative linear conversion grade scale 40% No

3

Examination: Final Examination 2 hour End of Trimester Exam No Alternative linear conversion grade scale 40% No

70


Carry forward of passed components
No
 
Resit In Terminal Exam
Spring Yes - 2 Hour
Please see Student Jargon Buster for more information about remediation types and timing. 
Feedback Strategy/Strategies

• Feedback individually to students, post-assessment
• Online automated feedback

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Name Role
Assoc Professor Anthony Cronin Lecturer / Co-Lecturer
Niyati Seth Tutor