WELCOME TO 1E1!
Calculus
Main Webpage:
http://www.maths.tcd.ie/~parnachev/ma1e01.html
Course 1E1
Engineering Mathematics 1
CALCULUS
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Dr Andrei Parnachev
Office 1.8 Hamilton building
Office hours: email for appointment
parnachev@maths.tcd.ie
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Class meetings:!
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Lectures: Monday, Tuesday, Thursday 11:00--11:50;
MacNeil Theater, Hamilton building!
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Tutorials:!
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A -- Wednesdays at 9am in M20 (Christopher Hobbs)!
B -- Wednesdays at 9am in M21 (Vanessa Koch)!
C -- Fridays at 9am in M20 (Christopher Hobbs)!
D -- Fridays at 9am in M21 (Vanessa Koch)!
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Functions: definition, domain and range, operations with functions, inverse function,
graphs, notions of rational, algebraic, and trigonometric functions; !
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Limits and Continuity: Two-sided, one-sided, and infinite limits, limit at infinity and
asymptotes; continuity, delta-epsilon language, intermediate-value and squeezing
theorems; !
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Differentiation of functions of one variable: the derivative function, techniques of
differentiation, implicit differentiation, related rates problems and the local linear
approximation; !
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Derivatives in graphing and applications: Analysis of functions, graphing polynomials
and rational functions, applied maximum and minimum problems and the NewtonRaphson method; !
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Integration: antiderivatives and introduction to integration, Riemann Sums, integration
by substitution and the Fundamental Theorem of Calculus; !
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Applications of the Definite Integral in Geometry: area between curves, volumes and
areas of solids of revolution and length of a plane curve. !
Grading: 20% homework, 80% final exam!
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HOMEWORK ONLY ACCEPTED ONLINE AT WILEY PLUS!
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YOU MUST REGISTER FOR THE COURSE AT WILEY PLUS!
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Book/Access Code package made specifically for this module !
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can be purchased at!
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Hodges and Figgis on Dawson Street !
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ISBN 9781118092484 !
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Calculus: Late Transcendentals 10th Edition Howard Anton, Irl Bivens, Stephen Davis !
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First homework will be announced later this week
http://edugen.wileyplus.com/edugen/class/cls416856/
Lecture 1: FUNCTIONS
Common
methods for representing functions are:
1. Numerically by tables
2. Geometrically by graphs
3. Algebraically by formulas
Examples:
b)
a)
y=x2
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INDEPENDENT AND DEPENDENT VARIABLES
For a given input x, the output of a function f is called the value of f at x or the image of x under f.
Sometimes we will want to denote the output by a single letter, say y, and write
y=f(x)
x is called the independent variable (or argument) of f
y is called the dependent variable of f
For now we will only consider functions in which the independent and dependent variables
are real numbers, in which case we say that f is a real-valued function of a real variable.
Example 2 The equation
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has the form in which the function f is given by the formula
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For each input x, the corresponding output y is obtained by substituting x in this formula. For example,
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GRAPHS OF FUNCTIONS
graph of f in the xy-plane is defined to be the graph of the equation y=f(x), i.e. plot points (x, f(x))
Note: No points with x<0
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THE ABSOLUTE VALUE FUNCTION
set of all allowable inputs (x-values) is called the domain of f
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set of outputs (y-values) that result when x varies over the domain is
called the range of f
Example:
V=(30-2x)(16-2x)x
1.1.5 DEFINITION. If a real-valued function of a real variable is defined by a formula, and if
no domain is stated explicitly, then it is to be understood that the domain consists of all real
numbers for which the formula yields a real value. This is called the natural domain of the
function.
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1.3 NEW FUNCTIONS FROM OLD
Example:
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COMPOSITION OF FUNCTIONS
Example:
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COMPOSITION OF FUNCTIONS
g o f, the composition of f and g
SYMMETRY