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Calculus MA180/MA185/MA190(Semester I)
- Limits and continuity (8 lectures),
- The derivative of a function (8 lectures),
- The anti-derivative of a function (8 lectures), and explains how these topics can be applied to problems such as
- existence of antipodal positions on the Earth's surface with equal air pressure,
- rates of change problems and maximization problems,
- prediction of how the world population of humans will grow.
- Lecturer: Prof Graham Ellis
- Lectures: Mon 1pm in IT250 and Tue 10am in the Anderson Theater.
- Tutorials: Workshops begin on Monday 17th September. Details can be found here.
- Recomended text: The MA180 calculus lectures are based on the textbook: "Calculus, early transcendentals " by James Stewart (Sixth Edition). Only so much of an explanation can be achieved in lectures, and this book can be used to reinforce (or maybe even clarify!) explanations given in lectures. It also contains many problems (some with fully worked solutions) on which you can practice. Even if you drop maths in second year, this will be a handy book for your scientific bookshelf. And if you continue with maths in second year then you'll be able to use the book again then.
- Problem sheet: available here.
- Module Website: Information and module documents will be posted to this site, which is linked from the Blackboard Calculus MA180/MA185/MA190(Semester I) pages. Blackboard will also be used for announcements and for posting grades.
MA180 and MA190 students:
- The end of semester exam is worth 60% of the total Semester I assessment. It will consist of three questions corresponding to the above three topics. A model exam paper is available here.
- The continuous assessment is worth 40% of the total Semester I assessment. It will consist of six online problem sheets which will be made available here. Submission deadlines are strict. There are about 10 questions per problem sheet and to score 100% on the Semester I CA component you need to submit 50 or more correct answers.
- The end of semester exam is worth 100% of the total score returned for the MA185 module. It will consist of three questions corresponding to the above three topics. A model exam paper is available here.
- The associated continuous assessment contributes towards 50% of the score returned for the MA187 Mathematical Skills module. The continuous assessment in Semester 2 contributes towards the remaining 50% of the MA187 assessment. The Semester I continuous assessment will consist of six online problem sheets which will be made available here. Submission deadlines are strict. There are about 10 questions per problem sheet and to score 100% on the Semester I CA component you need to submit 50 or more correct answers.
CLICKER OPINION POLLING may be used in some lectures.
WHAT IS MATHEMATICS?I'm not too sure of the answer. But whatever it is it is possibly something a bit larger than what was taught in your school mathematics classes. If you are interested in the question then you should browse this article by Fields Medallist William Thurston. He won the Fields Medal for his work in geometry. You could also take a look at the lovely little book A Mathematician's Apology by G.H. Hardy which is available online here.
WHAT ARE THE EMPLOYMENT PROSPECTS FOR A MATHS GRADUATE?Have a look at this link to answer this question.
STUDENT FEEDBACKI'll place student feedback here.
||We considered a stone being dropped from the top of the Eiffel Tower.
We assumed that the distance at time t is 4.9t2 (something
physicists tell us should be true). We used the formula y= 4.9t2
to begin a discussion of functions. We
then investigated the question: what is the speed of the stone at time t=2
The lecture touched on the notions of "continuous function", "limit" and "derivative". The remaining 23 lectures in Semester I will provide more details on these three fundamental notions.
For a more detailed summary of what calculus is all about, read the section "A Preview of Calculus" in Stewart, pages 2-9,
||We recalled that a function f:D--->C consists of a domain D, a codomain D and a rule for assigning precisely one element of the codomain to each element of the domain. When the domain and codomain are not explicitly specified then we just take D to be the largest subset of the reals for which the "function rule" makes sense, and we just take C to be the set of all real numbers. We recalled that functions can be represented by graphs and we studied some examples. During the examples we met concepts such as "horizonal asymptote", "vertical asymptote", "x-intercept", "y-intercept", "range of a function".
For more on the basics of functions read Stewart, Section 1.1.
Introduced the concept of a limit of a function f(x) as x tends to some number c. Gave some examples too.
For more details on the notion of a limit read Section 2.2 in Stewart.
For more details on the formal definition of a limit presented at the end of the lecture, read Section 2.4 in Stewart.
||Began with a recap of the definition of a limit of a function. Then stated and used a proposition which listed basic properties of limits. These properties are useful for making calculations.
The second half of the lecture was devoted to definitions and basic properties of the sine, cosine and tangent functions.
For more details and illustrative examples on the basic properties of limits read Section 2.3 in Stewart.
For more details on sines, cosines and tangents read Section 1.2 in Stewart.
||Began by stating the Sandwich Lemma (which can be rigorously derived from the formal definition of a limit). Illustrated how the Sandwich Lemma can be used to evaluate a limit.
Then introduced left-hand and right-hand limits. Stated a proposition which relates left-hand limits and right-hand limits to the usual notion of a limit at a point x=a. Gave some examples to illustrate these notions/relationships.
See Section 2.3 in Stewart for more information on the Sandwich Lemma (where it is called teh "Squeeze Theorem").
See Section 2.2 in Stewart for more details on left-hand and right-hand limits.
||Introduced the notion of a continuous function. The final example of a continuous function was f(x) = 1/x .
Then stated and illustrated the Intermediate Value Theorem.
For more details on the notion of continuity, and on the Intermediate Value Theorem, see Section 2.5 in Stewart.
Started by showing again the Intermediate Value Theorem from last lecture. Then used it to approximate the solutions to some polynomial equations. Also used it to `prove' that on any great circle on the Earth there exists a pair of opposite points with equal atmospheric pressure.
The fire alarm went off twenty minutes into the lecture and the lecture was aborted.
||Gave some examples of "limits at infinity". Then introduced the most important definition of this semester: the definition of the derivative of a function.
See Section 2.6 in Stewart for more details on limits at infinity.
See Section 2.8 for more details on the definition of the derivative of a function.
||Began with an explanation of why the derivative f'(x) of a function f(x) represents the slope to the tangent of the curve y=f(x) at the point x.
Then explained the rules for differentiating: (1) a sum of functions, (ii) a scalar product of a function, (iii) a product of two functions, (iv) a quotient of functions, (v) a composite of functions (the Chain Rule).
See Section 3.1 in Stewart for more details on the rules of differentiation and for lots more examples.
||Discusssed "rates of change applications" and solved two problems.
See Section 3.9 in Stewart for more examples of rates of change problems (called there "related rates" problems"). See also Sections 2.7 and 3.7 for examples of rates of change problems.
||Discussed max/min applications and solved one max/min problem about laying a cable from a lighthous to a point on the shore..
See Section 4.7 in Stewart for further examples of max/min problems.
||Used the derivative and the second derivative to help sketch the curve of a function. Talked about a curve being
"concave up" -- on the intervals where the acceleration is positive.
"concave down" -- on the intervals where the acceleration is negative
"points of inflection" -- a point where concavity changes
"critical points" -- points where the derivative is zero or not defined.
This lecture was given by John Burns.
See Section 4.3 in Stewart for more information on "concave up" and "concave down" functions.
||Summarized terminology on: continuity, differentiability, critical points, points of inflection. Worked through an example on determining conditions for a piecewise defined function to be differentiable. Ended up stating Rolle's Theorem. Gave an exercise which will be solved in class next time.
||Recalled Rolle's Theorem and used it to prove that a certain cubic equation has only one root. Then gave the generalization of Rolle's Theorem known as the Mean Value Theorem.
See Section 4.2 in Stewart for more details and examples on Rolle's Theorem and the Mean Value Theorem.
Started to talk about logarithms and exponents. Recalled the intuitive notion of a logarithm used in school, and "derived" two important properties of logarithms:
||Defined the natural logarithm Ln(x) (often written as loge(x) ) as the area under the curve y=1/t from t=1 to t=x for x≥1. For x between 0 and 1, defined Ln(x) to be the negative of the area under the curve y=1/t from t=1 to t=x. Then showed that Ln(x) has the properties required of a logarithm.
Ended up explaining how taking logarithms can simplify certain otherwise complicated calculations of derivatives.
See Appendix G (page A50) in Stewart for more details on the definition of a logarithm as the area under the curve y=1/t .
See Section 3.6 in Stewart for more examples of logarithmic differentiation.
||Explained that an injective function f:D-->R has an associated inverse function f-1:C-->D where C=f(D) is the range of f. Then gave a formula for the derivative of the inverse function. The function exp(x) or ex was introduced as the inverse of the natural logarithm function. The function f(x)=4x can be rigorously defined as the inverse to the function log4(x) = Ln(x)/Ln(4) .
Also defined the function y=sin-1(x) and then calculated its derivative.
See Section 3.5 in Stewart for more examples of derivatives of inverse functions.
||Introduced differential equations.
Solved the differential equation
See Section 9.1 of Stewart for more of an introduction to differential equations.