MATH 1LS3 - Rough summary of topics covered

Chapter 1: Introduction to Models and Functions

• 1.1 Introduction

• 1.2 Models in Life Sciences; understanding and interpretation

• 1.3 Variables, Parameters and Functions; qualitative descriptions, functions and relations, domain

• 1.4 Working with Functions; composition, inverses, conversion of units, horizontal line test, trans-formation: scaling (stretch/compress), translation (shift)

Chapter 2: Modelling Using Elementary Functions

• 2.1 Elementary Functions; proportional relationships, verbal descriptions and models, dependentparameter changing in relation to independent

• 2.2 Exponentials and Logarithms; laws of exponents and logs, exponential growth and doublingtime, exponential decay and half-life, semilog and double-log graphs

• 2.3 Trig functions; A(t) sin(λ(t− t0)) +B, graphing, special angles (π/4, π/3, π/6 and multiples),inverse trig functions

Chapter 4: Limits, Continuity and Derivatives

• 4.1 Change; average rate of change, slope of a secant line, delta notation

• 4.2 Limit of a Function; definition and meaning, left and right limits, calculation

• 4.3 Infinite Limits, Limits at Infinity; definition and meaning, horizontal asymptotes, behaviour ofpolynomials and exponentials at infinity, comparisons between functions

• 4.4 Continuity; definition and meaning, basic continuous functions, looking for discontinuities

• 4.5 Derivatives and Differentiability; definition of derivative as a limit, interpretation as slope oftangent line, instantaneous rate of change, situations where not differentiable, graphs and inter-pretation, critical points, intervals of increase/decrease, approximating

Chapter 5: Working with Derivatives

• 5.1 Derivatives of Powers, Polynomials and Exponentials; sums and differences, constant multiplerule

• 5.2 Derivatives of Products and Quotients; Product Rule and Quotient Rule

• 5.3 Chain Rule and Derivative of Log; chain rule, derivative of log and ln, relative rate of change

• 5.4 Derivative of trig and inverse trig functions; derivatives of sin, cos, tan, sec, csc, cot, arcsin,arctan

• 5.5 Implicit Differentiation, Logarithmic Differentiation, Related Rates;

• 5.6 Second Derivative, Curvature, Convexity; Concave up and down, point of inflection, graphing,approximation

• 5.7 Approximating Functions with Polynomials; tangent line approx, secant line approx, quadraticapprox, Taylor polynomials

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Chapter 6: Applications of Derivatives - part 1

• 6.1 Extreme Values; local or relative extrema, critical points, Fermat’s theorem, first deriv test,second deriv test, absolute extrema, Extreme Value Theorem,

• 6.4 l’Hopital’s Rule; for 0/0 or ∞/∞ only, handling other indeterminate forms (0 ∗ ∞,∞ −∞, 00, 1∞) Leading behaviour is not on the exam.

Chapter 7: Integrals and Applications

• 7.1 Differential equations; pure time, autonomous, general solutions, initial conditions, Euler’smethod

• 7.2 The Antiderivative; indefinite integral, basic integrals (power, sum, constant multiple, polyno-mials, exponential, log, trig and inverse trig), + C, guess and check

• 7.3 Definite Interal and Area; Riemann sums, definition of the definite integral, continuous sums/continuouslimit, signed areas, properties of definite integral,

• 7.4 Definite and Indefinite Integrals; Fundamental Theorem of Calculus (part 1), finding net change(integral function and FTC part 2 are not on the exam)

• 7.5 Techniques of Integration; Substitution, Integration by Parts, approximation with Taylor poly-nomials; What are these about? Why would you use these techniques? What are the potentialproblems?

• 7.6 Applications of the Integral; finding areas, areas between two curves, average value of a function,volume of revolution

• 7.7 Improper Integrals; limits at +∞, −∞, or both, integrals with a discontinuity in integrand(e.g. jumps or vertical asymptotes) and such integration as limit from left or right, convergenceand divergence

Chapter 3: Discrete Time Dynamical Systems

• 3.1 Discrete time dynamical systems; updating functions, solutions, initial condition, exponentialsolutions, composition, inverse, units

• 3.2 Analyzing Discrete Time Dynamical Systems (DTDS); cobwebbing, equilibria (graphically andalgebraically)

• 3.3 Modelling with DTDS; understanding and interpretting processes, per capita production, non-linear updating functions

Chapter 6: Application of Derivative - part 2, Application to DTDS

• 6.7 Stability of Discrete Time Dynamical Systems; stability of equilibria, evidence by cobwebbing,connection to slope

• 6.8 More complicated DTDS; Stability Theorem, more complicated per capita rates

See Dr. Lovric’s Checklist of Math Concepts for the Exam, as well as the other Exam information onthe web site.

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