Trinity College Dublin

Notes and handouts

Last updated: 13 May 2011

• 0. Course information and algebra review
  Arithmetic operations, powers, order of operations and equations. [I handed this out in the first lecture]
• 1. Functions, graphs and polynomials
  Functions, plotting graphs, slope and linear functions, direct proportion, the slope-intercept equation and the point-slope equation of a line, quadratic functions, other polynomials.
• 2. Calculus for polynomials
  Tangent lines, derivatives, antiderivatives, definite integrals, the fundamental theorem of calculus.
• 3. Interpretation and first applications of derivatives and integrals
  Average rates of change, instantaneous rates of change and derivatives, using antiderivatives to find populations from growth rates, accumulated changes and definite integrals, the average value of a function.
  [I handed out the first two pages of this in lecture 10]
• 4. Roots and fractional powers
  Square roots and nth roots, fractional powers, derivatives of powers and the extended power rule, antiderivatives and integrals of powers.
• 5. Trigonometric functions
  Angles and rotations, radian measure, the unit circle, graphs of cosine and sine, modelling periodic phenomena, derivatives and integrals.
• 6. Further differentiation and integration techniques
  The product rule, quotient rule and chain rule for differentiation. Integration by substitution and integration by parts.
• 7. Exponential and logarithmic functions
  Exponential growth and the exponential function, derivatives and integrals involving the exponential function. The natural logarithm, its properties and derivative. Applications to the uninhibited growth model, and radioactive decay.
• 8. Further applications of differentiation
  Solving maximum-minimum problems.

Errata

Here is a list of errors that have appeared in the notes at some point. Hopefully they're now corrected in the notes above.

Tutorial exercises

• Exercise sheet 1 - solutions
  simplifying expressions, the power laws, fractions, solving equations
• Exercise sheet 2 - solutions
  finding where a function is defined, equations of straight lines, direct proportion, quadratic functions and equations, graphs of functions
• Exercise sheet 3 - solutions
  differentiating polynomials, finding growth rates from a graph, antiderivatives and definite integrals of polynomials
• Exercise sheet 4 - solutions
  First applications of differentiation and integration to mathematical problems and population questions.
• Exercise sheet 5 - solutions
  Fractional powers: simplifying expressions, finding derivatives and integrals, and direct proportion.
• Exercise sheet 6 - solutions
  Converting between degrees and radians; finding and using a periodic model; derivatives and integrals of trigonometric functions.
• Exercise sheet 7 - solutions
  Finding derivatives using the product rule, the quotient rule and the chain rule. Finding antiderivatives using substitution and integration by parts.
• Exercise sheet 8 - solutions
  Simplifying, differentiating and integrating expressions with logarithms and the exponential function. The uninhibited population growth model, and radioactive decay.

New! Past exam solutions

Here are solutions to the calculus questions set in the summer exams in 2010 and in 2009.