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Resource type Sequences and limits - Numbas
3 questions. One question on limits of standard sequences. Other two on finding least $N$ such that $|a_n-L |lt 10^{-r},;;n geq N$ where $L$ is limit of $(a_n)$. Numbas resources have been made available under a Creative Commons licence by Bill Foster and Christian Perfect, School of Mathematics & Statistics at Newcastle University.
Resource type Integration as a summation
The second major component of the Calculus is called integration. This may be introduced as a means of finding areas using summation and limits. We shall adopt this approach in the present Unit. In later units, we shall also see how integration may be related to differentiation.
Resource type Integration as a summation
The second major component of the Calculus is called integration. This may be introduced as a means of finding areas using summation and limits. We shall adopt this approach in the present Unit. In later units, we shall also see how integration may be related to differentiation. (Mathtutor Video Tutorial) This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.
Resource type Integration as a summation
The second major component of the Calculus is called integration. This may be introduced as a means of finding areas using summation and limits. We shall adopt this approach in the present Unit. In later units, we shall also see how integration may be related to differentiation. (Mathtutor Video Tutorial) The video is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.
Resource type Limits of sequences
In this unit, we recall what is meant by a simple sequence, and introduce infinite sequences. We explain what it means for two sequences to be the same, and what is meant by the n-th term of a sequence. We also investigate the behaviour of infinite sequences, and see that they might tend to plus or minus infinity, or to a real limit, or behave in some other way.
Resource type Integration by substitution
There are occasions when it is possible to perform an apparently difficult piece of integration by first making a substitution. This has the effect of changing the variable and the integrand. When dealing with definite integrals, the limits of integration can also change. In this unit we will meet several examples of integrals where it is appropriate to make a substitution. (Mathtutor Video Tutorial) This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.
Resource type Integration by substitution
There are occasions when it is possible to perform an apparently difficult piece of integration by first making a substitution. This has the effect of changing the variable and the integrand. When dealing with definite integrals, the limits of integration can also change. In this unit we will meet several examples of integrals where it is appropriate to make a substitution. (Mathtutor Video Tutorial) The video is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.
Resource type Limits of sequences
In this unit, we recall what is meant by a simple sequence, and introduce infinite sequences. We explain what it means for two sequences to be the same, and what is meant by the n-th term of a sequence. We also investigate the behaviour of infinite sequences, and see that they might tend to plus or minus infinity, or to a real limit, or behave in some other way. (Mathtutor Video Tutorial) This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.
Resource type Limits of sequences
In this unit, we recall what is meant by a simple sequence, and introduce infinite sequences. We explain what it means for two sequences to be the same, and what is meant by the n-th term of a sequence. We also investigate the behaviour of infinite sequences, and see that they might tend to plus or minus infinity, or to a real limit, or behave in some other way. (Mathtutor Video Tutorial) The video is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.
Resource type Limits of functions
In this unit, we explain what it means for a function to tend to infinity, to minus infinity, or to a real limit, as x tends to infinity or to minus infinity. We also explain what it means for a function to tend to a real limit as x tends to a given real number. In each case, we give an example of a function that does not tend to a limit at all. (Mathtutor Video Tutorial) This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.
Resource type Limits of functions
In this unit, we explain what it means for a function to tend to infinity, to minus infinity, or to a real limit, as x tends to infinity or to minus infinity. We also explain what it means for a function to tend to a real limit as x tends to a given real number. In each case, we give an example of a function that does not tend to a limit at all. (Mathtutor Video Tutorial) The video is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.
Resource type Teaching of Chemical Thermodynamics using Available Data and an Innovative Approach
Analysis is made showing how Helmholtz and Gibbs energies conveniently interrelate enabling typical 2-D and 3-D curves to be drawn across a range of temperature for selected chemical equilibria. Opposing influences leading to a free energy minimum or an entropy maximum are given a physical explanation with the attainment of equilibrium and the choice of conditions made evident. Simplifying assumptions are emphasised and the examples show how the data are manipulated, limits evaluated and trends in equilibrium summarised by EXCEL charts.
Resource type Limits of functions
In this unit, we explain what it means for a function to tend to infinity, to minus infinity, or to a real limit, as x tends to infinity or to minus infinity. We also explain what it means for a function to tend to a real limit as x tends to a given real number. In each case, we give an example of a function that does not tend to a limit at all.