Concrete Solutions
About points...
We associate a certain number of points with each exercise.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
About difficulty...
We associate a certain difficulty with each exercise.
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
Question
Solution
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\(\LaTeX\)
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Exercise:
Find the concrete solutions of two-dimensional linear systems for the given eigenvalues eigenvectors and initial conditions. abcliste abc lambda_ - lambda_ - bf v_ leftmatrix matrixright bf v_ leftmatrix matrixright leftmatrixx_ y_ matrixright leftmatrix - matrixright abc lambda_ lambda_ - bf v_ leftmatrix matrixright bf v_ leftmatrix matrixright leftmatrixx_ y_ matrixright leftmatrix matrixright abc lambda_ i lambda_ -i bf v_ leftmatrix+i matrixright bf v_ leftmatrix-i matrixright leftmatrixx_ y_ matrixright leftmatrix matrixright abcliste
Solution:
The solutions can be written as leftmatrixx y matrixright a_ bf v_ e^lambda_ t+a_ bf v_ e^lambda t The coefficients a_ and a_ are given by the initial conditions: leftmatrixx_ y_ matrixright a_ bf v_ +a_ bf v_ abcliste abc a_+a_ - a_ Longrightarrow a_ a_ - The solution is xt result e^-t - e^- t yt result- e^- t abc a_+a_ a_+ a_ Longrightarrow a_ frac a_ -frac The solution is xt resultfrac e^t-frac e^-t yt resultfrac e^t-frac e^-t abc +ia_+-ia_ a_+ a_ Longrightarrow a_ -a_+frac Longrightarrow a_ +i+-ifrac-a_ i a_+frac-i Longrightarrow a_ frac+ifracifrac-i a_ frac+i The solution is xt frac-i+i e^iomega t + frac+i-ie^-iomega t fraclefte^iomega t+e^-iomega tright+fraclefte^iomega t+e^-iomega tright & quad -fracilefte^iomega t-e^-iomega tright+fracilefte^iomega t-e^-omega tright frac cosomega t-fraci i sinomega t resultcosomega t+sinomega t yt frac-i e^iomega t+frac+i e^-iomega t fraclefte^iomega t+e^-iomega tright & quad -fracilefte^iomega t-e^-iomega tright frac cosomega t-fraci i sinomega t resultcosomega t+sinomega t abcliste
Find the concrete solutions of two-dimensional linear systems for the given eigenvalues eigenvectors and initial conditions. abcliste abc lambda_ - lambda_ - bf v_ leftmatrix matrixright bf v_ leftmatrix matrixright leftmatrixx_ y_ matrixright leftmatrix - matrixright abc lambda_ lambda_ - bf v_ leftmatrix matrixright bf v_ leftmatrix matrixright leftmatrixx_ y_ matrixright leftmatrix matrixright abc lambda_ i lambda_ -i bf v_ leftmatrix+i matrixright bf v_ leftmatrix-i matrixright leftmatrixx_ y_ matrixright leftmatrix matrixright abcliste
Solution:
The solutions can be written as leftmatrixx y matrixright a_ bf v_ e^lambda_ t+a_ bf v_ e^lambda t The coefficients a_ and a_ are given by the initial conditions: leftmatrixx_ y_ matrixright a_ bf v_ +a_ bf v_ abcliste abc a_+a_ - a_ Longrightarrow a_ a_ - The solution is xt result e^-t - e^- t yt result- e^- t abc a_+a_ a_+ a_ Longrightarrow a_ frac a_ -frac The solution is xt resultfrac e^t-frac e^-t yt resultfrac e^t-frac e^-t abc +ia_+-ia_ a_+ a_ Longrightarrow a_ -a_+frac Longrightarrow a_ +i+-ifrac-a_ i a_+frac-i Longrightarrow a_ frac+ifracifrac-i a_ frac+i The solution is xt frac-i+i e^iomega t + frac+i-ie^-iomega t fraclefte^iomega t+e^-iomega tright+fraclefte^iomega t+e^-iomega tright & quad -fracilefte^iomega t-e^-iomega tright+fracilefte^iomega t-e^-omega tright frac cosomega t-fraci i sinomega t resultcosomega t+sinomega t yt frac-i e^iomega t+frac+i e^-iomega t fraclefte^iomega t+e^-iomega tright & quad -fracilefte^iomega t-e^-iomega tright frac cosomega t-fraci i sinomega t resultcosomega t+sinomega t abcliste
Meta Information
Exercise:
Find the concrete solutions of two-dimensional linear systems for the given eigenvalues eigenvectors and initial conditions. abcliste abc lambda_ - lambda_ - bf v_ leftmatrix matrixright bf v_ leftmatrix matrixright leftmatrixx_ y_ matrixright leftmatrix - matrixright abc lambda_ lambda_ - bf v_ leftmatrix matrixright bf v_ leftmatrix matrixright leftmatrixx_ y_ matrixright leftmatrix matrixright abc lambda_ i lambda_ -i bf v_ leftmatrix+i matrixright bf v_ leftmatrix-i matrixright leftmatrixx_ y_ matrixright leftmatrix matrixright abcliste
Solution:
The solutions can be written as leftmatrixx y matrixright a_ bf v_ e^lambda_ t+a_ bf v_ e^lambda t The coefficients a_ and a_ are given by the initial conditions: leftmatrixx_ y_ matrixright a_ bf v_ +a_ bf v_ abcliste abc a_+a_ - a_ Longrightarrow a_ a_ - The solution is xt result e^-t - e^- t yt result- e^- t abc a_+a_ a_+ a_ Longrightarrow a_ frac a_ -frac The solution is xt resultfrac e^t-frac e^-t yt resultfrac e^t-frac e^-t abc +ia_+-ia_ a_+ a_ Longrightarrow a_ -a_+frac Longrightarrow a_ +i+-ifrac-a_ i a_+frac-i Longrightarrow a_ frac+ifracifrac-i a_ frac+i The solution is xt frac-i+i e^iomega t + frac+i-ie^-iomega t fraclefte^iomega t+e^-iomega tright+fraclefte^iomega t+e^-iomega tright & quad -fracilefte^iomega t-e^-iomega tright+fracilefte^iomega t-e^-omega tright frac cosomega t-fraci i sinomega t resultcosomega t+sinomega t yt frac-i e^iomega t+frac+i e^-iomega t fraclefte^iomega t+e^-iomega tright & quad -fracilefte^iomega t-e^-iomega tright frac cosomega t-fraci i sinomega t resultcosomega t+sinomega t abcliste
Find the concrete solutions of two-dimensional linear systems for the given eigenvalues eigenvectors and initial conditions. abcliste abc lambda_ - lambda_ - bf v_ leftmatrix matrixright bf v_ leftmatrix matrixright leftmatrixx_ y_ matrixright leftmatrix - matrixright abc lambda_ lambda_ - bf v_ leftmatrix matrixright bf v_ leftmatrix matrixright leftmatrixx_ y_ matrixright leftmatrix matrixright abc lambda_ i lambda_ -i bf v_ leftmatrix+i matrixright bf v_ leftmatrix-i matrixright leftmatrixx_ y_ matrixright leftmatrix matrixright abcliste
Solution:
The solutions can be written as leftmatrixx y matrixright a_ bf v_ e^lambda_ t+a_ bf v_ e^lambda t The coefficients a_ and a_ are given by the initial conditions: leftmatrixx_ y_ matrixright a_ bf v_ +a_ bf v_ abcliste abc a_+a_ - a_ Longrightarrow a_ a_ - The solution is xt result e^-t - e^- t yt result- e^- t abc a_+a_ a_+ a_ Longrightarrow a_ frac a_ -frac The solution is xt resultfrac e^t-frac e^-t yt resultfrac e^t-frac e^-t abc +ia_+-ia_ a_+ a_ Longrightarrow a_ -a_+frac Longrightarrow a_ +i+-ifrac-a_ i a_+frac-i Longrightarrow a_ frac+ifracifrac-i a_ frac+i The solution is xt frac-i+i e^iomega t + frac+i-ie^-iomega t fraclefte^iomega t+e^-iomega tright+fraclefte^iomega t+e^-iomega tright & quad -fracilefte^iomega t-e^-iomega tright+fracilefte^iomega t-e^-omega tright frac cosomega t-fraci i sinomega t resultcosomega t+sinomega t yt frac-i e^iomega t+frac+i e^-iomega t fraclefte^iomega t+e^-iomega tright & quad -fracilefte^iomega t-e^-iomega tright frac cosomega t-fraci i sinomega t resultcosomega t+sinomega t abcliste
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