A.M. ONLINE

## KNOWLEDGE CHECKS

##### Below, you will find a few examples of these quizzes.
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$\,$Knowledge Check 20

$\,$A quick check on your understanding of:

First order differential equations

Second order differential equations

Separation of variables

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A first order differential equation is any equation of the form $$\displaystyle\frac{dy}{dx}= f(x,y)$$.

1 / 10

The equation $$\displaystyle\frac{dy}{dx} = 7$$ is a first order differential equation.

A first order differential equation is any equation of the form $$\displaystyle\frac{dy}{dx}= f(x,y)$$.

2 / 10

The equation $$\displaystyle\frac{dy}{dx} = \sqrt{2}\sin 5x^3$$ is a first order differential equation.

$$\displaystyle\frac{d^2y}{dx^2}$$ is referred to as a second derivative.

3 / 10

If an equation has both a $$\displaystyle\frac{d^2y}{dx^2}$$ term and a $$\displaystyle\frac{dy}{dx}$$ term, it is considered

$$e^{a+b} = e^ae^b$$

4 / 10

Which of the following equations is equivalent to $$y(x) = e^{x+A}$$?

What is the value of $$y$$ when $$x=0$$?

5 / 10

For the function $$y(x)=5x^2+A$$, where $$A$$ is an unknown constant, which of the following statements correctly describes a graph of this function?

We are free to choose the side(s) to put our constant(s) on.

6 / 10

If $$\displaystyle\frac{dy}{dx}=\frac{10x}{3y}$$, which of the following is also true? (Select all that apply.)

We can use separation of variables on equations of the form $$\displaystyle\frac{dy}{dx} = f(x)g(y)$$.

7 / 10

If we wish to solve the following differential equation using separation of variables, what must be the value of $$a$$?

\begin{align}\displaystyle\frac{dy}{dx} = 5x+y^a\end{align}

We are given the acceleration rate $$a$$ in terms of $$t$$ and are asked to find the object's displacement $$s$$ at a time $$t$$.

8 / 10

"The acceleration rate of an object is described by $$a(t)=5t$$. Find an equation for the object's displacement at time $$t$$."

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Which of the following equations should we solve?

We are given the acceleration rate $$a$$ in terms of $$t$$ and are asked to find the object's speed $$v$$ at a time $$t$$.

9 / 10

"The acceleration rate of an object is described by $$a(t)=5t$$. Find an equation for the object's speed at time $$t$$."

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Which of the following equations should we solve?

We are given the acceleration rate $$a$$ in terms of $$s$$ and are asked to find the object's speed $$v$$ at a certain displacement $$s$$.

10 / 10

"The acceleration rate of an object is described by $$a(t)=5s$$. Find an equation for the object's speed when it has been displaced by an amount $$s$$ from its original location."

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Which of the following equations should we solve?

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/10

$\,$Knowledge Check 40

$\,$A quick check on your understanding of:

Arithmetic series

Geometric series

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$$S_n = \dfrac{n}{2}[2a+(n-1)d]$$

1 / 10

The general term for a linear sequence is shown below

\begin{align}T_n = n+3\end{align}

What is the sum of the first six terms of this sequence?

$$S_n = \dfrac{n}{2}[2a+(n-1)d]$$

2 / 10

The general term for a linear sequence is shown below

\begin{align}T_n = n+3\end{align}

What is the sum of the first twenty terms of this sequence?

$$S_n = \dfrac{n}{2}[2a+(n-1)d]$$

3 / 10

The general term for a linear sequence is shown below

\begin{align}T_n = 5n-2\end{align}

What is the sum of the first six terms of this sequence?

$$S_n = \dfrac{n}{2}[2a+(n-1)d]$$

4 / 10

The general term for a linear sequence is shown below

\begin{align}T_n = 5n-2\end{align}

What is the sum of the first twenty terms of this sequence?

This is an arithmetic (linear) sequence.

5 / 10

If the first difference between successive terms is $$0.25$$, that sequence will converge.

$$S_n = \dfrac{a(1-r^n)}{1-r}$$

6 / 10

The general term for a geometric sequence is shown below

\begin{align}T_n = 2(3^{n-1})\end{align}

What is the sum of the first six terms of this sequence?

The common ratio for this sequence is $$3$$.

7 / 10

The general term for a geometric sequence is shown below

\begin{align}T_n = 2(3^{n-1})\end{align}

What does the corresponding series converge to?

$$S_n = \dfrac{a(1-r^n)}{1-r}$$

8 / 10

The general term for a geometric sequence is shown below

\begin{align}T_n = 10(0.2^{n-1})\end{align}

What is the sum of the first four terms of this sequence?

The common ratio for this sequence is $$0.2$$.

9 / 10

The general term for a geometric sequence is shown below

\begin{align}T_n = 10(0.2^{n-1})\end{align}

What does the corresponding series converge to?

This sequence has a common ration of $$2$$.

10 / 10

Which of the following statements are true about the following sequence? (Select all the apply.)

\begin{align}1,2,4,8,16,...\end{align}

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/10

$\,$Knowledge Check 50

$\,$A quick check on your understanding of:

Routing problems

Allocation of resources

Equipment replacement and maintenance

Stock control

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The states are the number of items in stock.

1 / 10

In a stock control question, if a state has a $$0$$ label, what does that imply?

We cross out a state if there are no possible actions from that state.

2 / 10

In a stock control question, we cross out a state if:

Can we have surplus stock?

3 / 10

In a stock control question, the largest value that a state can have is the largest value in our order book.

The weights of each action is the total amount of money made at each event.

4 / 10

For a routing problem, we minimise the value as we want the travelling costs to be as low as possible.

What events are the most profitable each day?

5 / 10

A promoter is setting up a food tent at some of the following events.

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Only one event can be attended per day. The expected profits at each event are shown below.

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If travelling costs are negligible, what events should the promoter attend to maximise profits?

The optimal events to attend are $$A$$, $$C$$ and $$E$$.

6 / 10

A promoter is setting up a food tent at some of the following events.

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Only one event can be attended per day. The expected profits at each event are shown below.

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If travelling costs are negligible, what is the maximum amount of profit that the promoter can make at these events?

A sink state exists only if it is the destination for all actions of that stage.

7 / 10

Rebecca has purchased a new bicycle at a cost of €$$500$$.

Her bicycle will depreciate in value over time. She keeps any bicycle for a maximum for $$4$$ years.

The maintenance cost to keep the bicycle roadworthy each year depends on the age of the bicycle.

Any time Rebecca sells her bicycle, she will repurchase the same bicycle brand new.

After $$10$$ years, the corresponding network will have a sink node only if:

Rebecca is expected to make a loss overall.

8 / 10

Rebecca has purchased a new bicycle at a cost of €$$500$$.

Her bicycle will depreciate in value over time. She keeps any bicycle for a maximum for $$4$$ years.

The maintenance cost to keep the bicycle roadworthy each year depends on the age of the bicycle.

Any time Rebecca sells her bicycle, she will repurchase the same bicycle brand new.

After $$10$$ years, Rebecca sells her bicycle as she now prefers jogging.

If we use dynamic programming to determine when and how often Rebecca should sell her bicycle to minimise the amount of money she spends within these 10 years, which of the following should we do?

James always uses at least $$100$$ kilograms to make bread each week.

9 / 10

James harvests $$500$$ kilograms of wheat on his farm each week.

Rather than directly selling it as wheat, he uses all of it to instead produce three products (bread, pasta and crackers) which he then sells.

The amount of profit (in euro) that he makes per $$100$$ kilograms used for each product is as shown.

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James always uses at least $$100$$ kilograms to make bread each week.

Unfortunately, due to a drought, James instead only harvested $$200$$ kilograms of wheat this week. What should he make to maximise his profit?

What product do we place in the first stage of the Guided Example?

10 / 10

James harvests $$500$$ kilograms of wheat on his farm each week.

Rather than directly selling it as wheat, he uses all of it to instead produce three products (bread, pasta and crackers) which he then sells.

The amount of profit (in euro) that he makes per $$100$$ kilograms used for each product is as shown.

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James always uses at least $$100$$ kilograms to make bread each week.

If we wish to use dynamic programming to determine how James should use his supply of wheat to maximise his profit, which product should we use in the first stage?