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Cambridge Prelim MX1 Textbook Marathon/Q&A (1 Viewer)

InteGrand

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Don't understand the graphing of y = log|x|

The answer have the typical log graph and then its reflection in the y axis. Why is that reflection included.

Also Logs can't be can be negative and positive, just not ZERO right?


 

Drongoski

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Don't understand the graphing of y = log|x|

The answer have the typical log graph and then its reflection in the y axis. Why is that reflection included.

Also Logs can't be can be negative and positive, just not ZERO right?
When you take |x|, negative x values become positive.

If f(x) = log |x| then

f(-x) = log |-x| = log |x| = f(x)

That means log|x| is an even function and its graph is symmetric about the y-axis.
 

Crisium

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Don't understand the graphing of y = log|x|

The answer have the typical log graph and then its reflection in the y axis. Why is that reflection included.

Also Logs can't be can be negative and positive, just not ZERO right?
ln(1) is zero

Something interesting to note with negative logs though, whenever you're solving an inequality such as

(x)ln(0.4) < 5

When you divide both sides by ln(0.4) remember to swap around the inequality because it is a negative number

I've rarely seen them do something like this but they often do it in 2 unit to bring out the better students
 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

I understand how the log base b b^x = x but am unsure about b^log base b x = x

Also could you help with this question:

Use the identity: a = e^log a to write each expression as a power of e. THus differnetiate it.

x2^x
 

InteGrand

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

I understand how the log base b b^x = x but am unsure about b^log base b x = x

Also could you help with this question:

Use the identity: a = e^log a to write each expression as a power of e. THus differnetiate it.

x2^x


 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

I have sketched y = log ( 1 + x^2)

Now I need to sketch y = log { (1 + x^2) / 2 }

I meant to use log law. I am not sure which one. I answer show it is a shift log 2 but am not sure why.
 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

I get the answer = 2x ( xlog2 + 1 )

But the answer in the book says = 2^x ( xlog2 + 1)

Not sure why it is to the power and not times 'x' . Is 2x meant to = e^ln2^x instead of what I used 2x = e^(ln2)x ??
 

InteGrand

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

I get the answer = 2x ( xlog2 + 1 )

But the answer in the book says = 2^x ( xlog2 + 1)

Not sure why it is to the power and not times 'x' . Is 2x meant to = e^ln2^x instead of what I used 2x = e^(ln2)x ??
 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Show that y = Ae^kx + C is a solution of dy/dx = k(y - C)

Do I just sub in y for LHS and RHS ???
 

rand_althor

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Show that y = Ae^kx + C is a solution of dy/dx = k(y - C)

Do I just sub in y for LHS and RHS ???
You differentiate y= Ae^{kx} + C to get dy/dx = k(y-C).
 
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rand_althor

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

So I can't assume that y=Ae^{kx}+C, and then use that assumption to show that it works? My thinking was, if y is equal to that, and we sub it in to dy/dx and then integrate, and end up with y=Ae^{kx}+C again, then it is a solution.
 

InteGrand

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

So I can't assume that y=Ae^{kx}+C, and then use that assumption to show that it works? My thinking was, if y is equal to that, and we sub it in to dy/dx and then integrate, and end up with y=Ae^{kx}+C again, then it is a solution.
 
Last edited:

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

What is the natural domain of y = log(log x)?
 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

why is that? Why is it not the usual x > 0 ??
 

VBN2470

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Note that , so taking the logarithm of this will give us an undefined output. We need to take values of such that which happens only when (since log is an increasing function). Sketch a graph of this function to verify what I have said.
 

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