Use the tangent line to estimate the gradient of ##y=2^x## when ##x=1##

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  • #1
chwala
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Homework Statement:
Draw the tangent to the curve at the point where ##x=1##. Use this tangent to claculate an estimate to the gradient of ##y=2^x## when ##x=1##
Relevant Equations:
gradient
Ok this is a question that i am currently marking...the sketch is here;

1669462338489.png


In my mark scheme i have points ##(1,2)## and ##(3,5)## which can be easily picked from the graph to realize an estimate of ##m=1.5## where ##m## is the gradient ...of course i have given a range i.e ##1.6≥m≥1.2##

Now to my question. hmmmmm :wink:

A student picked the points ##(1,2)## and ##(0.9,1.8)## getting ##m=2## ...the difference from actual is quite big...but the points are picked from their straight line...am i missing something here...

Actual gradient using differentiation would be given by;

##\dfrac{dy}{dx}= 2^x\ln 2##

##\dfrac{dy}{dx}[x=1]= 2^1\ln 2=1.386##

Your insight welcome.
 
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Answers and Replies

  • #2
pasmith
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Is the tangent line drawn by the student, or is it printed on the paper?

In any case, I think [itex](0.9, 1.8)[/itex] is visibly below the tangent line in the picture.
 
  • #3
chwala
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Is the tangent line drawn by the student, or is it printed on the paper?

In any case, I think [itex](0.9, 1.8)[/itex] is visibly below the tangent line in the picture.
Drawn by the student...we would not expect the students to draw this accurately 100%. How do i deal with this? should i increase range of expected values? my thinking is; i cannot penalise the student for having picking those points from their tangent line...and the calculation as per their picked points is correct.
 
  • #4
pasmith
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I dont think (0.9, 1.8) is on their tangent line: it's visibly below it. That should result in some loss of marks. If they'd gone with (0.9, 1.85) then they might have a better case.
 
  • #5
chwala
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I just checked the accurate graph on desmos...points ##(0.9,1.85)## would have been fine...
 
  • #6
chwala
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I dont think (0.9, 1.8) is on their tangent line: it's visibly below it. That should result in some loss of marks. If they'd gone with (0.9, 1.85) then they might have a better case.
Thanks...i agree...i will emphasis the need to try and pick obvious points on the graphs to mitigate this. Cheers great weekend mate!
 
  • #7
BvU
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An additional comment: as a physicist, I would emphasize the large error introduced by picking such a small ##\Delta x##: any error in ##\Delta y## as read off from the graph is multiplied by 10 !

##\ ##
 
  • #8
WWGD
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Notice in general, unless your function is itself linear to start with, the approximation is only local. But if youre just given the point (1,2), there are infinitely-many lines that go through it. The point (0.9, 1.8) defines just one of infinitely-many such lines. Not sure if this addresses your question.
 
  • #9
chwala
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Notice in general, unless your function is itself linear to start with, the approximation is only local. But if youre just given the point (1,2), there are infinitely-many lines that go through it. The point (0.9, 1.8) defines just one of infinitely-many such lines. Not sure if this addresses your question.
Not sure ...we need points that are on the tangent line and as discussed above, it's clear that ##(0.9,1.8)## is not a point on the tangent line to the given curve ##y=2^x##.
 
  • #10
WWGD
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Not sure ...we need points that are on the tangent line and as discussed above, it's clear that ##(0.9,1.8)## is not a point on the tangent line to the given curve ##y=2^x##.
Do you mean at the point (1,2)?
 
  • #12
WWGD
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Yes.
Well, the points (1,2) and (0.9, 1.8) defines the line y2=2x, which contrasts with the line
y1-1=1.386(x-2).
Maybe a bound/estimate for |y1-y2| would help explain why the choice of (0.9, 1.8) was not as good. Additionally , comparing it with the choice of {(1,2),(3,5)} which would define the line y3-2=2(x-1).

Maybe we can find a generic value for
|y-y1|, for y=mx+b.
Hope I'm not too far of.
 

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