Equation of a tangent, with a slope of 'e'.

In summary, you are trying to solve for x using the derivative of y=lnx. You need to find the slope of the tangent at some point, and then use that slope to solve for x.
  • #1
Morphayne
13
0
Help my guys. I'm so hopelessly lost on this question...

Problem:

Determine the equation of the tangent to the graph of y=ln x that has slope e.

The answer in the book is y=ex-2.

Please help. :smile:
 
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  • #2
Where is your work? Show some and I will help.
 
  • #3
Well I just wrote y=xe as my answer but when I checked my answer it was wrong. I'm guessing that I need to do something using the derivative of y=ln x, but that is as far as know. I'm pretty weak with all this work involving Euler's constant.
 
  • #4
y=ex can't be the answer. The general equation of a line is y=mx+b. You are given that the slope is e. What else do you know about the graph of the function and the line tangent to it at some point.

If you think that you need to use the derivative of the function then try using it.

and the last hint:

What does ln(e) equal?
 
  • #5
I don't understand.:confused: Do I take the answer of y=ln e and plug it into y=ln x? If so how would I solve for x? Can someone please help me with a solution? I have an exam in 4 hours and there will be at least 3 questions like this worth(all together) 30 marks out of a possible 70.:frown: I'm in panic mode.:bugeye:

Please. I need this.:cry:
 
  • #6
This question has almost nothing to do with "e", it has to do with knowing how to find the equation of the line tangent to the graph. Unless you understand how to get the solution it won't help you when the question is phrased differently on the exam.

here is the last hint:
if the line is tangent to the curve then it shares the x,y coordinates as well as the slope.

you shouldn't need anymore than that.
 
  • #7
But I don't have any x,y coordinates. If I did, I wouldn't be stuck.
 
  • #8
Morphayne said:
But I don't have any x,y coordinates. If I did, I wouldn't be stuck.

You haven't figured out for what value of x that the slope of the tangent is e. That's why you don't have coordinates. Start by doing that. What's the slope of the tangent to the curve y=ln(x)? For what value of x is that equal to e?
 
  • #9
I don't know how to solve it:

y = ln x

-sub. y = e:

e = ln x

Now, here is where I'm stuck. I solve for x right? But how do I do that?
 
  • #10
y doesn't equal e.

You said initially that you need to use the derivative of your function, try using it.

Do you remember what the derivative of a function represents?
 
  • #11
So;

y = ln x

y' = 1/x

The derivative equation gives us the slope of any tangent, at any point along the given function. What do I do from here?

Should I sub. y' = e?

y' = 1/x

e = 1/x

xe = 1

x = 1/e ?
 
  • #12
why can't x = 1/e? (in fact it is)

suppose x = 1/e is correct, how do you get the y coordinate? and finally how do you get the equation of the line?
 
  • #13
To get the y-coordinate, I must sub. x = 1/e into y = ln x. Finally, to get the equation of the tangent I would use y = m(x-p)+q and sub. in the appropriate values. Right?

Did I get the part with the derivative right?
 
  • #14
yes and yes.
 

1. What is the equation of a tangent with a slope of 'e'?

The equation of a tangent with a slope of 'e' is y = ex + b, where b is the y-intercept.

2. How is the slope 'e' related to the equation of a tangent?

The slope 'e' is the coefficient of the x-term in the equation of a tangent. It represents the instantaneous rate of change at a specific point on the curve.

3. Can the equation of a tangent with a slope of 'e' be used to find the derivative of a function?

Yes, the equation of a tangent with a slope of 'e' can be used to find the derivative of a function. The derivative is equal to the slope of the tangent line at a specific point on the curve.

4. How is the equation of a tangent with a slope of 'e' different from a regular tangent line?

The equation of a tangent with a slope of 'e' is specifically used to find the derivative of a function at a specific point on the curve, whereas a regular tangent line is used to find the slope of a curve at a specific point.

5. Can the equation of a tangent with a slope of 'e' be used to find the limit of a function?

Yes, the equation of a tangent with a slope of 'e' can be used to find the limit of a function. The limit is equal to the derivative of the function at a specific point on the curve.

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