Exponential Fuctions (1 Viewer)

[kevda1st]

Im having trouble with exponential fuctions....Im finding it difficult understanding and finding limits and asymtopes..
Any tips?? or good sites??


Here are some questions im having trouble with
For Y=Xe^-X
i) Find any zeroes of the function and examine its sign
ii) Examine the function's behaviour as X approaches (infinity) and X approaches - (infinity)

For the function y=1-e^-x
ii) what is the equation of the asymptote of this curve??

 

[Timothy.Siu]

Im having trouble with exponential fuctions....Im finding it difficult understanding and finding limits and asymtopes..
Any tips?? or good sites??


Here are some questions im having trouble with
For Y=Xe^-X
i) Find any zeroes of the function and examine its sign
ii) Examine the function's behaviour as X approaches (infinity) and X approaches - (infinity)

For the function y=1-e^-x
ii) what is the equation of the asymptote of this curve??

i) zeroes? x=0
ii)x--> infinity y---> 0
x--->negative infinity y--->negative infinity

asymptote is y=1

 

[jet]

For these questions:
i)y=xe^(-x)
Let y=o. therefore x=0 or e^(-x)=0.
But, e^(-x) cannot equal zero.
Hence x=0 is the only zero of the function.
So, for x<0, y<0.
for x>0, y>0
x=0, y=0.

ii) As x approaches infinity e^(-x) approaches zero (this is just general knowledge of the graph of e^(+x)), hence y=0 is an asymptote.

As x approaches negative infinity, e^(-x) approaches infinity, therefore y approaches negative infinity.

iii)y=1-e^(-x)
Consider this the transformation of the curve y=e^(x) which has asymptote y=0.
The first step is to transform it to y=e^(-x), which is a rotation about the y-axis, so the asymptote is the same.

Next it is transformed to y=-e^(-x), which is a rotation about the y axis, maintaining the asymptote y=0.

Next we transform the function upwards by 1 unit, giving y=1-e^(-x), moving he asymptote upwards 1 unit to y=1.
EDIT: Trebla's method of this step is probably easier. mine is a more geometrical interpretation.

 

[Trebla]

Im having trouble with exponential fuctions....Im finding it difficult understanding and finding limits and asymtopes..
Any tips?? or good sites??


Here are some questions im having trouble with
For Y=Xe^-X
i) Find any zeroes of the function and examine its sign
ii) Examine the function's behaviour as X approaches (infinity) and X approaches - (infinity)

For the function y=1-e^-x
ii) what is the equation of the asymptote of this curve??

To find limits, simply sub in a number very close to the value x approaches and it gives you a good idea of how the exponential functions behave.


For y = xe^-x

i) Zeroes of the function are when y = 0, i.e. xe^-x = 0, which occurs at x = 0.

ii) As x --> ∞, e^-x --> 0 (i.e. e^-x gets very very small)
hence, xe^-x --> 0 (Even though x is very large, the multiplier of e^-x shrinks this very large number to virtually zero)
You can try subbing in x = 10000000 in the equation to see this...

As x --> - ∞, e^-x --> ∞ (i.e. e^-x gets very very large)
hence, xe^-x --> - ∞ (as x becomes a large negative number and e^-x becomes a large positive number so when you multiply the two, you get a large negative number)
You can try subbing in x = - 100000000 in the equation to see this...


For y = 1 - e^-x
i) Since e^-x can never be zero, an asymptote exists as y = 1.
Alternatively you can interpret it as x --> ∞, y --> 1
Again, try subbing in x = 100000000 in the equation if you can't see this...
Notice that as x --> - ∞, y --> - ∞ (check by subbing in a large negative number), so the curve approaches the asymptote only at very large positive values of x.