How do you use the first and second derivatives to sketch #y = e^(1-2x)#?

Answer 1

See argument below.

#y=e^(1-2x)#
#dy/dx = d/dx e^(1-2x)#

Apply chain rule

#dy/dx = e^(1-2x) * d/dx (1-2x)#

Apply power rule

#dy/dx=-2 e^(1-2x) #
Similarly, #(d^2y)/dx^2 = +4e^(1-2x)#
For extrema points of #y, dy/dx =0#
However, in this case, #-2 e^(1-2x) <0 forall x in RR#
Consider, #lim_(x->+oo) -2 e^(1-2x) =0 and lim_(x->+oo) e^(1-2x) =0#
Also consider, #(d^2y)/dx^2 =+4e^(1-2x) >0 forall x in RR#
Hence, it seems reasonable to deduce that #y# tends to its minimum value of #0# as #x# tends to #+oo#
This helps us visualise the graph of #y# below.
The other important point we need for the graph is at #x=0#
Here, #y = e^(1-0) = e#
We see the point #(0,e)# on the graph below.

graph{e^(1-2x) [-5.69, 5.406, -1.722, 3.827]}

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Answer 2

To sketch ( y = e^{1-2x} ) using the first and second derivatives:

  1. Find the first derivative: ( \frac{dy}{dx} = -2e^{1-2x} ).
  2. Find critical points by setting the first derivative equal to zero and solving for ( x ).
  3. Find the second derivative: ( \frac{d^2y}{dx^2} = 4e^{1-2x} ).
  4. Determine the concavity of the function by analyzing the sign of the second derivative.
  5. Sketch the curve based on the behavior of the first and second derivatives, including critical points and concavity.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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