How do you sketch the graph #y=(2e^x)/(1+e^(2x))# using the first and second derivatives?
By writing the function as:
we can also see that the function is even and at the limits of the domain of definition we have:
Evaluate now the first and second derivatives:
graph{ (2e^x)/(1+e^(2x)) [3.59, 3.59, 1.794, 1.796]}
By signing up, you agree to our Terms of Service and Privacy Policy
To sketch the graph ( y = \frac{2e^x}{1 + e^{2x}} ) using the first and second derivatives:

Find the first derivative ( y' ): [ y' = \frac{d}{dx} \left( \frac{2e^x}{1 + e^{2x}} \right) ] [ y' = \frac{2e^x(1 + e^{2x})  2e^x(2xe^{2x})}{(1 + e^{2x})^2} ] [ y' = \frac{2e^x + 2e^{3x}  4xe^{3x}}{(1 + e^{2x})^2} ]

Find the critical points by setting ( y' = 0 ): [ 2e^x + 2e^{3x}  4xe^{3x} = 0 ]

Solve for ( x ) to find critical points.

Determine the intervals where ( y' > 0 ) and ( y' < 0 ) to identify increasing and decreasing segments.

Find the second derivative ( y'' ): [ y'' = \frac{d}{dx} \left( \frac{2e^x}{1 + e^{2x}} \right)' ]

Evaluate ( y'' ) at the critical points found earlier to determine concavity.

Plot the critical points, increasing/decreasing intervals, and concavity to sketch the graph.

Determine the behavior of the function as ( x ) approaches positive and negative infinity to complete the sketch.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided 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 onesided 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 onesided 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 onesided 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.
 How do you sketch the graph by determining all relative max and min, inflection points, finding intervals of increasing, decreasing and any asymptotes given #f(x)=xx^(2/3)(5/2x)#?
 How do you sketch the graph #y=x^4x^3x# using the first and second derivatives?
 For what values of x is #f(x)=2x^26x+4# concave or convex?
 How do you sketch the curve #y=x^2/(x^2+9)# by finding local maximum, minimum, inflection points, asymptotes, and intercepts?
 How do you find the exact relative maximum and minimum of the polynomial function of #p(x) =80+108xx^3 #?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7