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 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.
- Is #f(x)=-x^3-x^2+x# concave or convex at #x=4#?
- What is the second derivative of #f(x)= sec^2x#?
- Using the second derivative test, how do you find the local maximum and local minimum for #f(x) = x^(3) - 6x^(2) + 5#?
- For what values of x is #f(x)= -x^4-9x^3+2x+4 # concave or convex?
- Is #f(x)=x^3-x+e^(x-x^2) # concave or convex at #x=1 #?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7