sketch the graph of the following. Show all your work and include all the important points and asymptotes.

also find y' and y'' ?

y= (x^3)/(x+1)^2

should be no problem. That's just Algebra II. Anyway,

y = x^3/(x+1)^2
y' = x^2(x+3)/(x+1)^3
y" = 6x/(x+1)^4

see the graph and some analysis here:

http://www.wolframalpha.com/input/?i=x^3%2F%28x%2B1%29^2

To sketch the graph of the function y = (x^3)/(x+1)^2, we will follow these steps:

Step 1: Find the vertical asymptote(s):

The vertical asymptotes occur when the denominator of the function becomes zero. In this case, when x + 1 = 0. So, the vertical asymptote is x = -1.

Step 2: Find the x-intercept(s):

The x-intercepts occur when the numerator of the function becomes zero. In this case, when x^3 = 0. So, the only x-intercept is at x = 0.

Step 3: Find the y-intercept:

The y-intercept occurs when x = 0. Plugging this value into the function, we get y = (0^3)/(0+1)^2 = 0.

Step 4: Determine the behavior as x approaches positive and negative infinity:

For large negative values of x, the function approaches 0 because the x^3 term in the numerator becomes negligible compared to the (x+1)^2 denominator. Therefore, the graph approaches the x-axis as x approaches negative infinity.

For large positive values of x, the function approaches positive infinity because both the numerator and denominator grow exponentially, but the numerator grows faster. Therefore, the graph approaches positive infinity as x approaches positive infinity.

Step 5: Determine the concavity and find any inflection points:

To find the concavity, we need to find the second derivative of the function. Let's find the first derivative and second derivative of y.

Given: y = (x^3)/(x+1)^2

To find y', we use the quotient rule:

y' = [(3x^2)(x+1)^2 - (x^3)(2(x+1)(1))]/[(x+1)^4]

Simplifying y', we get:

y' = [3x^2(x+1)^2 - 2x^3(x+1)] / (x+1)^4

Expanding and simplifying further:

y' = [3x^4 + 6x^3 + 3x^2 - 2x^4 - 2x^3] / (x+1)^4

y' = [x^4 + 4x^3 + 3x^2] / (x+1)^4

To find y", we differentiate y' with respect to x:

y" = [(4x^3 + 12x^2 + 6x)(x+1)^4 - (x^4 + 4x^3 + 3x^2)(4(x+1)^3)] / (x+1)^8

Simplifying further:

y" = [(4x^3 + 12x^2 + 6x)(x+1)^4 - (4x^4 + 12x^3 + 12x^2 + 4x^3 + 12x^2 + 9x)(x+1)^3] / (x+1)^8

y" = [(4x^7 + 12x^6 + 6x^5 + 4x^6 + 12x^5 + 6x^4 + 4x^4 + 12x^3 + 6x^2) - (4x^7 + 4x^6 + 12x^5 + 4x^6 + 12x^5 + 9x^4 + 4x^6 + 12x^5 + 9x^4 + 12x^3 + 3x^2)] / (x+1)^8

Simplifying y", we get:

y" = [ -3x^4 - 3x^2 - 12x^3 - 3x^2] / (x+1)^5

y" = [-3x^4 - 12x^3 - 6x^2] / (x+1)^5

Now that we have the first and second derivatives, we can proceed to sketch the graph.

Step 6: Sketching the graph:

- Vertical Asymptote: x = -1
- x-intercept: x = 0, y = 0
- Concavity and inflection points: To find the concavity and inflection points, we can analyze the sign of y". We see that y" is negative for x values less than -1, indicating a concave down section. Similarly, y" is positive for x values greater than -1, indicating a concave up section. Therefore, we have an inflection point at x = -1.

With this information, we can now sketch the graph of the function y = (x^3)/(x+1)^2, indicating the important points, asymptotes, concavity, and inflection point.