# How do you use the first and second derivatives to sketch #f(x)=x^2/(x^2+5)#?

See the explanation below

graph{x^2/(x^2+5) [-5, 5, -2.5, 2.5]}

We then evaluate the behavior at the limits of the domain:

Now we calculate the first derivative using the quotient rule:

Calculating the second derivative:

The points of inflection are then the roots of:

(i) the denominator is always positive

Then:

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A few additional graphing notes

In addition to the excellent answer that you see earlier in this thread, I would like to add a few (general) graphing tips.

Graphing techniques in Algebra provide an excellent idea of what the graph looks like. Calculus only provides specific details about relative extremes and concavity.

The graph of a rational function may change from increasing to decreasing (or vice-versa) only at a critical value -- a relative extreme, removable discontinuity, or vertical asymptote. You can already determine much of this information with algebra, but you will not know the locations of relative extremes (maxes or mins) without Calculus.

The graph of a rational function may change concavity only at an inflection point. You will not know these values without Calculus.

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To sketch ( f(x) = \frac{x^2}{x^2 + 5} ) using the first and second derivatives:

- Find the first derivative of ( f(x) ) (denoted as ( f'(x) )).
- Find the critical points by solving ( f'(x) = 0 ).
- Determine the intervals of increase and decrease of ( f(x) ) using the sign of ( f'(x) ).
- Find the second derivative of ( f(x) ) (denoted as ( f''(x) )).
- Determine the intervals of concavity of ( f(x) ) using the sign of ( f''(x) ).
- Locate any inflection points by solving ( f''(x) = 0 ).
- Plot the critical points, inflection points, and any vertical asymptotes of ( f(x) ).
- Sketch the curve by considering the behavior of ( f(x) ) in each interval determined in steps 3 and 5, along with the end behavior as ( x \to \pm \infty ).

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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.

- How do you find the exact relative maximum and minimum of the polynomial function of #2x^3 -23x^2+78x-72#?
- How do you sketch the graph #y=x^3+2x^2+x# using the first and second derivatives?
- What is the second derivative of #e^(2x)#?
- How do you find the local maximum and minimum values of #f ' (x) = (x^2 -9)#?
- How do you determine whether the function #f(x) = x/(x^2 - 5)# is concave up or concave down and its intervals?

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