How do you sketch the curve #y=x^2/(x^2+9)# by finding local maximum, minimum, inflection points, asymptotes, and intercepts?
The minimum is
The points of inflexions are
The intercept is
The horizontal asymptote is
The curve is symmetric about the yaxis
The derivative of a quotient is
We start by calculating the first derivative
We can build a chart
Now, we calculate the second derivative
We can build the chart
graph{(y(x^2)/(x^2+9))(y1)=0 [7.02, 7.024, 3.51, 3.51]}
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To sketch the curve ( y = \frac{x^2}{x^2+9} ), follow these steps:

Intercepts:
 yintercept: Set ( x = 0 ) and solve for ( y ).
 xintercept: Set ( y = 0 ) and solve for ( x ).

Asymptotes:
 Vertical asymptotes occur where the denominator equals zero, so set ( x^2 + 9 = 0 ) and solve for ( x ).
 Horizontal asymptotes occur as ( x ) approaches positive or negative infinity. Use limits to find them.

Local Maximum and Minimum:
 Find critical points by taking the derivative of ( y ), setting it equal to zero, and solving for ( x ).
 Use the second derivative test or first derivative test to determine if these points are local maxima or minima.

Inflection Points:
 Find the second derivative of ( y ) and set it equal to zero to find inflection points.
Once you have found all these points, sketch the curve by plotting the intercepts, asymptotes, local maximum and minimum points, and inflection points, and then connecting them smoothly.
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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.
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