How do you sketch the curve #y=x/(x^29)# by finding local maximum, minimum, inflection points, asymptotes, and intercepts?
Local Maximum/Minimum
Start by finding the first derivative.
Therefore, there are no critical numbers and thus no local max/min.
Points of inflection
Now find the second derivative. I used a derivative calculator for this, because it's very long to do by hand.
Asymptotes
This will have horizontal and vertical asymptotes.
Vertical
We factor the denominator:
Horizontal
Intercepts
We can now more or less trace the graph. Many people, including myself, like to prepare a table of values before graphing, but I'll leave that choice up to you.
graph{y = x/(x^2  9) [28.87, 28.86, 14.43, 14.44]}
Hopefully this helps!
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Find the intercepts:
 yintercept: Set (x = 0) to get (y = 0).
 xintercepts: Set (y = 0) to solve for (x).

Find asymptotes:
 Vertical asymptotes: Set the denominator (x^2  9) equal to zero and solve for (x). These are at (x = 3) and (x = 3).
 Horizontal asymptote: Compare degrees of numerator and denominator. Since both are of degree 1, divide the leading coefficients: (y = \frac{1}{1}), so (y = 1) is the horizontal asymptote.

Find critical points for extrema and inflection:
 Find (f'(x)) and set it equal to zero to find critical points.
 (f'(x) = \frac{2x^2 + 9}{(x^2  9)^2}).
 Critical points are at (x = 3/√2) and (x = 3/√2).

Test for local extrema:
 Use the second derivative test or first derivative test at the critical points.
 Second derivative (f''(x) = \frac{2(3x^2  13)}{(x^2  9)^3}).
 Evaluate at the critical points to determine nature of extrema.

Test for inflection points:
 Set (f''(x) = 0) to find possible inflection points.
 Solve (3x^2  13 = 0) to get (x = ±\sqrt{\frac{13}{3}}).

Plot the points:
 Plot intercepts, asymptotes, local max/min, and inflection points.
 Sketch the curve using this information.
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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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