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Perhaps this is a silly question, but I haven't been able to find a clear answer anywhere as to what exactly the steps are for using derivatives to find the shape of a graph (I'm having difficulty even expressing what I mean properly). I've done my best to take information from my textbook and on the internet and create a list of what I think is supposed to be going on.

So here it goes:

Original function:

  • used to find $f'(x)$
  • produces the y-value for a critical number found using $f'(x)$
  • produces the y-value for the inflection point found using $f''(x)$

1st Derivative:

  • finds $f''(x)$
  • finds critical numbers
  • shows what intervals a function is increasing or decreasing on by using the roots of $f'(x)$
  • First derivative test: finds a local maximum or minimum

2nd Derivative:

  • finds where $f(x)$ is a maximum or minimum by evaluating $f''(x)$ by the roots of $f'(x)$ (which takes place of the first derivative test)
  • by using the critical numbers of $f'(x)$, $f''(x)$ can find where the graph is concave up or concave down.
  • gives where the inflection points are by finding where $f''(x)=0$, and then evaluating $f(x)$ at these numbers to find the point.

Any help I can get on this will be greatly appreciated!

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That looks pretty good, though I would add three more items in your "Original function" section.

  • produces the y-intercept of the graph by finding $f(0)$
  • produces the x-intercepts of the graph by solving $f(x)=0$
  • gives the domain of the graph by finding where $f(x)$ is undefined

That then covers what I teach in my calculus class. There are always more things you could check, such as $f'(0)$ and $f''(0)$, and perhaps a few more points of $f(x)$ for special values of $x$ that come to mind by looking at the formula for $f(x)$. (Example: look at $x$ a few multiples of $\pi$ if $f(x)$ includes trigonometric functions.)

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To find the shape of graph is also to plot it. You have got the basic approach. I repeat the essential steps:

Find as many pairs of points

$$ (x, f(x)) ... \tag {1}$$

as possible. Plot these points on x- and y- axis parallel lines respectively.

This fixes the domain.

Solve $ f'(x) = 0 $ and mark x values with short vertical lines on x-axis.

Calculate corresponding y values for these solved x- values. These points are

$$ (x_{extr},y_{extr}) ... \tag {2} $$

These fix up the range.

Draw a smooth curve through these points.

Increase the number of points ( reduce x-interval) if curve not sufficiently

smooth.

After these basic first steps the role of second derivative is to confirm

only,whether the extremum point is maximum or minimum. So it has no primary

role in making the graph, but helps to verify your plot.

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