Let $f(x)=%0.$
Below is what you might see in a grapher if you draw the graph withou specifying any window at all. This graph may or may not show all the important features. To see and locate the features accurately we suggest you use the Zweig grapher where you can vary the window appropriately or use a Zoom box to examine the interesting portions up close.
$f'(x) =$ BOX*
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$f''(x) =$ BOX*
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Which of the following shows some (perhaps not all) of the interesting features of the graph of $f'?$ (Note that you will likely need a close view of the graph of $f$ to see these features clearly, so use the Zweig grapher.)
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#[Step 1: The x- and y-intercepts:][Paso 1: las intersecciones- x e y:]#
#[Now enter the value(s) of the intercepts, accurate to four decimal places, separated by commas if there are more than one, or dne if there are none. ][Ahora ingresa el (los) valor(es) de las intersecciones con una precisión de cuatro decimales, separados por comas si hay más de uno, o ne si no hay ninguna.]#
#[$x$-intercept(s)][Interseccion(es)-$x$]#: BOX
#[$y$-intercept(s)][Interseccion(es)-$y$]#: BOX
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Step 2: Extrema:
Now enter a comma-separated list of all the absolute and relative extrema in the following form:
($x$-coordinate, $y$-coordinate, relmin/relmax/absmin/absmax)
with coordinates accurate to four decimal places. For instance, if there is a relative maximum at $(-1.2345,0.6789)$ and an asolute minimum at $(2.2345,-3.6789)$, enter
(-1.2345,0.6789,relmax),(2.2345,-3.6789,absmin)
or dne if there are no extrema.
Hint To locate the extrema accurately, use the graph of the derivative to see where it is zero or singular.
#[Extrema][Extremos]#: BOX
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Step 3: Points of inflection:
Now enter the coordinates, accurate to four decimal places, of all points of inflection, separated by commas if there are more than one, or dne if there are none.
For instance, you could enter (0.1234,-8.9012), (-8.1234,4.9012) for two point with those coordinates.
Hint To locate the points of inflection, use the graph of the second derivative to see where that crosses the $x$-axis.
#[Points of inflection][Puntos de inflexión]#: BOX
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Step 4: Behavior near singular points:
Now enter a comma-separated list of showing the behavior of the function near all singular points in the form:
$\left(a,\quad \lim_{x \to a^-}f(x),\quad \lim_{x \to a^+}f(x)\right)$
with all numbers accurate to four decimal places. For an infinite limit, enter -inf for −∞, and inf for ∞.
For instance, if there is a singular point at $x = -1.2345$ with the curve approaching $3.4567$ on the left and $-\infty$ on the right as
(-1.2345,3.4567,-inf)
If there are no sigular points, enter dne
#[Singular points][Puntos Singulares]#: BOX
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#[Step 5: Behavior at infinity:][Paso 5: Comportamiento en el infinito:]#
#[Now enter the value(s) of the given limits, accurate to four decimal places. For $\infty$ use inf and for $-\infty$ use -inf][Ahora ingresa el (los) valor(es) de los límites dados con una precisión de cuatro decimales. Para $\infty$ usa inf y para $-\infty$ usa -inf]]#
$\displaystyle \lim_{x \to -\infty}f(x) =$ BOX
$\displaystyle \lim_{x \to -\infty}f(x) =$ BOX
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