This adaptive game tutorial will help you master this topic in a way that adapts to your ability as you practice the questions.
If this is your first time studying the material, you may find yourself requiring more help, and as a result your game scores could end up lower, or you could even "die." Do not be discouraged! Just press "New game" to play the game as many times as you need in order to require less help and improve your scores.
You can also try the non-game version (link on the side), which does not score you, is not randomized or adaptive and gives you all the answers, but will not give you additional practice at the level you might need.
To complete this game you must answer all the questions correctly; you will be allowed to have a number of tries for some questions.
The questions are randomized: Expect to see lots of differences differences every time you load the page.
Press "Scores" at any time to show the scores you currently have.
The game is automatically saved. Relaunching the game on the same computer with the same browser will bring up the saved version.
Press "New game" to discard the saved game and start a new game.
Caution: Clicking on the little pictures that appear from time to time on the left can have unexpected consequences! You might want to experiment with them; this is a game after all!
It is easy enough to use graphing technology to draw a graph, but we need to use calculus to understand some of the features we are seeing and tell us where to look, and also to help us draw a curve by hand without technology.
To illustrate this point, take a look at the following graph, which has all kinds of interesting-looking features shown with different colors and markers. Each of the buttons shows a particular feature, as listed below.
Features of the graph of $\bold{y = f(x)}$
x- and y-intercepts : $x$-intercepts: In the equation $y = f(x)$, set $y=0$ and solve for $x$. $y$-intercepts: In the equation $y = f(x)$, set $x=0$ and solve for $y$.
Maxima and minima : Use the techniques of the %%maxmintut to locate and classify the absolute and relative maxima and minima.
Points of inflection : Use the techniques of the %%inflectiontut to locate and classify the points of inflection.
Behavior near singular points of $\bold{f}$ : If $a$ is a singular point of $f$, consider $\lim_{x \to a^-}f(x)$ and $\lim_{x \to a^+}f(x)$ to see how the graph of $f$ behaves as $x \to a$.
Behavior at infinity : Consider $\lim_{x \to -\infty}f(x)$ and $\lim_{x \to \infty}f(x)$ to see how the graph of $f$ behaves far to the left and right.
Note It is sometimes difficult or impossible to solve all of the equations that come up in Steps 1, 2, and 3. As a consequence, we might not be able to say exactly where the $x$-intercepts, extrema, or points of inflection are. When this happens we can use graphing technology to assist us in determining accurate numerical approximations.
In the following quiz, use the graphs of $f'$ and $f''$ to identify the locations of (sometimes hard to see!) features of a given graph.
Next, you should calculate the location of the important features of a curve you are already given.
Finally, you are given a more complicated function whose features to identify and localize using graphing technology. We suggest you use the Zweig grapher.
Now try the exercises in Section 12.4 in Finite Mathematics and Applied Calculus.
or move ahead to the next tutorial by pressing "Next tutorial" on the sidebar.
Function evaluator and grapher