Let's start with simple substitution into an equation:
a) Find y at x=3 for y = 2x^2 + 2x + 2
The "|" is important between the equation y = 2x^2 + 2x + 2 and x=3
Let's now find x if we know y. This is a little harder as we have to solve the equation.
b) Find x at y=14 for y = 2x^2 + 2x + 2
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWdIsDAoDsUmLIz-hh_p8uTVg4DBqjRSR4t0qdvGGQ5C0Kt6Cu4BocSUUoCDltyf0b27jbMqrqaZLDtdraZoqvxWC2xjB86uQmV35Hia2NftkLfHSacS-dTypG078w00sfAryNbmY6dl4/s1600/find+x+at+y.png)
The easiest way to do this is to type 14 = 2x^2 + 2x + 2, highlight it using your stylus (this is important!) and then go
interactive->advanced->solve->ok
Note that it find both possible solutions (unlike using numsolve with the incorrect range specified)
Let's find the first derivative. For this use the 2D template in the soft keyboard
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrPqxXs4dbPW8iRGMDB-xnRQc61Z_Z7St4iFCfikiLB214SUGhuOXLnjnXmKeASDEd-5Y7UeS4_tIVDnlYzRTuW_ZuviLe8vHZJVMHOdNbf0ItPQmBGNkMKHo-h5Oyy7hIwNsrvIpjl7w/s1600/Screen+shot+2012-04-04+at+7.27.42+AM.png)
c) Find the 1st derivative of y=2x^2 + 2x + 2
Note that I removed the "y=" this time. I differentiated the expression on purpose as it makes the next part easier.
Finding the 1st derivative/gradient/instantaneous rate of change at a point is also easy.
d) Find the 1st derivative of y = 2x^2 + 2x + 2 at x = 3
As you can see, it is a mix between c) and a)
Last but not least we can find a point for a particular gradient.
e) Find x at y' = 14 for y = 2x^2 + 2x + 2
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKtHH9VStBqQ4RcBFW2kWJsfUfHvBpE-q1tyHnt_iByu4jnwtKjt9acBtKY2ZBXn9kCINbVUNLHtV4Y03FoOU8Aywqjw0rEpClBlKZ2NWfwQAqXP-lDoRijJFS8iob-tB21259l5bobPQ/s1600/Find+gradient+at+x.png)
To find y itself we could repeat a)
It's very much a case of thinking what you need and then finding it. As you can see it can all be done with one line on a CAS calculator, things that would take multiple steps on paper. TanLine is also a useful function that can be investigated and used to quickly find tangents.
Viola!
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