# Interactive video lesson plan for: Linearization and differentials Part I cU4l6 calculus ap ab bc Newton tangent TI-89 tutorial

#### Activity overview:

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were going to be going over to examples on linearization and differentials before we get started on the first example let's go over what the linearization means again first so the definition of linearization definition just in three is F of X is differentiable at X equals a then equation of the tangent line the equation of the tangent line which is written as L of X equals F of a plus write a times X minus a defined still linearization can define the linearization so what this is basically telling us is that of the tangent line is a good approximation of the function at the point immediately close to the specified X value okay so basically using insufficiently equals to appoint on any curve that curve looks like a line as long as the care is differentiable at that point the tangent line will be a good approximation of the bugs of the function right so if you already know how to find the equation of the tangent line then your you know how to find the linearization of the function because the tangent line is in fact be equation of the linearization of the function right so let's go ahead and take a look at some examples want one or two of find the linearization of why equals X to the third plus two X acts X equals one can't and that's but a parts ineffably the parts you to represents represents your results graphically your results graphically and states the significance the significance of your results take I will also show you how to use a calculator to find the equation of the tangent line of the linearization of the function in this example to write okay so let's go ahead and do this for so went to do is write down my function of function is X equals X to the third plus two X okay now let's some apply to formula1 to fight for much this function in our open focus of the sum specified X values here so for four X equals a well what do we know let's write down the formula first before I stop find the species so the formula tells is that linearization L of X so basically the function can be approximately given by L of X which is equal to F of a plus F time of a times X minus a right so this is a formula that we going to be using so if X equals a write X equals a bent F of a is going to be what the can every a to the third plus today and and the what is and primacy and families what you get we differentiate his function and substitute a okay so if I differentiate this and primacy a differentiate his function is can three a square plus two okay so this functionthe grounds the value of the Valley of the function at any specified X Valley a difficult be brought the derivative on the slope of the tangent line of any specific right so this question we are to find the linearization of X equals want so let's set X is the value of one okay to that's basically what's what a is so is going to be one right sell is one F of a is simply going to be at of one okay what is F of one is what you get the plug-in wanted to this function so you have one to the third plus two times one which is one plus two and that's the okay and then you have F time of a and primacy is going to be F of time of one which is about which is the derivative of dysfunction at a equals one right X equals one so that yields three times one square whenever differentiate first plus two using the auto corporations will have three plus two which is five right so now that write down the equation of the on the linearization on equation of the tangent line a just to show you a connection you have actually seen this equation before update when shady connection here so we of the linearization is F of a plus times X minus a sold this is the equation of the line okay the points look for the equation of a line that you learned algebra one two two this is simply why equals why one plus and times X minus X one exactly the same okay so this basically is why one in this is the slope I just a real quick algebra connection or you okay so look alike but we're going to put all this together in this formula to give us the linearization function so we going to have of the linearization of the function L of X equals F of a which is three plus a from of a which is five times X minus a a is one simplify this so we have L of X equals three plus five X minus five in this simplifies into by X minus two so there goes the linearization of the function the function X of the third was to X okay right so initially how to do this with your calculator but before do that lets graph this by hand and then check the accuracy of our results using calculator so I will make a table the right here the people of values a distinctly points and the sketch in the function so let's X and Y we use negative one zero in one and we know why is equal to Exodus

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