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Linear approximation and concavity Carry out the following steps for the given functions $f$ and points a.a. Find the linear approximation $L$ to the function $f$ at the point a.b. Graph $f$ and $L$ on the same set of axes.c. Based on the graphs in part (b), state whether linear approximations to $f$ near $x=a$ are underestimates or overestimates.d. Compute $f^{\prime \prime}(a)$ to confirm your conclusions in part (c).$$f(x)=\sqrt{2} \cos x ; a=\frac{\pi}{4}$$

a.) $L(x)=-x+1+\frac{\pi}{4}$ , c.) Overestimate

Calculus 1 / AB

Chapter 4

Applications of the Derivative

Section 6

Linear Approximation and Differentials

Derivatives

Differentiation

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Oregon State University

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to solve this problem. Let's start with partner. And the first step we have to take is to find f prime of X. Well, the derivative of the square root of two times co sign of X is just going to be negative. Square root of two time signing backs. Fruit of co sign is negative sign and the route to is going to stay. No, we need to find ethnic enough prime of a death of A which is going to equal f of pi over four is going to equal that's going equal the square root of two times co sign of high over four, which is going to equals square root of two times uh, square root of 2/2 which these this spirit to Times Square to to close to 2/2 is one nexus find f prime a day or a crime of pi over four. This is gonna look very similar. This is going to be the negative times square root of two times the Sinus over four. And for those you who know your unit circle really well, you know that the coastline of power for in the sign of pie before are the same thing. So this is going to equal. Let's just say negative, Ruutu. Negative, right. Two times route to over to which is just going to be negative one. Now we have enough information to find Ella Vex l of X is going to equal every day one plus f prime obey or in this case, would be a minus one times X minus A or X minus However, four. Simplifying this and just rearranging and to make a local cleaner is going to give us negative X plus one plus pi over four. This is easier to keep this. This is exact. You could put a decimal she wanted to, but we'll just leave like that for now. Now, let's do part B, which is the graph of these two functions. Now, this graph is not gonna be anywhere near perfect. But please just bear with me here. The coast sign times were two. Let's call this point Scary of two just for fun and three co sign is going to look like this and I'm only gonna be drawing on the positive side of the access. I guess I could go like this, but we really only care about the positive access because our A value is positive. It's called that a next we're going to draw are L of X function, which is going to come down here, touch the graph briefly and then skirt down into infinity and script of engine 50 going this way. So they say they touch right here. That's not exactly where they touched, but it's close enough. It's close enough for us to see the point. So this leads into Parts Seat determined. This is an underestimate or overestimate. Well, at and near the point A L of X is going to be over the coastline function that we have here. So at this point right here, which is let's call this slightly past a l of X is above the coastline, so that tells us that we're dealing with an overestimate. And let's justify that. Let's find our second derivatives. Well, let's remind ourselves that F prime of X is negative. Square root of two times Sign of X. I forgot unequal sign. Sorry about that. At the final, X equals the negative square root of two times Sign of X. So now when we take another derivative of this or F double prime of X, we're going to get well, The derivative of Negative Sign is negative coastline. So this negatives going mistaking her and the route to is going to say there as well. So we're gonna have negative route to co sign attacks. So what does this tell us? Since that's hard to tell. But that is supposed to be a negative sign right there. So since our our double derivative is negative at point A, that means that at the point, a f is con cave down, which tells us that which is the proof that we're looking for that l of X is going to be an overestimate.

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