Linear approximation Find the linear approximation to the foll...

Question

Linear approximation Find the linear approximation to the following functions at the given point a.

g(t) = √(2t + 9); a = -4

Accepted Answer

Welcome back, everyone. In this problem, we want to determine the linear approximation of H of Y equals root 4Y minus 5 at the point A equals 2. A says it's 2 route 3 divided by 3 Y plus route 3 divided by 3. B says it's 2/3 of Route 3Y minus 1/3 of Route 3. C says it's 2/3 of Route 3Y plus 7/3 of Route 3, and the D says it's 2/3 of Route 3 Y minus 7/3 of Route 3. Now what do we know about linear approximation that can help us to determine the linear approximation for H of Y? Well, recall, OK, that the formula for linear approximation tells us that the approximation L of Y will be equal to H of A plus the derivative of H of A multiplied by Y minus A. Now, in our problem, we know that A equals 2, but we'll need to figure out H A and H A in order to find our linear approximation. So let's go ahead and do that. First, let's start with H of A. Let's evaluate H of A. So solving for HFA. Which we know A equals 2, so we're really looking for H of 2. That means we're gonna substitute H of 2 H. Sorry, we're gonna substitute 2 for Y into H of Y. Because no, that means when we do that, when Y equals 2, then H equals the square root of 4 multiplied by 2 minus 5. That's 8 minus 5, which equals root 3. So we have our value for H of 2. Next, we need to solve, OK, for the derivative of H of 2, OK? In other words, we're gonna differentiate H of X. And then substitute. Sorry, we're going to differentiate H of Y and then substitute Y for 2. Now what's the derivative of H of Y? Well, We already know that H of Y equals the square root of 4 Y minus 5, OK? And we could rewrite this as 4 Y minus 5 raise to the power of a half. So that means the derivative of H with respect to Y. OK. Is going to be differentiated using the chain rule, which means we're gonna bring down our po half, subtract 1 from the alter function. So that's 4 Y minus 5 raises to the negative half multiplied by the derivative of the inner function. When we differentiate 4 Y minus 5 with respect to Y, it equals 4. Now we could rewrite this then as 2, because here 2 goes into 21 time, 2 goes into 42 times. So that's 2 multiplied by 4 minus 5 to the negative half or we can rewrite it, OK, with its root, so that's 2 divided by the square root of 4 Y minus 5 because a negative power means it's going to be in the denominator. We're going to reciprocate our expression. So now that we have the derivative of H of Y, this implies then that if we substitute 24 Y into the derivative, it's gonna be equal to 2 divided by root 4 multiplied by 2 minus 5, and we know that that's gonna be 8 minus 5, so it's 2 divided by route 3. Now that we have a of 2. And we have H2, we can solve for our linear approximation. Because no substituting. OK. A equals 2, sorry, H2 H A equals root 3. OK. And H A equals 2 divided by root 3. Then, OK, our linear approximation. Is gonna be equal to route 3 + 2 divided by route 3 multiplied by Y minus 2. Now if we go ahead here. And expand and, and expand this a little bit more. We're gonna have route 3, OK, plus 2 route 3 divided by 3, multiplied by Y minus 2, OK? If we multiply it by a conjugate here, try to rationalize. And then we can take this even a step further by expanding our bracket for 2/3 of route 3 across the expression Y minus 2. And when we do that, then we're gonna have route 3, OK. Plus 2/3 of Route 3 Y. OK, so that properly here. Minus 4/3 of Route 3. I know when we simplify that even further, then we get it to be equal to 2/3 of Route 3, why? Minus 1/3 of Route 3. Therefore, the linear approximation of H of Y at route 44 minus 5 at A equals 2 is 2/3 of route 3Y minus 1/3 of route 3. If we look back on our answer twice, that means B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Linear approximation Find the linear approximation to the following functions at the given point a.g(t) = √(2t + 9); a = -4

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Linear approximation Find the linear approximation to the following functions at the given point a.

g(t) = √(2t + 9); a = -4