Find the linearization of ƒ(x) = √(1 + x) + sin x - 0.5 at x =...

Question

Find the linearization of ƒ(x) = √(1 + x) + sin x - 0.5 at x = 0.

Accepted Answer

Welcome back, everyone. Find the linearization of the given function at the specified value of X. G of X equals 22 root of x + 4 minus tangent of X at X equals 0. Where given 4 answer choices A L X equals 4 minus 1/2 X. B says 4 + 1/2 X C 4 minus 2 X, and D4 + 2 X. So let's recall that we can linearize a function using the formula L of X equals G at the specified point A multiplied by x minus A plus G A. And now let's break it down. So first of all, formula says that we want to evaluate G at A, and this means that our first step is to differentiate the G of X. This means that we want to find the derivative of 2 square root of X + 4 minus tangent of X because this is the original function that we are trying to linearize, right? So we can find the derivative using the power rule to begin with because we are differentiating 2 multiplied by X + 4, raise the power of 1/2. Using the properties of exponents, we are converting the radical into an exponent, and then we want to subtract the derivative of tangent of X. So now we're going to get 2 multiplied by 1/2 multiplied by X + 4, raise the power of -12. So here we are applying the power rule, and according to the chain rule, we also have to multiply by the X + 4 derivative, which is equal to 1, so it has no effect. And then the derivative of tangent is squad of X using the table of basic derivatives. And now if we simplify, to multiplied by 1/2 gives us 1, so we can simply rewrite the answer as 1 divided by square root of. X + 4 minus C2. Of X Now that we have our derivative, all that we have to do is evaluate G at point A, and our A is equal to 0 because it says at x equals 0. So what we're going to do is simply use our derivative and replace every X with 0. So we get 1 divided by a square root of 0 + 4. Minus c 2d of 0. This gives us 1 divided by square root of 4, which is equal to 1/2. And 2 squad of 0 is equal to 1, so we get 1/2 minus 1, which is -12. Well done. Our next step is to evaluate G, the original function at the same point. G adds 0. So now we're taking the original function 2 square root of x + 4. X now becomes 0, so we get 2 square root of 0 + 4, and we are subtracting tangent at 0. So this gives us 2 multiplied by square root of 4 which is. Or And 0 is 0, right? And now knowing those, we can write the L of X expression because it says G A. Our G 0 is -12. We are multiplying that by x minus A, which is X minus 0, and then we're adding G of 0, which is 4. From here, we can just simplify and we can see that we have 4. Minus 1/2 X. Well done. This is our final answer, and now if we consider the answer choices, we can conclude that the correct option is A. Thank you for watching.

Find the linearization of ƒ(x) = √(1 + x) + sin x - 0.5 at x = 0.

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