Quadratic approximations

[Technology Exercise...

Question

Quadratic approximations

[Technology Exercise] e. Find the quadratic approximation to h(x) = √(1 + x) at x = 0. Graph h and its quadratic approximation together. Comment on what you see.

Accepted Answer

Hello. In this video, we are going to be determining the quadratic approximation to G of X equal to the square root of 4 + X at the value X is equal to 0. We are also told that the quadratic approximation of G of X is defined as P X equal to C0 plus C1 multiplied by X minus M, plus C2 multiplied by X minus M squad, where X is equal to M. We are told that PF M is equal to G M P M is equal to G M, and P is equal to G M. So our goal for this problem is to solve for the coefficients in the PFX approximation. Where X is equal to M. Let's go ahead and start by solving for the C sub-zero coefficient. Since we know that X is equal to M, if we plug in M into our function, that is going to give us P M equal to C0 plus C1 multiplied by M minus M plus C2 multiplied by M minus M squared. Now, every M minus M is going to simplify to be zero. So we are left with PFM equal to C subzero. Now the problem tells us that PF M is equal to GM. This means that the coefficiency sub-zero is going to equal to G with the value M plugged into it. Now, the problem tells us that our G of X function is the square root of 4 + X, and we are told that this is evaluated at the value X is equal to 0. So, if we plug in 0 into the G of X function, G of 0 is equal to the square root of 4 + 0, which is going to leave us with the square root of 4, which simplifies to be 2. So C subzero is going to equal to 2, and that is the first coefficient we are looking for. Now, what we are going to do is we are going to use the fact that P M is equal to GM. So, in order to solve for the next coefficient, we will take the derivative of our PFX function. That is going to give us P X equal to C1 plus 2 C2 multiplied by X minus M. If we plug in M into the function, we get P M equal to C 1 + 2 C2 multiplied by M minus M, which is 0. So P M is equal to C1, and from the problem, we are also told that this is equal to G M. What we need now is we need the derivative of the square root 4 + X. The derivative of the G function is going to be G of X equal to 1 divided by 2 multiplied by the square root of 4 plus X. And again, we are going to evaluate this at X is equal to 0. So G 0 is equal to 1 divided by 2 multiplied by the square root of 4 + 0, which is 4. The square root of 4 simplifies to be 2, so we have 1 divided by 2 multiplied by 2, which is 1/4, and this means that our C1 coefficient is going to be 1/4. Now, we are going to solve FA's final coefficient by using the fact that the 2nd derivative of P is equal to the 2nd derivative of G. So let's go ahead and take the derivative of PX. That is going to give us P X, which is equal to 2 C2. Now, since P X is equal to G X 2 C2 is equal to G of X, and if we solve for C2, we get C2 is equal to G X divided by 2. So we need the second derivative of the G of X function, which is a derivative of the first derivative. So if we take the derivative of G X, we will get G of X is equal to 1. 1 divided by 4 multiplied by the quantity, 4 + X raised to the 3 halves power. And again, this is going to be evaluated at X is equal to 0. So if we plug in X is equal to 0 into the second derivative, G 0 is equal to -1 divided by 4 multiplied by 4 rays to the 3 halves power. 4 raised to the 3 halves power is going to simplify to be 8. So we have -1 divided by 4 multiplied by 8, which further simplifies to be -1 divided by 32. Now, this is the value of G 0, but keep in mind that we need to take this value and divide it by 2 in order to solve for C2. So C2 is equal to -1 divided by 32, all divided by 2, which simplifies to be -1 divided by 64. And this is going to be the final coefficient for our quadratic approximation. So to recap, we have C0 equal to 2, C1 equal to 1/4, and C2 equal to -1 divided by 64. Now what we are going to do is we are going to take these coefficients, plug them into the approximation, and reorder the terms, and that is going to give us P X is equal to -1 divided by 64, multiplied by X minus M squared. Then we are going to add 1/4. Multiplied by X minus M. And finally, we are adding our final coefficient, which is 2. And this is going to be the polynomial approximation of our given function. So I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

Quadratic approximations

[Technology Exercise] e. Find the quadratic approximation to h(x) = √(1 + x) at x = 0. Graph h and its quadratic approximation together. Comment on what you see.

Key Concepts

Quadratic Approximation

Taylor Series

Graphing Functions

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