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Let fbe a functions that has derivatives of all orders for all real numbers_ Assume f (0) = -1,f' (0) = 3,f" (0) = 2 and f" (0) = 4 a) Write the thir...

Question

Let fbe a functions that has derivatives of all orders for all real numbers_ Assume f (0) = -1,f' (0) = 3,f" (0) = 2 and f" (0) = 4 a) Write the third degree Taylor Polynomials for f (2) about x 0_ b) Use the polynomial from (a) to approximate f (1) _ The fourth derivative of f (c) satisfies the inequality | f(4) (.)| < 3 for all x in the interval [0, 1]. Use the Lagrange error bound to find the size of the error in the approximation of f (1).a) The third degree Taylor polynomi

Let f be a functions that has derivatives of all orders for all real numbers_ Assume f (0) = -1,f' (0) = 3,f" (0) = 2 and f" (0) = 4 a) Write the third degree Taylor Polynomials for f (2) about x 0_ b) Use the polynomial from (a) to approximate f (1) _ The fourth derivative of f (c) satisfies the inequality | f(4) (.)| < 3 for all x in the interval [0, 1]. Use the Lagrange error bound to find the size of the error in the approximation of f (1). a) The third degree Taylor polynomial is b) The approximation of f (1) is The Lagrange Error bound is



Answers

Use the second Taylor polynomial of $f(x)=\sqrt{x+1}$ at $x=0$ to approximate $f(0.1)$ and find a bound for the error in the approximation.

Mhm. In this exercise A Echo zero and the function is 1 -1 to the -1. So it's derivatives and higher order derivatives, minors act to the miners two. two times 1 -2 Extra -3. Yeah, six times 1 -12 -4. And their function values of 0112, six respected. Hence the third trailer park phenomenal. He calls one plus X plus X squared plus that's cubed and Its value at .2 because one two for eight. Now let's find a bound for the arrow. In this approximately first we need your foreign capital. M Which is greater than or Echo two. The absolute value of calf four parks, which is the force order derivative Of this function which equals 24 Tom's one miners. Erikss Geographics. And we know that these echoes. Yeah four porn tube which is 24 over one Miner's points to to face power enhance the residual had appointed to Nikos cat joy. Um Over four Factorial Times Point to to the Force, Which equals five hour 2024, which is approximately point 00 for it is free. And the real value of this function at 0.2 because 1/1 minus point to Which is one point too far.

This exercise echo zero. And the functioning FX. He calls 88 or -X. And hence its derivatives miners. To a power miners there. E two miners there, miners detail miners acts and their function value selves there or 1. -1. A -1 parents. The third taylor ponte phenomenal is one minus X plus X. Squared over two miners. That's cubed over six. And its value at appointed to because 307 over 375 which is approximately 0.81 867 Next to find a bound for the arrow. In this approximation when you first find a capital air Such as it is Greater than Echo two. The absolute value. Uh huh. The fourth order to really jump which equals me to the miners X. Amazon. We know that um yukos he to the miners zero Page A Close one. So the procedure and appointed to you calls Captain William Over four factorial Times pointed to to the 4th Which is approximately two point sorry 6.66 67 times turned to the miners face.


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