Question

In Taylor’s theorem for expansion of function "5+4(x–1)2–3 (x–1)3+(x–1)4 "
power of x, the value of h is _____
A) -1
B) 1​

Answer 1

The value of h by Taylor's theorem is -1

In Taylor's theorem, the value of h represents the change in the independent variable (x) from the point at which the Taylor series is expanded (a) to the point at which the approximation is made (x).

In the given function

5 + 4(x - 1)² - 3(x - 1)³ + (x - 1)⁴, it is clear that the point at which the Taylor series is expanded is x = 1 because the terms (x-1)², (x-1)³ and (x-1)⁴ are present in the function.

Therefore, we can say that a = 1 and the value of h = x - 1.

So, the answer is A) -1

• Taylor's theorem is a fundamental result in calculus that states that any smooth function can be approximated by a polynomial function called its Taylor series. The theorem is named after Brook Taylor, who first formulated it in 1715, and Joseph-Louis Lagrange, who expanded on it in 1762. The Taylor series of a function f(x) is an infinite sum of terms, each of which is a polynomial function of x.

• The basic form of Taylor's theorem states that for any function f(x) that is infinitely differentiable at a point a, there exists a polynomial function P(x) such that:

• f(x) = P(x) + Rn(x) where Rn(x) is the remainder term and P(x) is the nth degree Taylor polynomial.

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Answer 2

Answer:

According to Taylor's theorem, the value of h is -1.

Step-by-step explanation:

What is taylor theorem?

The value of h in Taylor's theorem indicates the change in the independent variable (x) from the point when the Taylor series is extended (a) to the point where the approximation is made (x).

In the provided function,

5 + 4(x - 1)² - 3(x - 1)³ + (x - 1) (x - 1) 4, it is evident that the point at which the Taylor series is enlarged is x = 1 since the function contains the components (x-1)2, (x-1)3, and (x-1)4.

As a result, we may state that a = 1 and h = x - 1.

As a result, the answer is (A) -1.

Taylor's theorem is a basic conclusion in calculus that claims that any smooth function may be approximated by a polynomial function known as its Taylor series. Brook Taylor, who initially proposed the theorem in 1715, and Joseph-Louis Lagrange, who built on it in 1762, are both named after it. A function f(xTaylor )'s series is an infinite sum of terms, each of which is a polynomial function of x. Taylor's theorem in its most basic version asserts that for each indefinitely differentiable function f(x) at a point a, there exists a polynomial function P(x) such that: f(x) = P(x) + Rn(x), where Rn(x) is the remainder term and P(x) is the nth degree Taylor polynomial.

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