Question

Let be a real function of x, where x ∈ R. Complete the following expression to the 4th order: (x + ∆x) =

Answer 1

To complete the expression to the 4th order, we can use the binomial theorem and binomial coefficients. The expanded expression is x^4 + 4x^3(∆x) + 6x^2(∆x)^2 + 4x(∆x)^3 + (∆x)^4.

Step-by-step explanation:

To complete the expression to the 4th order, we need to expand it using the binomial theorem. The binomial theorem states that:

(a + b)^n = C(n,0)a^n + C(n,1)a^(n-1)b + C(n,2)a^(n-2)b^2 + ... + C(n,n)b^n

In this case, 'a' is x and 'b' is ∆x. Therefore, we have:

(x + ∆x)^4 = C(4,0)x^4 + C(4,1)x^3(∆x) + C(4,2)x^2(∆x)^2 + C(4,3)x(∆x)^3 + C(4,4)(∆x)^4

Using the binomial coefficients (C(n, k) = n! / (k!(n-k)!), we can simplify the expression:

(x + ∆x)^4 = x^4 + 4x^3(∆x) + 6x^2(∆x)^2 + 4x(∆x)^3 + (∆x)^4

Learn more about Expanding Expressions