Question

Use the basic definition (Ad-hoc Calculations) to show that (a) Homework. f(x)=x⁴+2x³−7x²+3x−6∈Θ(x⁴) for n∈R+.

Answer 1

To show that f(x)=x⁴+2x³−7x²+3x−6∈Θ(x⁴) for n∈R+, apply ad-hoc calculations to rewrite the function and demonstrate that it is proportional to x⁴ for large values of x.

Step-by-step explanation:

To show that f(x) = x⁴ + 2x³ − 7x² + 3x − 6 ∈ Θ(x⁴) for n ∈ R+, we need to use the basic definition of ad-hoc calculations.

Let's consider the highest power of x in the function f(x), which is x⁴. We can rewrite the function as f(x) = x⁴ * [1 + 2/x − 7/x² + 3/x³ − 6/x⁴].

As x approaches infinity, all the terms inside the square brackets approach 0, since the powers of x keep decreasing. Therefore, the function f(x) ≈ x⁴ * 1 = x⁴ for large values of x. This shows that f(x) is in Θ(x⁴).