Find a good numerical approximation to F(5) for the function with the properties that F′(x)=e−x2/5 and F(0)=3.
Using the Fundamental Theorem of Calculus, we know that:
F(x) = ∫0x F'(t) dt + F(0)
We can approximate F(5) by using numerical methods to approximate the definite integral:
F(5) ≈ ∫0^5 e^(-t^2/5) dt + 3
One common numerical method for approximating integrals is the trapezoidal rule. Using the trapezoidal rule with a step size of h = 0.5, we have:
F(5) ≈ (0.5/2) [e^(-0/5) + 2e^(-0.5^2/5) + 2e^(-1^2/5) + 2e^(-1.5^2/5) + 2e^(-2^2/5) + 2e^(-2.5^2/5) + 2e^(-3^2/5) + 2e^(-3.5^2/5) + 2e^(-4^2/5) + e^(-4.5^2/5) + e^(-5^2/5)]
Using a calculator or computer program, we can evaluate this expression to get:
F(5) ≈ 1.877
Therefore, a good numerical approximation to F(5) is approximately 1.877.