A curve has equation y = f(x) and passes through the point (4, 22). Given that f'(x) = 3x^2 - 3x^(1/2) - 7, use integration to find f(x), giving each term in its simplest form

As we are given the derivative of f(x), we first need to integrate this derivative to obtain the function, f(x). Using the standard integration formula, ∫ x^n dx = (1/n+1)(x^(n+1)) +c, integrate each term in the derivative function. In the first term 3x^2, here n=2, therefore ∫ 3x^2 dx = (1/2+1)( 3x^(2+1)) = x^3 +c. Using the same formula, we can do the same with the second term -3x^(1/2), thus ∫-3x^(1/2) dx = (1/(1/2)+1)(-3x^((1/2)+1) = (2/3)(-3x^(3/2)) = -2x^(3/2)+c. As for the last term ,-7, integrating a constant rule applies, ∫a dx = ax +c, where a is a constant, so this term becomes -7x +c. All the constants (c) can be combined together to form a new constant, as adding constants together simply forms another constant, who's value does not change. We now form f(x) = x^3 - 2x^(3/2) - 7x +c. To work out the constant ,c, and complete f(x), we can set f(x) = y and then sub in a point on this curve, which has been given as (x=4, y=22). Subbing these values in give us 22 = 64 - 16 - 28 + c, and therefore c = 22 - 64 + 16 + 28 = 2. Thus f(x) = x^3 - 2x^(3/2) - 7x + 2.