Question

Find the function f whose tangent has slope (x^3-2/x^2+2) for each value of x and whose graph passes through the point ( 1, 3 ).

Answer 1

The tangent slope is equal to the derivative, so for each value $f'(x) = x^3-\dfrac{2}{x^2}+2$ . Therefore,

$f(x) = \int (x^3-\dfrac{2}{x^2}+2)\,dx = \dfrac14 x^4+ \dfrac2x +2x + c.$

Let us determine the constant c. We know that the graph of the function passes through the point (1, 3), so

$3 = \dfrac14\cdot1^4+ \dfrac21 +2\cdot1 + c \; \Rightarrow \; c = -1.25.$

So $f(x) == \dfrac14 x^4+ \dfrac2x +2x -1.25.$