Answer to Question #140985 in Calculus for Vanessa Samson

Question #140985
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 ).
1
Expert's answer
2020-10-29T20:35:10-0400

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."


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