"Find the function with the given derivative whose..." was answered by our community

enrosca0fx

Answered question

2023-01-18

Find the function with the given derivative whose graph passes through the point p

g^{'} (x) = \frac{1}{x^{2}} + 2 x, p (- 1, 1)

Elgari8lx

Beginner2023-01-19Added 7 answers

So we given:

\int g^{'} (x) d x = g (x) = \int (x^{- 2} + 2 x) d x = - x^{- 1} + x^{2} + C_{0}

but

g (- 1) = 1 \Rightarrow - {(- 1)}^{- 1} + {(- 1)}^{2} + C_{0} = 1

and then

C_{0} = - 1

so the sought function is

g (x) = - x^{- 1} + x^{2} - 1

Delaney Escobar

Beginner2023-01-20Added 1 answers

g^{'} (x) = \frac{1}{x^{2}} + 2 x

, then

g (x) = \int (\frac{1}{x^{2}} + 2 x) d x

\int (x^{- 2} + 2 x) d x = - x^{- 1} + x^{2} + c = - \frac{1}{x} + x^{2} + c

g (x) = - \frac{1}{x} + x^{2} + c

g (x)

passes through point P

(- 1, 1)

therefore:

1 = - \frac{1}{- 1} + {(- 1)}^{2} + c

1 = 1 + 1 + c \Rightarrow c = - 1

g (x) = - \frac{1}{x} + x^{2} - 1

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