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The differential equation dy/dx=x

I.will produce a family of quadratic functions.

II.will produce a family of linear functions.

III.will produce a family of exponential functions.


ANSWER OPTIONS

I only

II only

III only

None of these

(please explain)

Verified Answer :

apologiabiology

dy/dx
if we find the general antiderivitive
y=(1/2)x²+C
C is a constant
so then it is quaratic
I is answer

I don't think the teacher wanted you to solve it like this so

dy/dx means the slope at x
the slope at x=x is x
might be confusing, but think a little bit
when x increases, the slope at x increases
sounds like an exponential or quadratic
when x decreases, the slope also decreses
when x becomes negative, the slope is negative,
so when x=0, slope is 0
when x>0, slope>0
when x<0, slope<0
we got a parabola
this is quadratic

it isn't exponential because the sloepe in exponential will decrease but never reach zero (only approach it)




answer is I

first option

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