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# The expression dy/dx=x/y gives the slope at any point on the graph of the function f(x) where f(1) = 2.

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The expression dy/dx=x/y gives the slope at any point on the graph of the function f(x) where f(1) = 2.

A.    Write the equation of the tangent line to f(x) at point (1, 2).

B.    Write an expression for f(x) in terms of x.

C.    What is the domain of f(x)?

D.    What type of figure is f(x)?

E.    Using the axes provided, sketch a slope field for the given differential equation at the nine points indicated. off-topic
May 12, 2020
edited by Guest  May 12, 2020
edited by Guest  May 12, 2020

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I don't know  E  [ been too long since I took differential equations ]  but  I think I might be able to help you with the rest

(D)  I believe  that the  function is a  hyperbola  with an equation of

y^2  -  x^2   =  3

Using implicit differentiation we have  that

2y y'  -  2x  =  0

2y y'  =  2x

y' =   2x / 2y

y'  =  x / y

(A )    at   (1,2)  the slope is   (x  / y)  =  (1/2)

So  the equation of the tangent line at this point is

y = (1/2) (x - 1) + 2

y = (1/2)x  + 3/2

(B)  In terms of x

f(x)  = ±√[3 + x^2]

(C)  the domain of f(x)  =  (-inf, inf)

Here's the graph here  : https://www.desmos.com/calculator/fdxb2d50kv   May 12, 2020