(Solved):   Q.1 Q.2 Q.3 Q.4 Q.5 PLEASE 1. For each \( F(x, y)=0 \), find \( d y / d x \) for each of ...

 

Q.1
Q.2
Q.3
Q.4
Q.5 PLEASE

1. For each \( F(x, y)=0 \), find \( d y / d x \) for each of the following: (a) \( y-6 x+7=0 \) (b) \( 3 y+12 x+17=0 \) (c) \( x^{2}+6 x-13-y=0 \) 2. For each \( F(x, y)=0 \) use the implicit-function rule to find \( d y / d x \) : (a) \( F(x, y)=3 x^{2}+2 x y+4 y^{3}=0 \) (b) \( F(x, y)=12 x^{5}-2 y=0 \) (c) \( F(x, y)=7 x^{2}+2 x y^{2}+9 y^{4}=0 \) (d) \( F(x, y)=6 x^{3}-3 y=0 \) 3. For each \( F(x, y, z)=0 \) use the implicit-function rule to find \( \partial y / \partial x \) and \( \partial y / \partial z \) : (a) \( F(x, y, z)=x^{2} y^{3}+z^{2}+x y z=0 \) (b) \( F(x, y, z)=x^{3} z^{2}+y^{3}+4 x y z=0 \) (c) \( F(x, y, z)=3 x^{2} y^{3}+x z^{2} y^{2}+y^{3} z x^{4}+y^{2} z=0 \) 4. Assuming that the equation \( F\left(U, x_{1}, x_{2}, \ldots, x_{n}\right)=0 \) implicitly defines a utility function \( U=f\left(x_{1}, x_{2}, \ldots, x_{n}\right) \) : (a) Find the expressions for \( \partial U / \partial x_{2}, \partial U / \partial x_{n}, \partial x_{3} / \partial x_{2} \), and \( \partial x_{4} / \partial x_{n} \). (b) Interpret their respective economic meanings. 5. For each of the given equations \( F(y, x)=0 \), is an implicit function \( y=f(x) \) defined around the point \( (y=3, x=1) \) ? (a) \( x^{3}-2 x^{2} y+3 x y^{2}-22=0 \) (b) \( 2 x^{2}+4 x y-y^{4}+67=0 \)