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please help me with calculus. -confused scream-
#math​

Please Help Me With Calculus Confused Screammath class=

Sagot :

[tex]\underline{\underline{\large{\red{\mathcal{✒GIVEN:}}}}}[/tex]

A curve has an equation

[tex]\bullet \: \: \rm{x^{3}−4xy+y3=0}[/tex]



[tex]\underline{\underline{\large{\red{\mathcal{REQUIRED:}}}}}[/tex]

The equation of the tangent to the curve at the point (0, -3)



[tex]\underline{\underline{\large{\red{\mathcal{SOLUTION:}}}}}[/tex]

Hi, Brainly User!

If you are having difficulties in Math especially in calculus, let me help you!

1. To find the equation of the tangent to the curve at the point (0, -3) , we first need to find dy/dx using implicit differentiation:

[tex]\small{\tt{ {3x}^{2} - 4x \dfrac{d}{y} - 4y + 3 {y}^{2} \dfrac{dy}{dx} = 0}}[/tex]

[tex]\tt{(3 {y}^{2} - 4x) \dfrac{dy}{dx} = 4y - {3x}^{2} }[/tex]

[tex]\tt{ \dfrac{dy}{dx} = \dfrac{4y - {3x}^{2} }{3 {y}^{2} - 4x } }[/tex]

2. Now we evaluate [tex]\rm{\dfrac{dy}{dx}}[/tex] at the point (0, -3):

[tex]\tt{ \dfrac{dy}{dx} = \dfrac{4( - 3) - 3(0 {)}^{2} }{3( - 3 {)}^{2} - 4(0) } }[/tex]

[tex]\tt{ \dfrac{dy}{dx} = - \dfrac{12}{27} }[/tex]

[tex]\tt{ \dfrac{dy}{dx} = - \dfrac{4}{9} }[/tex]

3. The slope of the tangent at (0, -3) is -4/9.

The equation of the tangent line is given by:

[tex]\small{\boxed{ \bm{{ \red{y - y_{1} = m(x - x_{1}) }}}}}[/tex]

[tex]\tt{y - ( - 3) = - \dfrac{4}{9} (x - 0)}[/tex]

[tex]\tt{y + 3 = - \dfrac{4}{9} x}[/tex]

[tex]\boxed{ \tt{ \purple{ \large{y = - \dfrac{4}{9} x - 3}}}}[/tex]

Final Answer:

Thus, the equation of the tangent line to the curve at the point (0,−3) is

[tex] \rm{ \purple{ \large{D. \: y = - \dfrac{4}{9} x - 3}}}[/tex].