Question

my favorite calc 1 question:

The "Folium of Descartes" has the following equation, \( x^3+y^3-6xy = 0 \)

a) sketch the tangent line at the point \( (\frac{4}{3},\frac{8}{3}) \)

b) Find the slope of the tangent line to the graph at \( (\frac{4}{3},\frac{8}{3}) \)

c) Find the equation of the tangent line at the point \( (\frac{4}{3}, \frac{8}{3}) \)

Answer 1

besides figuring out what the graph looks like, the rest are easy to solve using:
implicit differentiation (b), and point-slope form (c).

Answer 2

WAIT WAAAAAAAAAAT

Answer 3

http://youtu.be/jG4DYZ5uuE0

Answer 4

The "Folium of Descartes" has the following equation, \( x^3+y^3-6xy = 0 \)

a) sketch the tangent line at the point \( (\frac{4}{3},\frac{8}{3}) \)

b) Find the slope of the tangent line to the graph at \( (\frac{4}{3},\frac{8}{3}) \)

c) Find the equation of the tangent line at the point \( (\frac{4}{3}, \frac{8}{3}) \)

Answer 5

b) Find the slope of the tangent line to the graph at \( (\frac{4}{3},\frac{8}{3}) \)

\( \frac{d}{dx}(x^3+y^3-6xy) = \frac{d}{dx}0 \\
3x^2+3y^2 y' -6(y+xy') = 0 \\
3x^2 + 3y^2y' - 6y - 6xy' = 0 \\
3y^2y'-6xy'=-3x^2+6y \\
y'(3y^2-6x) = -3x^2+6y \\
\sf \large y' = \frac{-3x^2+6y}{3y^2-6x} \\
clean~up \\
\large y' = \frac{-3(x^2-2y)}{-3(-y^2+2x)} \rightarrow y'=\frac{x^2-2y}{-y^2+2x} \)

plug-in the coordinate \( \sf \large(\frac{4}{3}, \frac{8}{3}) \)
\[
y' =\large \frac{(\frac{4}{3})^2-2(\frac{8}{3})}{-(\frac{8}{3})^2+2(\frac{4}{3})} \rightarrow \large \frac{\frac{16}{9}-\frac{16}{3}}{-\frac{64}{9}+\frac{8}{3}}\\
\large \frac{\frac{-32}{9}}{\frac{-40}{9}} \rightarrow \frac{-32}{-40} \rightarrow \color {blue}{\frac{4}{5}}
\]

c) Find the equation of the tangent line at the point \( (\frac{4}{3}, \frac{8}{3}) \)
\( y-y_1 = m(x-x_1) \\
y \large - \frac{8}{3} = \frac{4}{5}(x-\frac{4}{3}) \\
y = \large \frac{4}{5}x-\frac{16}{15}+\frac{40}{15} \\
\color{blue}{y = \large \frac{4}{5}x+\frac{24}{15}}
\)

Answer 6

So weird...

Answer 7

so how does the graph look like!

Answer 8

to sketch the tangent line i need to sketch the graph of that equation first