Difference between revisions of "009A Sample Final A, Problem 4"

Revision as of 21:52, 26 March 2015

4. Find an equation for the tangent line to the function $-x^{3}-2xy+y^{3}=-1$ at the point $(1,1)$ .

Foundations:
Since only two variables are present, we are going to differentiate everything with respect to $x$ in order to find an expression for the slope, $m=y'=dy/dx$ . Then we can use the point-slope equation form $y-y_{1}=m(x-x_{1})$ at the point $\left(x_{1},y_{1}\right)=(1,1)$ to find the equation of the tangent line.
Note that implicit differentiation will require the product rule and the chain rule. In particular, differentiating $2xy$ must be treated as
$(2x)\cdot (y),$
which has as a derivative
$2\cdot y+2x\cdot y'=2y+2x\cdot y'.$

Finding the slope:
We use implicit differentiation on our original equation to find
$-3x^{2}-2y-2x\cdot y'+3y^{2}\cdot y'=0.$
From here, I would immediately plug in (1,1) to find y ':
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -3-2-2y'+3y'=0} , or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y' = 5.}

Writing the Equation of the Tangent Line:
Now, we simply plug our values of x = y = 1 and m = 5 into the point-slope form to find the tangent line through (1,1) is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y-1=5(x-1),}
or in slope-intercept form
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=5x-4.}

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 |Since only two variables are present, we are going to differentiate everything with respect to <math style="vertical-align: 0%">x</math> in order to find an expression for the slope, <math style="vertical-align: -21%">m = y' = dy/dx</math>. Then we can use the point-slope equation form <math style="vertical-align: -21%;">y-y_{1} = m(x-x_{1})</math> at the point <math style="vertical-align: -21%">\left(x_1,y_1\right) = (1,1)</math> to find the equation of the tangent line.
 |-
-|Note that implicit differentiation will require the product rule <u>''and''</u> the chain rule.  In particular, differentiating 2''xy'' must be treated as
+|Note that implicit differentiation will require the product rule <u>''and''</u> the chain rule.  In particular, differentiating <math style="vertical-align: -18%">2xy</math> must be treated as
 |-
 |&nbsp;&nbsp;&nbsp;&nbsp; <math>(2x)\cdot (y),</math>