MATH104 December 2013
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Question 01 (f)

If $\displaystyle y=xy+x^{2}+1$, find the equation of the tangent line to this curve at the point $\displaystyle (1,1)$.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

We can use implicit differentiation to find the slope of the tangent line (and subsequently the equation of the tangent line). Don't forget to apply the product rule to the xy term!

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Solution

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Using implicit differentiation with respect to x, we see that
$\displaystyle {\begin{aligned}{\frac {dy}{dx}}&={\frac {d}{dx}}(xy+x^{2}+1)\\&=(1\cdot y+x\cdot {\frac {d}{dx}}\,y)+2x+0\\&=y+x\,{\frac {dy}{dx}}+2x\\&={\frac {y+2x}{1x}}\\\end{aligned}}$
and substituting in the point $\displaystyle (1,1)$, we have
$\displaystyle {\begin{aligned}{\frac {dy}{dx}}&={\frac {y+2x}{1x}}\\&={\frac {1+2(1)}{1(1)}}={\frac {1}{2}}\\\end{aligned}}$
We use the slopepoint formula to determine that the equation of the tangent line at $\displaystyle (1,1)$ is given by
 ${\begin{aligned}y1&={\frac {1}{2}}(x(1))\\y1&={\frac {1}{2}}x{\frac {1}{2}}\\y&={\color {blue}{\frac {1}{2}}x+{\frac {1}{2}}}\end{aligned}}$

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MER QGH flag, MER QGQ flag, MER QGS flag, MER QGT flag, MER Tag Implicit differentiation, MER Tag Tangent line, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag

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