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KonstantinChe [14]
2 years ago
5

Use the definition mtan=lim

Mathematics
1 answer:
Leokris [45]2 years ago
4 0

(a) mtan refers to the slope of the tangent line. Given <em>f(x)</em> = 9 + 7<em>x</em> ², compute the difference quotient:

\dfrac{f(x+h)-f(x)}h = \dfrac{(9+7(x+h)^2)-(9+7x^2)}h = \dfrac{(9+7x^2+14xh+7h^2)-9-7x^2}h = \dfrac{14xh+7h^2}h

Then as <em>h</em> approaches 0 - bearing in mind that we're specifically considering <em>h</em> <em>near</em> 0, and not <em>h</em> = 0 - we can eliminate the factor of <em>h</em> in the numerator and denominator, so that

m_{\rm tan} = \displaystyle \lim_{h\to0}\frac{f(x+h)-f(x)}h = \lim_{h\to0}\frac{14xh+7h^2}h = \lim_{h\to0}(14x+7h) = 14x

and so the slope of the line at <em>P</em> (0, 9), for which we take <em>x</em> = 0, is 0.

(b) The equation of the tangent line is then <em>y</em> = 9.

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