(a) mtan refers to the slope of the tangent line. Given f(x) = 9 + 7x ², compute the difference quotient:
[tex]\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[/tex]
Then as h approaches 0 - bearing in mind that we're specifically considering h near 0, and not h = 0 - we can eliminate the factor of h in the numerator and denominator, so that
[tex]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[/tex]
and so the slope of the line at P (0, 9), for which we take x = 0, is 0.
(b) The equation of the tangent line is then y = 9.
