What is the tangent of f(x)=2x-x^2,(a,0) ?

The tangent of f(x)=2x-x^2,(a,0) is

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tangent of f(x)=2x-x^2,(a,0)

tangent of

f (x) = 2 x - x^{2}, (a, 0)

y = - \frac{( - 2 a ^{2} + 4 a + 2 a a ( a - 2 ) - 2 a ( a - 2 ) ) a ( a - 2 )}{a ^{2} - 2 a} x + \frac{2 a ( - a ^{2} + 2 a + a a ( a - 2 ) - a ( a - 2 ) )}{a - 2} or y = \frac{( - 2 a ^{2} + 4 a - 2 a a ( a - 2 ) + 2 a ( a - 2 ) ) a ( a - 2 )}{a ^{2} - 2 a} x - \frac{2 a ( - a ^{2} + 2 a - a a ( a - 2 ) + a ( a - 2 ) )}{a - 2}

Solution steps

Slope of the tangent line at

(a, 0) : m = \frac{- x ^{2} + 2 x}{x - a}

Find the slope of

f (x) = 2 x - x^{2} : m = 2 - 2 x

Solve

2 - 2 x = \frac{- x ^{2} + 2 x}{x - a} : x = a - a (a - 2), x = a + a (a - 2)

Find the tangent line at

x = a - a (a - 2) : y = - \frac{( - 2 a ^{2} + 4 a + 2 a a ( a - 2 ) - 2 a ( a - 2 ) ) a ( a - 2 )}{a ^{2} - 2 a} x + \frac{2 a ( - a ^{2} + 2 a + a a ( a - 2 ) - a ( a - 2 ) )}{a - 2}

Find the tangent line at

x = a + a (a - 2) : y = \frac{( - 2 a ^{2} + 4 a - 2 a a ( a - 2 ) + 2 a ( a - 2 ) ) a ( a - 2 )}{a ^{2} - 2 a} x - \frac{2 a ( - a ^{2} + 2 a - a a ( a - 2 ) + a ( a - 2 ) )}{a - 2}

y = - \frac{( - 2 a ^{2} + 4 a + 2 a a ( a - 2 ) - 2 a ( a - 2 ) ) a ( a - 2 )}{a ^{2} - 2 a} x + \frac{2 a ( - a ^{2} + 2 a + a a ( a - 2 ) - a ( a - 2 ) )}{a - 2} or y = \frac{( - 2 a ^{2} + 4 a - 2 a a ( a - 2 ) + 2 a ( a - 2 ) ) a ( a - 2 )}{a ^{2} - 2 a} x - \frac{2 a ( - a ^{2} + 2 a - a a ( a - 2 ) + a ( a - 2 ) )}{a - 2}

\frac{d}{d x} ((3 x^{2} + 1)^{40} (4 x - 7)^{50})

\int \frac{x}{27 + 6 x - x ^{2}} d x

tan^{40} (sec (cos (x)))

\frac{4}{s ^{2} ( s + 1 )}

n \to \infty lim (\frac{3 ^{n + 1}}{( n ) !})