Tool to calculate the tangent to a curve, to a function, at a given point (infinity close to this point) and to find its tangent line equation as a function of the variable x.

Tangent to a Curve - dCode

Tag(s) : Geometry

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In geometry, **a tangent** to a curve is a straight line that approaches / caresses / touches the curve at this point so as to form an angle equal to 0.

The equation of **the tangent** at $ x = a $ is calculated from the equation of the curve $ f(x) $, by applying a limit calculation and a derivative calculation.

Calculate the limit $$ \lim_{h \to 0} \frac{f(a+h)-f(a)}{h} $$

If the limit is indeterminate, then there is no tangent at this point (the function is not differentiable in $ x = a $)

If the limit is infinite ($ +\infty $ or $ -\infty $), then the line of equation $ x = a $ is **a tangent** in $ a $ (and also an asymptote)

Otherwise (the limit is finite, it has a value) then calculate $ f'(x) $ the derivative of $ f (x) $ in order to obtain the equation of **the tangent** in $ a $ which is defined by the formula $$ y = (x-a) \times f'(a) + f(a) $$

__Example:__ Determinate **the tangent** equation of $ f(x) = x^2 $ at the point $ x = 1 $, first calculate the limit $$ \lim_{h \to 0} \frac{(1+h)^2-1^2}{h} = \lim_{h \to 0} \frac{2h+h^2}{h} = \lim_{h \to 0} 2+h = 2 $$ this limit is therefore finite, the function is differentiable and its derivative is $ f'(x) = 2x $, so **the tangent** equation is $$ y = (x-a) \times f'(a) + f(a) \\ y = (x-1) \cdot 2 \cdot 1 + 1^2 \\ y = 2x-2+1 \\ y = 2x-1 $$

If it is already known that the function is differentiable in $ a $, then the calculation of the limit is not necessary and the formula can be applied directly.

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