hi everyone what is the definition of the tangent straight line to the curve without relying to the derivitve because the derivative is the slope and thanx for all


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Question Number 89698 by M±th+et£s last updated on 18/Apr/20

$h i e v e r y o n e$ $w h a t i s t h e d e f i n i t i o n o f t h e t a n g e n t$ $s t r a i g h t l i n e t o t h e c u r v e w i t h o u t r e l y i n g$ $t o t h e d e r i v i t v e$ $b e c a u s e t h e d e r i v a t i v e i s t h e s l o p e$ $a n d t h a n x f o r a l l$
Commented by mr W last updated on 18/Apr/20

$t h e d e r i v a t i v e i s t h e s l o p e, t h i s i s$ $c o r r e c t . b u t w h o s e s l o p e ? i t i s t h e$ $s l o p e o f t h e t a n g e n t l i n e! r e a d t h e$ $d e f i n i t i o n o f d e r i v a t i v e i n y o u r b o o k,$ ${y o u}^{'} l l s e e .$
Commented by M±th+et£s last updated on 18/Apr/20

$t h a t i s r i g h t t h a n k y o u b u t i w a n t t h e t a n g e n t l i n e$ $d e f i n i t i o n n o t t h e s l o p e o f t h e t a n g e n t$ $l i n e$
Commented by Kunal12588 last updated on 18/Apr/20

$i f y o u k n o w t h e s l o p e o f t a n g e n t a n d a p o i n t$ $f r o m w h e r e t h e t a n g e n t i s p a s s i n g . w e c a n f i n d$ ${e q}^{n} o f t a n g e n t$
Commented by M±th+et£s last updated on 19/Apr/20

$t h a n k y o u s i r i k n o w h o w c a n i f i n d$ $t h e {e q}^{n} b u t i w a s a s k i n g a b o u t t h e t a n g e n t$ $l i n e d e f i n t i o n$
	Answered by MJS last updated on 18/Apr/20
	$the tangent line to a curve in a given point is defined as the limit of secants. let y=f(x) and P= ((p),((f(p))) ) further let O= (((p−h)),((f(p−h))) ) and Q= (((p+h)),((f(p+h))) ) then the lines l_h : X=O+λOQ^(⇀ ) or { ((x=p−h+2λh)),((y=f(p−h)+λ(f(p+h)−f(p−h)))) :} with h>0 and λ∈R are secants to the curve y=f(x). let h→0 ⇒ l_0 is the tangent in P if the curve is concave or convex in P, l_0 touches the curve in P. if the curve has a turning point (= inflection) in P, l_0 intersects the curve in P$
	$the tangent line to a curve in a given point$ $is defined as the limit of secants .$ $let y = f (x) and P = (\begin{matrix} p \\ f (p) \end{matrix})$ $further let O = (\begin{matrix} p - h \\ f (p - h) \end{matrix}) and Q = (\begin{matrix} p + h \\ f (p + h) \end{matrix})$ $then the lines l_{h} : X = O + λ \overset{⇀}{O Q} or$ ${\begin{cases} x = p - h + 2 λ h \\ y = f (p - h) + λ (f (p + h) - f (p - h)) \end{cases} with$ $h > 0 and λ \in R are secants to the curve$ $y = f (x) .$ $let h \to 0 \Rightarrow l_{0} is the tangent in P$ $if the curve is concave or convex in P, l_{0}$ $touches the curve in P . if the curve has a$ $turning point (= inflection) in P, l_{0}$ $intersects the curve in P$
	Commented by M±th+et£s last updated on 18/Apr/20

	$i {d o n}^{'} t k n o w h o w c a n i t h a n k f o r t h i s$ $g o d b l e s s y o u s i r$
	Commented by MJS last updated on 18/Apr/20

	${you}^{'} re welcome$ $‘ ‘ i t i s m o r e b l e s s e d t o g i v e t h a n i t i s t o {r e c i e v e}^{″}$
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