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

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

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

Commented by mr W last updated on 18/Apr/20

the derivative is the slope, this is correct. but whose slope? it is the slope of the tangent line! read the definition of derivative in your book, you′ll see.

$${the}\:{derivative}\:{is}\:{the}\:{slope},\:{this}\:{is} \\ $$$${correct}.\:{but}\:{whose}\:{slope}?\:{it}\:{is}\:{the} \\ $$$${slope}\:{of}\:{the}\:{tangent}\:{line}!\:{read}\:{the} \\ $$$${definition}\:{of}\:{derivative}\:{in}\:{your}\:{book}, \\ $$$${you}'{ll}\:{see}. \\ $$

Commented by M±th+et£s last updated on 18/Apr/20

that is right thank you but i want the tangent line definition not the slope of the tangent line

$${that}\:{is}\:{right}\:{thank}\:{you}\:{but}\:{i}\:{want}\:{the}\:{tangent}\:{line} \\ $$$${definition}\:{not}\:{the}\:{slope}\:{of}\:{the}\:{tangent} \\ $$$${line} \\ $$

Commented by Kunal12588 last updated on 18/Apr/20

if you know the slope of tangent and a point from where the tangent is passing. we can find eq^n of tangent

$${if}\:{you}\:{know}\:{the}\:{slope}\:{of}\:{tangent}\:{and}\:{a}\:{point} \\ $$$${from}\:{where}\:{the}\:{tangent}\:{is}\:{passing}.\:{we}\:{can}\:{find}\: \\ $$$${eq}^{{n}} \:{of}\:{tangent} \\ $$

Commented by M±th+et£s last updated on 19/Apr/20

thank you sir i know how can i find the eq^n but i was asking about the tangent line defintion

$${thank}\:{you}\:{sir}\:{i}\:{know}\:{how}\:{can}\:{i}\:{find} \\ $$$${the}\:{eq}^{{n}} \:{but}\:{i}\:{was}\:{asking}\:{about}\:{the}\:{tangent} \\ $$$${line}\:{defintion}\: \\ $$

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

$$\mathrm{the}\:\mathrm{tangent}\:\mathrm{line}\:\mathrm{to}\:\mathrm{a}\:\mathrm{curve}\:\mathrm{in}\:\mathrm{a}\:\mathrm{given}\:\mathrm{point} \\ $$$$\mathrm{is}\:\mathrm{defined}\:\mathrm{as}\:\mathrm{the}\:\mathrm{limit}\:\mathrm{of}\:\mathrm{secants}. \\ $$$$\mathrm{let}\:{y}={f}\left({x}\right)\:\mathrm{and}\:{P}=\begin{pmatrix}{{p}}\\{{f}\left({p}\right)}\end{pmatrix} \\ $$$$\mathrm{further}\:\mathrm{let}\:{O}=\begin{pmatrix}{{p}−{h}}\\{{f}\left({p}−{h}\right)}\end{pmatrix}\:\mathrm{and}\:{Q}=\begin{pmatrix}{{p}+{h}}\\{{f}\left({p}+{h}\right)}\end{pmatrix} \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{lines}\:{l}_{{h}} :\:{X}={O}+\lambda\overset{\rightharpoonup\:} {{OQ}}\:\mathrm{or} \\ $$$$\begin{cases}{{x}={p}−{h}+\mathrm{2}\lambda{h}}\\{{y}={f}\left({p}−{h}\right)+\lambda\left({f}\left({p}+{h}\right)−{f}\left({p}−{h}\right)\right)}\end{cases}\:\mathrm{with} \\ $$$${h}>\mathrm{0}\:\mathrm{and}\:\lambda\in\mathbb{R}\:\mathrm{are}\:\mathrm{secants}\:\mathrm{to}\:\mathrm{the}\:\mathrm{curve} \\ $$$${y}={f}\left({x}\right). \\ $$$$\mathrm{let}\:{h}\rightarrow\mathrm{0}\:\Rightarrow\:{l}_{\mathrm{0}} \:\mathrm{is}\:\mathrm{the}\:\mathrm{tangent}\:\mathrm{in}\:{P} \\ $$$$ \\ $$$$\mathrm{if}\:\mathrm{the}\:\mathrm{curve}\:\mathrm{is}\:\mathrm{concave}\:\mathrm{or}\:\mathrm{convex}\:\mathrm{in}\:{P},\:{l}_{\mathrm{0}} \\ $$$$\mathrm{touches}\:\mathrm{the}\:\mathrm{curve}\:\mathrm{in}\:{P}.\:\mathrm{if}\:\mathrm{the}\:\mathrm{curve}\:\mathrm{has}\:\mathrm{a} \\ $$$$\mathrm{turning}\:\mathrm{point}\:\left(=\:\mathrm{inflection}\right)\:\mathrm{in}\:{P},\:{l}_{\mathrm{0}} \: \\ $$$$\mathrm{intersects}\:\mathrm{the}\:\mathrm{curve}\:\mathrm{in}\:{P} \\ $$

Commented by M±th+et£s last updated on 18/Apr/20

i don′t know how can i thank for this god bless you sir

$${i}\:{don}'{t}\:{know}\:{how}\:{can}\:{i}\:{thank}\:{for}\:{this} \\ $$$${god}\:{bless}\:{you}\:{sir} \\ $$

Commented by MJS last updated on 18/Apr/20

you′re welcome “it is more blessed to give than it is to recieve”

$$\mathrm{you}'\mathrm{re}\:\mathrm{welcome} \\ $$$$``{it}\:{is}\:{more}\:{blessed}\:{to}\:{give}\:{than}\:{it}\:{is}\:{to}\:{recieve}'' \\ $$

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