Interpretation — instantaneous rate of change, slope of tangent: Question bank

4.1.11 · D5 · Maths › Calculus I — Limits & Derivatives › Interpretation — instantaneous rate of change, slope of tang

True or false — justify

Derivative

f^{'} (a)

ek ratio (ek fraction) hota hai point

x = a

par.

False. Ye ek ratio

\frac{f ( a + h ) - f ( a )}{h}

ka limit hai;

h = 0

par ratio khud

\frac{0}{0}

hota hai, jo undefined hai. Derivative woh single number hai jiske paas ratio approach karta hai, koi actual fraction nahi.

Agar

[a, b]

par average rate of change

5

hai, toh

f^{'} (a) = 5

Generally False. Average ek poore interval par ek secant slope hota hai; instantaneous rate limit

h \to 0

hota hai. Ye sirf straight lines mein coincide karte hain, jahan slope kabhi change nahi karta.

Ek tangent line curve ko exactly ek hi point par touch kar sakti hai.

False. "Tangent" ka matlab hai

f^{'} (a)

ke zariye locally slope match karna, na ki "ek baar touch karna."

y = x^{3}

ka tangent origin par, yaani

y = 0

, seedha wahan se curve ke through guzarta hai.

Agar

f

a

par continuous hai, toh

f^{'} (a)

zaroor exist karta hoga.

False. Continuity zaroori hai lekin kaafi nahi.

f (x) = ∣ x ∣

0

par continuous hai phir bhi ek corner hai, isliye koi single tangent slope exist nahi karta. Dekho Differentiability and continuity.

Agar

f^{'} (a)

exist karta hai, toh

f

a

par continuous hai.

True. Differentiability continuity ko force karti hai: ek jump ya hole ki wajah se

f (a + h) - f (a)

0

tak shrink nahi hoga, isliye

f^{'} (a)

define karne wala limit exist hi nahi kar sakta.

f^{'} (a) = 0

ka matlab hai function

a

ke aaspaas kahin bhi change nahi ho raha.

False. Iska matlab hai us exact point par instantaneous rate zero hai (ek flat tangent). Function dono taraf rapidly change ho sakta hai — ek peak ya valley mein

f^{'} = 0

hota hai lekin woh constant se bahut dur hota hai.

Secant slope hamesha tangent slope ko overestimate karta hai.

False. Secant tangent se steep hai ya nahi, ye curve ke bend (concavity) aur

h

ke sign par depend karta hai. Koi universal direction nahi hai. Dekho Average rate of change.

Dono limit forms

lim_{h \to 0} \frac{f ( a + h ) - f ( a )}{h}

aur

lim_{x \to a} \frac{f ( x ) - f ( a )}{x - a}

same value dete hain jab derivative exist karta hai.

True. Ye same expression hai substitution

x = a + h

(isliye

h = x - a

) ke saath. Jaise

h \to 0

x \to a

; aur kuch nahi badalta.

Negative derivative ka matlab hai function ka graph

x

-axis ke neeche hai.

False. $f^{'}$ ka sign slope direction (decreasing) describe karta hai,

f

ka sign nahi. Ek curve axis ke upar high baith sakta hai aur phir bhi neeche ki taraf slope kar sakta hai, giving a positive value lekin negative derivative.

Spot the error

\frac{f ( a + h ) - f ( a )}{h}

h = 0

equals

\frac{f ( a ) - f ( a )}{0} = \frac{0}{0} = 0

Error ye hai:

\frac{0}{0}

indeterminate hai,

0

nahi. Limit lene se pehle algebraically simplify karna padta hai (cancel the

h

) — "Simplify Before Limit."

"Velocity distance over time hoti hai, isliye instantaneous velocity

\frac{s ( 2 )}{2}

hai."

t = 0

t = 2

tak ki average velocity hai, instantaneous nahi. Instantaneous velocity hai

lim_{h \to 0} \frac{s ( 2 + h ) - s ( 2 )}{h} = s^{'} (2)

. Dekho Velocity and acceleration.

x = a

par tangent line

y = f^{'} (a) x

hai."

Point missing hai. Ek line ko ek point aur ek slope dono chahiye:

y - f (a) = f^{'} (a) (x - a)

. Unki version bhool jaati hai ki curve

(a, f (a))

se guzarti hai, origin se nahi.

∣ x ∣

ka derivative

x = 0

par

0

hai kyunki graph wahan apne lowest point par pahunchta hai."

Minimum hona derivative guarantee nahi karta. One-sided slopes

- 1

(left) aur

+ 1

(right) hain; ye agree nahi karte, isliye two-sided limit — aur hence

f^{'} (0)

— exist nahi karta.

"Jaise

h \to 0

secant ban jaata hai tangent, isliye small

h

ke liye secant slope tangent slope ke barabar hota hai."

Kisi bhi actual small

h \neq = 0

ke liye secant slope sirf approximately tangent slope ke barabar hota hai. Equality sirf limit mein hoti hai, kisi real nonzero

h

par nahi.

"Kyunki

f^{'} (a)

do points

a

aur

a + h

use karta hai, ye ek interval ki property hai."

Do points sirf scaffolding hain; limit unhe single point

a

par collapse kar deta hai. Derivative ek point par local property hai, interval property nahi.

Why questions

Hum pehle secant kyun banate hain tangent slope directly compute karne ki bajaye?

Secant do genuine points use karta hai, isliye uska rise-over-run bina kisi division by zero ke honestly computable hai. Tangent tak pahunchne ke liye ek limiting process chahiye, jo hume secant ko shrink karke milta hai.

Limit lene se pehle

h

ko cancel kyun karna padta hai?

\frac{0}{0}

ki problem poori tarah us shared factor of

h

mein rehti hai. Cancelling illegal division ko hatata hai jabki

h \neq = 0

, ek aisa expression chodta hai jiska limit hum substitution se evaluate kar sakte hain.

Wahi number

f^{'} (a)

"slope of tangent" aur "rate of change" dono kyun mean karta hai?

Tangent

a

ke paas curve ka best straight-line model hai; uska slope

\frac{rise}{run} = \frac{change in output}{change in input}

hai, jo exactly ek rate of change hai. Same idea, do costumes.

Derivative ko two-sided limit kyun chahiye?

Ek well-defined tangent slope ko dono directions se identically approach karna chahiye. Agar left aur right alag limits dein (ek corner), toh curve ko graze karne wali koi single line nahi hai, isliye koi derivative nahi.

Average rate ek curved graph ke behaviour ko describe karne ke liye kaafi kyun nahi hai?

Curve par slope point to point change hota hai, isliye

[a, b]

par ek average saari variation chhupa deta hai. Sirf limit

h \to 0

hi har specific instant par rate pin down karta hai. Dekho Derivative as a function.

a

ko vary karne se

f^{'} (a)

ek poora naya function

f^{'} (x)

kyun ban jaata hai?

Limit recipe har us point par kaam karti hai jahan exist karti hai, har input ke liye ek slope output karti hai. Un saare outputs ko collect karne se ek function milta hai — derivative as a function, jise phir Power Rule jaisi shortcuts directly compute karti hain.

Edge cases

Ek sharp corner par (jaise

∣ x ∣

0

), exactly kya fail hota hai?

One-sided difference quotients alag values (

- 1

aur

+ 1

) ko approach karte hain, isliye two-sided limit exist nahi karta aur koi unique tangent slope nahi hota.

Ek vertical tangent par (e.g.

y = x^{1/3}

x = 0

), kya

f^{'} (0)

exist karta hai?

Koi finite value nahi: difference quotient

+ \infty

tak blow up karta hai. Tangent vertical hai (undefined slope), isliye

f^{'} (0)

ek real number ke roop mein exist nahi karta, chahe curve smooth dikhti ho.

Agar

f

ek constant function hai, toh

f^{'} (a)

har jagah kya hai, aur kyun?

Zero. Har difference quotient

\frac{c - c}{h} = 0

hota hai sabhi

h \neq = 0

ke liye, isliye limit

0

hai — ek horizontal, flat tangent, "not changing" se match karta hai.

Agar

f

ek straight line

y = m x + b

hai, toh average aur instantaneous rates kaise compare hote hain?

Ye identical hain aur har jagah

m

ke barabar hain. Difference quotient har

h

ke liye

m

tak simplify hota hai, isliye koi limiting subtlety nahi — ye woh ek case hai jahan average equals instantaneous.

Jump discontinuity par difference quotient ka kya hota hai?

Jaise

h \to 0

jump ke across,

f (a + h) - f (a)

0

tak shrink nahi karta, isliye ek tiny

h

se divide karne par quotient

\pm \infty

par chala jaata hai. Derivative exist nahi karta — discontinuity differentiability ko khatam kar deti hai.

Connections

Average rate of change — secant slope jiske against yahan har "true/false" contrast karta hai.
Limits — formal definition — kyun "approaches" rigorous hai, hand-waving nahi.
Derivative as a function — "letting $a$ vary" edge/why item.
Power Rule — woh shortcut jo ye limits justify karte hain.
Differentiability and continuity — corner, jump, aur vertical-tangent cases.
Velocity and acceleration — average-vs-instantaneous velocity trap.