Secant to tangent
"Secant to tangent" is a free, interactive Class 12 Maths simulation for JEE, built on the lesson "Differentiability" in Continuity and Differentiability. P is fixed at x = c and Q sits at x = c + h. The orange line through them is a chord (secant), with slope [f(c + h) − f(c)]/h. Slide Q toward P from both sides and watch the slope. F′(c) is the limit of chord slopes as h → 0. It exists only when the slopes from the left and from the right settle on the same finite number. A smooth curve gives one tangent; a corner like |x| at 0 gives two different slopes. JEE Main asked 48 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
From the lesson Differentiability in Continuity and Differentiability. Change the values and watch what happens.
In JEE Main
42 shifts analysed
48 questions from Continuity and Differentiability across the 42 JEE Main shifts Lumi has analysed (2017–2026).
The idea behind it
NCERT §5.3
- The derivative of f at c is f′(c), the limit of [f(c + h) − f(c)]/h as h → 0, provided it exists. Geometrically it is the slope of the tangent: the limit of the slopes of chords through (c, f(c)).
- f is differentiable at c when the left-hand derivative (h → 0⁻) and the right-hand derivative (h → 0⁺) are both finite and equal.
- Rules from Class XI: (u ± v)′ = u′ ± v′, (uv)′ = u′v + uv′, and (u/v)′ = (u′v − uv′)/v² where v ≠ 0. Standard results: (xⁿ)′ = nxⁿ⁻¹, (sin x)′ = cos x, (cos x)′ = −sin x, (tan x)′ = sec²x.
- Theorem: if f is differentiable at c, it is continuous at c. Proof idea: f(x) − f(c) equals the chord slope times (x − c), which tends to f′(c) × 0 = 0.
- So every differentiable function is continuous. The converse fails: |x| is continuous at 0, but its left-hand derivative there is −1 and its right-hand derivative is +1, so it is not differentiable at 0.
- Corners and jumps both spoil differentiability. |x − 1| is not differentiable at 1, and [x] is not differentiable at 1 or 2 (it is not even continuous there).
- A function is differentiable on [a, b] when it is differentiable at every point, using the one-sided derivative at each end.
Questions
3 answers
- What does the "Secant to tangent" simulation show?
- P is fixed at x = c and Q sits at x = c + h. The orange line through them is a chord (secant), with slope [f(c + h) − f(c)]/h. Slide Q toward P from both sides and watch the slope.
- Does JEE Main ask questions from Continuity and Differentiability?
- Yes. JEE Main asked 48 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
- Is this simulation free?
- Yes. It is free and runs in the browser on a phone or a laptop. It belongs to the lesson "Differentiability" in Continuity and Differentiability (Class 12 Maths).
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