Calculus · Tangents & Derivatives
Drag the point of tangency, compare secant and tangent slopes, locate stationary points, and connect differentiation to optimization.
NCERT alignment: Class 12; chapter Applications of Derivatives. Exam practice: JEE Main, JEE Advanced.
Learning objectives
Students should use the live simulator to test a quantitative prediction, read measurements and explain the limits of the model. Do not rely on the animation alone: calculate the expected result before adjusting controls.
Theory, equations and interpretations
Secant and tangent
The secant line joins two graph points. Its slope approaches the tangent slope as h tends to zero.
Important relation: mₛ = [f(x₀+h)−f(x₀)]/h
Definition of derivative
For differentiable functions this limit is the instantaneous rate of change, or tangent slope.
Important relation: f′(x₀) = lim(h→0) [f(x₀+h)−f(x₀)]/h
Power and trigonometric rules
The lab differentiates the selected cubic, quadratic, or sine analytically. Sine uses radians.
Important relation: d(xⁿ)/dx = nxⁿ⁻¹; d(sin x)/dx = cos x
Stationary points
At a stationary point the tangent is horizontal. A sign change in f′ helps classify a local maximum or minimum.
Important relation: f′(x)=0 is necessary at an interior differentiable extremum
Practical investigations
- Shrink the secant: Hold x₀=0.5 on the cubic and reduce h from 2 to 0.1; compare both slopes.
- Locate minima: Switch to quadratic. Slide x₀ to 1 and explain why f′(1)=0.
- Classify a cubic: For the cubic, find both points with horizontal tangents (x=−1 and x=1) and classify them.
- Sine slope: Switch to sine and compare f′(0)=a with f′(π/2)=0 (x≈1.57).
Worked exam-style questions and answers
Q1. Derivative of x³−3x is
Options: A. 3x²−3; B. x²−3; C. 3x²+3; D. x³−3
Answer: A. Differentiate term by term: 3x²−3.
Q2. Slope of y=x² at x=2 is
Options: A. 2; B. 4; C. 6; D. 8
Answer: B. f′(x)=2x, hence f′(2)=4.
Q3. Stationary point of y=(x−1)² is at
Options: A. −1; B. 0; C. 1; D. 2
Answer: C. Derivative 2(x−1) vanishes at x=1.
Q4. lim(h→0) [sin h]/h equals
Options: A. 0; B. 1; C. ∞; D. −1
Answer: B. Fundamental trigonometric limit (h in radians).
Q5. The derivative of sin x is
Options: A. −cos x; B. sin x; C. cos x; D. −sin x
Answer: C. d(sin x)/dx=cos x for radians.
Q6. At a local differentiable interior maximum, necessarily
Options: A. f′=0; B. f′>0; C. f′<0; D. f=0
Answer: A. Fermat theorem: f′(x)=0 at an interior differentiable extremum.
Common misconceptions and limitations
The derivative is exact for the three displayed differentiable families. The secant difference quotient approximates the tangent only as h→0. This laboratory does not claim to test discontinuities or cusps.
All learning content is available free of charge. The simulation is a simplified educational model. Check assumptions and units before applying a result. NANDAX SimStudio is not affiliated with NCERT or examination boards.
Teacher classroom use
Ask students to form a hypothesis, vary one parameter at a time, keep a record of graph readings, explain any change and compare results with the analytic relation. The on-screen worksheet offers questions with an answer key.