Continuity and Differentiability — Class 12 CBSE Mathematics revision notes
A 3-page visual revision sheet for Continuity and Differentiability, written for Class 12 CBSE Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
What this Continuity and Differentiability sheet covers
- Continuity at a point
- condition 1: f(a) is defined, a real number actually comes out
- condition 2: the limit approached with x below a matches the limit approached with x above a
- condition 3: that common limit equals the function's own value at a
- What the pictures show
- Three ways continuity fails at x=a
- Check continuity of a piecewise function at x=1
- Algebra of continuous functions
- f and g both continuous at a: their sum, difference and product are too
- their quotient is continuous at a as well, needs g(a) not zero
- a continuous function of a continuous function stays continuous
- Differentiability
- Chain rule
- y depends on u, and u depends on x
- one more layer chains the same way
- Standard derivatives: inverse trig, exponential, log
- d/dx of sin inverse x
- d/dx of tan inverse x
- d/dx of e to the x
- d/dx of natural log of x, for x>0
- Differentiate y = x^x using logarithmic differentiation
- Rolle's theorem
- conditions: f continuous on [a,b], differentiable on (a,b), f(a)=f(b); conclusion holds for some c strictly between a and b
- Lagrange's mean value theorem
- conditions: f continuous on [a,b], differentiable on (a,b), no equal-endpoint condition needed; conclusion holds for some c strictly between a and b
- Verify Rolle's theorem
- Reading the chapter together
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Continuity and Differentiability the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.