Continuity and Differentiability — Flashcards
Punjab Board · Class 12 · Mathematics
25 flashcards for Continuity and Differentiability (Punjab Board Class 12 Mathematics) to test yourself on key terms and facts.
Interactive on Super Tutor
Studying Continuity and Differentiability? Get the full interactive chapter.
Quizzes, flashcards, AI doubt-solver and a step-by-step study plan — built for flashcards and more.
Free trial, no card needed.
Learn better with visuals Super Tutor pairs illustrations like this with notes and quizzes for Continuity and Differentiability.
Check if f(x) = 2x + 3 is continuous at x = 1.
Answer
To check continuity at x = 1, verify that lim(x→1) f(x) = f(1). Step 1: Find f(1) = 2(1) + 3 = 5 Step 2: Find lim(x→1) f(x) = lim(x→1) (2x + 3) = 2(1) + 3 = 5 Step 3: Compare: lim(x→1) f(x) = 5 = f…
Show that f(x) = |x| is continuous at x = 0.
Answer
For f(x) = |x| = {-x if x < 0; x if x ≥ 0}, check continuity at x = 0. Step 1: Find f(0) = |0| = 0 Step 2: Find left-hand limit: lim(x→0⁻) f(x) = lim(x→0⁻) (-x) = 0 Step 3: Find right-hand limit: l…
Find all points of discontinuity of f(x) = [x] (greatest integer function) on [0, 3).
Answer
For f(x) = [x], analyze at each integer point in [0, 3). At x = 0: - Left limit: lim(x→0⁻) [x] doesn't exist in domain - Right limit: lim(x→0⁺) [x] = 0 - f(0) = 0 → Continuous At x = 1: - Left limit…
Find the value of k so that f(x) = {kx² if x ≤ 2; 3 if x > 2} is continuous at x = 2.
Answer
For continuity at x = 2, we need lim(x→2⁻) f(x) = lim(x→2⁺) f(x) = f(2). Step 1: Find left-hand limit: lim(x→2⁻) f(x) = lim(x→2⁻) kx² = k(2)² = 4k Step 2: Find right-hand limit: lim(x→2⁺) f(x) = 3 (…
When do you use the definition of continuity? Give the three-part condition.
Answer
Use the continuity definition when asked to verify or prove a function is continuous at a point c. The THREE CONDITIONS that must ALL be satisfied: 1. f(c) must be defined (the function exists at x …
Prove that f(x) = |x| is NOT differentiable at x = 0.
Answer
To show non-differentiability, prove the left and right derivatives are not equal. For f(x) = |x| = {-x if x < 0; x if x ≥ 0} Step 1: Find left-hand derivative at x = 0: f'(0⁻) = lim(h→0⁻) [f(0+h) -…
What is the relationship between continuity and differentiability? (Theorem 3)
Answer
THEOREM: If f is differentiable at c, then f is continuous at c. Important understanding: - Differentiability IMPLIES continuity - Continuous ≠ Differentiable (converse is FALSE) In symbols: f'(c) e…
Apply the Chain Rule to find d/dx[sin(x²)].
Answer
The Chain Rule: If f = v∘u, then df/dx = (dv/dt)(dt/dx) where t = u(x). For f(x) = sin(x²): Step 1: Identify composition - Inner function: u(x) = x² - Outer function: v(t) = sin(t) - Therefore: f(x)…
+17 more flashcards
Practise AllFrequently Asked Questions
What are the important topics in Continuity and Differentiability for Punjab Board Class 12 Mathematics?
How many flashcards are available for Continuity and Differentiability?
How should I revise Continuity and Differentiability for the Punjab Board Class 12 board exam?
Sources & Official References
Content is aligned to the official syllabus. Refer to the board website for the latest curriculum.
More resources for Continuity and Differentiability
Practice Quiz
Test yourself with a quick quiz
Important Questions
Exam-style questions with answers
Revision Notes
Key points for last-minute revision
Formula Sheet
The chapter's formulas in one place
Chapter Summary
Understand the chapter at a glance
Concept Maps
See how topics connect
Syllabus
What topics to cover
For serious students
Get the full Continuity and Differentiability chapter — start free.
Quizzes, flashcards, an AI doubt solver and a study plan for Punjab Board Class 12 Mathematics. Free to start, no card needed.