Single-variable calculus and applications
| English | 中文 | Pinyin |
|---|---|---|
| continuity/kɒntɪˈnjuːɪti/ | 连续性 | lián xù xìng |
| convergence/kənˈvɜːdʒəns/ | 收敛 | shōu liǎn |
A decision before an answer
- A function can be continuous at zero while having no derivative there. A graph’s corner tests that distinction.
- Your goal: Use limits, continuity and differentiability.
Read the relationship
- A limit describes values near a point, without requiring the function to be defined there. Continuity adds the requirement that the function value exists and agrees with the limit. Algebraic cancellation is valid only away from the cancelled zero, but can reveal a removable limit. One-sided limits must agree for a two-sided limit. For quotient limits, check the denominator and hypotheses before applying a rule; 0/0 is an indeterminate form, not an answer.
- Apply derivatives, integrals and the fundamental theorem.
f(x)=|x| at 0 is:
Left and right derivatives are −1 and 1.
Use the defining rule
- A derivative is a limit of difference quotients and implies continuity; the converse fails, as |x| at zero shows. Product, quotient and chain rules describe different structures: the derivative of f(g(x)) is f′(g(x))g′(x), not a product of unrelated values. Differentiating a composition a second time generally produces two terms. Check domains, nonzero denominators and differentiability assumptions before using a symbolic expression as a derivative.
- Recognise Riemann sums and distinguish them from infinite series.
The series sum 1/n for n≥1:
The harmonic series diverges; the term test alone is insufficient.
Check the conditions
- The fundamental theorem says that the derivative of ∫ from a to x of a continuous integrand f(t) is f(x). With a variable upper bound h(x), multiply by h′(x); with two variable bounds, subtract the corresponding lower-bound contribution. A definite integral also arises as a limit of Riemann sums. Rewrite the sum as (1/n)Σf(k/n) before identifying the integral on [0,1], rather than treating n-dependent terms as constants.
- Recognise Riemann sums and distinguish them from infinite series.
For F(x)=integral from 0 to x² of e^t dt, the fundamental theorem and chain rule give F′(x)=e^(x²)·2x. At x=1, F′(1)=2e. The upper limit is x², so omitting 2x misses its rate of change.
Derivative of ln x at x=2 is ____.
d(ln x)/dx=1/x for x>0.
Apply the task format
- For example Σ from k=1 to n of n/(n²+k²) equals (1/n)Σ1/(1+(k/n)²), tending to ∫₀¹1/(1+t²)dt=π/4. This limit uses continuity and the partition width 1/n. A series over an unbounded number of terms is a different limit: terms tending to zero do not alone guarantee convergence. For Σ1/k the partial sums diverge; compare, estimate or apply a valid series test instead of using the necessary term condition as sufficient.
- Recognise Riemann sums and distinguish them from infinite series.
A-level calculus is useful prerequisite material but does not cover the undergraduate analysis and applications tested here.
Which answer fits this case?
Use limits, continuity and differentiability
Every continuous real function is differentiable.
Absolute value at zero is a counterexample.
Keep the distinctions
- continuity 连续性 — Agreement of a function value with its limit.
- convergence 收敛 — Approach to a finite limiting value.
- Use limits, continuity and differentiability.
- Apply derivatives, integrals and the fundamental theorem.
- Recognise Riemann sums and distinguish them from infinite series.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.