Secants, limits and first-principles power derivatives
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| difference quotient/ˈdɪfrəns ˈkwəʊʃənt/ | thương sai phân |
What happens to their average rate as the interval shrinks?
- A distance sensor gives two positions a short time apart.
- This lesson studies difference quotient 差商: The change in function output divided by a nonzero change in input.
The method
- For h≠0, the secant gradient is [f(x+h)−f(x)]/h.
- The derivative f′(x) is its limit as h tends to zero, if the two-sided finite limit exists.
- It gives the tangent gradient and instantaneous rate.
- The variable h may approach zero from either sign; simplify before taking the limit.
Why can h=0 not be substituted directly into [f(x+h)-f(x)]/h?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Cancelling h is valid for nonzero h: for x² the quotient is exactly 2x+h, whose limit as h→0 is 2x. The derivative is that limit, not the value at h=0.
A worked case
For f(x)=x², [(x+h)²−x²]/h=2x+h, so f′(x)=2x. At x=1, h=1,0.5,−0.5 give secant gradients 3,2.5,1.5, tending to 2 from both sides. The tangent through (1,1) is y−1=2(x−1). For f(x)=x³, expanding gives 3x²+3xh+h², hence f′(x)=3x². A constant gives zero and f(x)=x gives one. For any positive integer n, the binomial expansion leaves nx^(n−1) plus terms containing h, so the limit is nx^(n−1). This argument alone does not prove the rule for rational powers.
Secants, limits and first-principles power derivatives
For h≠0, the secant gradient is [f(x+h)−f(x)]/h
Connect a derivative calculation to its limit or gradient sign interpretation.
Find the derivative of x² at x=1.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The simplified difference quotient tends to 2x; at x=1 this is 2.
Where it goes wrong
- Substituting h=0 into the unsimplified quotient gives 0/0, not a gradient.
- Cancelling h is valid for h≠0 before the limit.
- Continuity alone is insufficient: at x=0, |x| has right quotient 1 and left quotient −1, so no derivative there.
- The tangent is a local line, not an exact replacement for the entire curve.
Not true: A difference quotient can be evaluated by setting h=0 before simplification. Setting h=0 first gives 0/0; simplify and cancel h, then let h tend to 0.
For x³, find f′(2).
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The cubic quotient tends to 3x²; at x=2 this is 12.
Setting h=0 first gives 0/0.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. Setting h=0 first gives 0/0; simplify and cancel h, then let h tend to 0.
Applying it
- Write the difference quotient, expand, cancel only with h≠0 stated, and take the limit.
- Interpret derivative units as output units divided by input units.
- When a question says first principles, merely quoting nx^(n−1) does not establish the limit.
- At a corner check both approach directions.
For x² at x=1 with h=0.5, find the secant gradient.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
2x+h at x=1,h=0.5 gives 2.5.
Match each first-principles concept to its meaning.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The limit must agree from both sides; continuity alone, as |x| shows, does not guarantee differentiability.
In your course
- AQA A-level Mathematics (7357), specification section G.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Difference quotient: The change in function output divided by a nonzero change in input.
- For h≠0, the secant gradient is [f(x+h)−f(x)]/h.
- Watch for: substituting h=0 into the unsimplified quotient gives 0/0, not a gradient.