Calculus Volume 1 · 免费预览
中值定理
中值定理把平均变化率与瞬时变化率联系起来，保证存在一点使切线与割线平行；由此可推出单调性、导数为零的常函数刻画以及函数的比较。
1. Rolle's theorem
Rolle's theorem
2. The mean value theorem
The mean value theorem
3. Consequences for monotonicity
Consequences for monotonicity
4. Bounding differences
Bounding differences
5. Rolle's theorem hypotheses
Which trio of hypotheses makes the conclusion of Rolle's theorem valid on [a, b]?
6. Mean value theorem conclusion
The mean value theorem applied to f on [a, b] guarantees a point c in (a, b) satisfying which equation?
7. Speeding ticket logic
A highway patrol photographs a car at mile 0 and again at mile 120 exactly 2 hours later. A mean value theorem argument supports what conclusion?
8. Zero derivative consequence
A differentiable function f satisfies f prime of x = 0 for every x in an interval. What does the mean value theorem imply about f on that interval?
9. Derivative sign and monotonicity
On an open interval, f prime of x is positive at every point. Which conclusion follows from the mean value theorem?
10. Proof reduction
The standard proof of the mean value theorem reduces it to Rolle's theorem by which construction?
11. Secant slope computation
For f(x) = x^2 on [1, 3], the secant slope (f(3) - f(1))/(3 - 1) equals what number? Answer with a number.
12. Where tangent parallels secant
For f(x) = x^2 on [1, 3], the mean value theorem point c satisfies f prime of c = 2c = 4. What is c? Answer with a number.
13. MVT hypothesis count
The mean value theorem requires continuity on the closed interval and differentiability on the open one. How many distinct hypotheses are there? Answer with a number.
14. Lipschitz bound
f(0) = 3 and |f prime of x| is at most 5 everywhere. By the mean value theorem bound, |f(2) - 3| is at most what number? Answer with a number.