Calculus Volume 1 · 免费预览
导数与图像的形状
一阶导数的符号表揭示图像何处上升、何处下降，二阶导数的符号表揭示图像何处向上弯曲、何处向下弯曲；两者结合渐近线，便可拼合出完整的定性图像。
1. Increase, decrease, and local extrema
Increase, decrease, and local extrema
2. Concavity and the second derivative
Concavity and the second derivative
3. The full curve-sketching protocol
The full curve-sketching protocol
4. A worked sketch
A worked sketch
5. Sign chart reading
The derivative f prime is positive on (-3, 0) and negative on (0, 2). What does f do on those intervals?
6. Second derivative and concavity
On an interval, f double prime of x is negative everywhere. What does that say about the graph of f there?
7. Inflection point condition
Which condition identifies an inflection point of f at an interior point c?
8. Cubic local maximum
For f(x) = x^3 - 3x^2 - 9x + 5 with f prime of x = 3(x+1)(x-3), which point is the local maximum?
9. Concavity of the cubic
For the same cubic, f double prime of x = 6x - 6. Where is the graph concave up?
10. Vanishing second derivative at a non-inflection point
The function f(x) = x^4 has f double prime of 0 = 0, yet the origin is NOT an inflection point. Why not?
11. Cubic local maximum value
For f(x) = x^3 - 3x^2 - 9x + 5, the local maximum occurs at x = -1. What is the value f(-1)? Answer with a number.
12. Inflection point location
The graph of f has f double prime of x = 6x - 6. At what x does the inflection point occur? Answer with a number.
13. Sign chart interval count
The real line is partitioned by two critical points, x = -1 and x = 3. How many intervals of constant sign for f prime result? Answer with a number.
14. Decreasing stretch endpoints
The cubic f falls on the interval (-1, 3). What is the length of this decreasing stretch? Answer with a number.