Calculus Volume 1 · 免费预览
应用优化问题
优化把实际问题转化为一元函数在闭区间上的最值问题，再用闭区间法求出其绝对极值；而真正关键的工作在于建立约束和确定定义域这些建模步骤。
1. From situation to function
From situation to function
2. Constraints and domains
Constraints and domains
3. Interpreting and verifying
Interpreting and verifying
4. Optimization in economics
Optimization in economics
5. First modeling step
In solving an applied optimization problem, which step comes first?
6. Why the domain matters
Why must the domain of an applied optimization function be checked before differentiating?
7. Box problem optimum
The open box from a 12 unit square sheet has volume V(x) = x(12 - 2x)^2 on [0, 6]. What cut size x maximizes the volume?
8. Fence constraint elimination
A rectangular field against a river uses 100 meters of fence on three sides, area A = x y with 2x + y = 100. What is the area as a function of x alone?
9. Endpoint versus interior optimum
Under what conditions does an applied optimization's optimum occur at an endpoint rather than an interior critical point?
10. Verifying the optimum
In the box problem, besides the endpoint comparison, which additional check certifies x = 2 as a maximum?
11. Box maximum volume
The open box volume V(x) = x(12 - 2x)^2 on [0, 6] is maximized at x = 2. What is the maximum volume? Answer with a number.
12. Fence optimal width
The fenced area A(x) = x(100 - 2x) has derivative 100 - 4x. At what x is the area maximized? Answer with a number.
13. Domain right endpoint
In the box problem the domain of V(x) = x(12 - 2x)^2 is [0, 6]. What is the right endpoint of the domain? Answer with a number.
14. Second derivative check
For A(x) = x(100 - 2x), the second derivative is constant. What is its value? Answer with a number.