Calculus Volume 1 · 免费预览
微积分预览
微积分回答两个古老的问题：求瞬时变化率的切线问题，以及求累积量的面积问题。二者都是极限问题，而整门课程正是围绕它们展开，逐步给出答案。
1. Two ancient questions
Two ancient questions
2. The tangent problem and velocity
The tangent problem and velocity
3. The area problem
The area problem
4. Why limits unify the course
Why limits unify the course
5. What the tangent problem asks for
What exactly does the tangent problem ask the calculus student to find?
6. Why substitution fails for instantaneous velocity
For a position curve s(t), the average velocity between t = a and t = b is (s(b) - s(a))/(b - a). What goes wrong if we set b = a to get the velocity at the instant a?
7. What improves as rectangles thin
In the rectangle approximation of the area under a curve, what happens as the strips are made thinner and more numerous?
8. The unifying idea of calculus
Which single concept do both the derivative and the definite integral reduce to?
9. Speedometer reading
A car's speedometer reports the number that calculus calls instantaneous velocity. Which construction produces that number from the position function?
10. Area versus tangent
Which founding problem of calculus corresponds to computing the area of a region bounded by a curve?
11. Area of a triangle by limit thinking
A triangle of base 4 and height 8 has area one half base times height. What is its area? Answer with a number.
12. Secant slope on a line
The position is s(t) = 5t. What is the average velocity, the secant slope, between t = 1 and t = 3? Answer with a number.
13. The indeterminate form
Setting b equal to a in the difference quotient produces the meaningless fraction 0 over 0. What two-letter name is given to this form? Answer in lowercase.
14. Number of founding problems
The preview names two founding problems of calculus, the tangent problem and the area problem. How many founding problems are there? Answer with a number.