Calculus Volume 1 · 免费预览
积分公式与净变化定理
积分表可推广到复合函数，而净变化定理把变化率的定积分解释为总变化，从而统一了位移、累积成本以及一切由变化率求总量的计算。
1. Extending the formula table
Extending the formula table
2. The net change theorem
The net change theorem
3. Worked applications
Worked applications
4. The substitution preview
The substitution preview
5. Reverse chain rule pattern
The integral of x (x^2 + 1)^3 dx evaluates in one line to (x^2 + 1)^4/8 + C. Which pattern made this immediate?
6. Logarithmic integral pattern
Which integral has the form u prime over u and therefore evaluates to a logarithm?
7. Net change theorem statement
The net change theorem expresses the total change of a quantity Q with rate Q prime over [a, b] as what?
8. Displacement versus distance
A particle's velocity is negative on part of [a, b]. How does the displacement integral differ from the distance traveled?
9. Rate to total direction
In applications, which direction does the net change theorem most commonly enable?
10. Integral of marginal cost
A factory's marginal cost is C prime of q = 5 + 0.2q dollars per unit. The cost of increasing production from q = 0 to q = 10 is the integral of C prime, which equals what number? Answer with a number.
11. Units audit
A flow rate is measured in liters per minute and integrated over a time span in minutes. According to the net change theorem, what units must the resulting integral carry?
12. Displacement computation## practice-fill Displacement computation
Velocity v(t) = 2t on [0, 3] gives displacement the integral of 2t dt. What displacement results? Answer with a number.
13. Splitting for distance
To compute distance traveled from a velocity that changes sign once on [a, b], the integral of the absolute value must be split. Into how many pieces? Answer with a number.
14. Pumped volume
Water flows into a tank at rate r(t) = 20 - 0.5t liters per minute for t from 0 to 20 minutes. What total volume flows in? Answer with a number.