Calculus Volume 2 · 免费预览
参数曲线的微积分
参数曲线的求导运用链式法则，dy/dx 等于 dy/dt 除以 dx/dt；积分则利用速度以及经调整以适应参数的旋转曲面公式，计算弧长和表面积。
1. Parametric differentiation
Parametric differentiation
2. Velocity and speed
Velocity and speed
3. Arc length parametrically
Arc length parametrically
4. Surface area of revolution
Surface area of revolution
5. Parametric slope formula
For a parametric curve x = x(t), y = y(t), the slope dy/dx equals which ratio?
6. Vertical tangent condition
A parametric curve has a vertical tangent where which condition holds?
7. Circle circumference by parametric arc length
The parametric arc length integral for the unit circle x = cos t, y = sin t over [0, 2 pi] gives what integrand?
8. Cycloid arch length
The arc length of one cycloid arch works out to 8r. Compared to the rolling circle's circumference 2 pi r, the arch is about how many times longer?
9. Singular point condition
At a parametric curve's singular point, both dx/dt and dy/dt vanish. What behavior typically occurs there?
10. Projectile peak speed
At the projectile's peak, the vertical velocity vanishes and the speed equals the horizontal component v0 cos theta. As a function name, what trig function appears? Answer in lowercase.
11. Speed integrand
The arc length integrand is the speed root of (x')^2 plus (y')^2. As a single word, what physical quantity is it? Answer in lowercase.
12. Second derivative denominator
The second derivative d^2y/dx^2 divides d/dt of dy/dx by dx/dt. As a derivative, what is the denominator? Answer like dx/dt.
13. Surface formula axis change
Rotating a parametric curve about the y axis instead of the x axis changes the surface area formula how?
14. Surface band radius
Rotating the parametric curve about the x axis gives surface area 2 pi integral of y times speed dt. As a single letter, what is the band's radius? Answer with one lowercase letter.