C.8 · פאזה טריגונומטרית ועקומות פרמטריות
GRE Subject Test · GRE Mathematics נושא 23 3:17 קריאת קול באנגלית · תרגום אנגלי + סינית שרוף בתוך הסרטון
פרקים
תרגום טקסט
This unit distinguishes amplitude, angular frequency, phase angle and horizontal shift; eliminates a parameter while keeping its domain and direction; and uses parametric derivatives without assuming a vertical tangent is stationary.
本单元区分振幅、角频率、相位角和水平平移；在保留定义域和方向的前提下消去参数；并在不假设竖直切线即驻点的情况下使用参数导数。
For A cos of omega x minus phi, the amplitude is A, the period 2 pi over omega, and the horizontal shift phi over omega: the phase is not itself the shift.
对 A cos(ωx − φ)，振幅是 A，周期是 2π/ω，水平平移是 φ/ω：相位本身不是平移。
y equals minus 2 cos 3x becomes 2 cos of 3x minus pi, with period 2 pi over 3.
y = −2cos 3x 可写成 2cos(3x − π)，周期为 2π/3。
Its minima include x equals 0, and its maxima include x equals pi over 3.
它的极小值点包括 x = 0，极大值点包括 x = π/3。
Eliminating t describes a point set but can lose restrictions and direction.
消去 t 得到一个点集，但可能丢掉限制和方向。
x equals cos cubed t, y equals sin cubed t gives an astroid with cusps on the axes.
x = cos³t，y = sin³t 给出在坐标轴上有尖点的星形线。
It starts at 1, 0, passes 0, 1 at pi over 2, and runs counterclockwise.
它从 (1, 0) 出发，在 π/2 时经过 (0, 1)，逆时针运动。
Restricting t to 0, pi over 2 gives only the first-quadrant arc.
把 t 限制在 只得到第一象限的弧。
When dx dt is nonzero, the slope is dy dt over dx dt.
当 dx/dt ≠ 0 时，斜率是 (dy/dt)/(dx/dt)。
A horizontal tangent needs dy dt zero with dx dt nonzero; a vertical one reverses those.
水平切线需要 dy/dt = 0 且 dx/dt ≠ 0；竖直切线则相反。
If both vanish, take a limit: for t squared, t cubed, the slope 3t over 2 tends to zero.
若两者都为零，就取极限：对 (t², t³)，斜率 3t/2 趋于零。
Zero velocity is not by itself an extremum.
速度为零本身不代表极值。
For x equals 2 cos t, y equals sin t, elimination gives x squared over 4 plus y squared equals 1.
对 x = 2cos t，y = sin t，消参得 x²/4 + y² = 1。
At pi over 4, dx dt is minus root 2 and dy dt is root 2 over 2, so the slope is minus a half.
在 π/4 处，dx/dt = −√2，dy/dt = √2/2，所以斜率为 −1/2。
The full interval traces the ellipse once counterclockwise.
完整区间逆时针描出椭圆一次。
Phase angle and horizontal shift differ by the frequency factor.
相位角与水平平移相差一个频率因子。
An implicit equation can add untraced parts of a curve.
隐式方程可能加入参数并未描出的部分。
If both parameter derivatives vanish, use local reasoning instead of calling the point an extremum.
若两个参数导数都为零，要用局部推理，而不是把该点称为极值。
Amplitude three and period pi.
振幅为 3，周期为 π。
Writing 3 cos of 2x minus pi gives phase pi but a rightward shift of pi over 2: phase and shift differ.
写成 3cos(2x − π) 得相位 π，但向右平移 π/2：相位与平移不同。
The ellipse is traced by 3 cos t, 2 sin t, starting at 3, 0 and moving upward.
椭圆由 (3cos t, 2sin t) 描出，从 (3, 0) 出发向上运动。
Elimination gives y squared equals x cubed, for x from 0 to 1.
消参得 y² = x³，x 从 0 到 1。
Both parameter derivatives vanish at zero, but for nonzero t the slope 3t over 2 tends to zero, so the cusp has a horizontal tangent.
两个参数导数在零处都为零，但对非零 t，斜率 3t/2 趋于零，所以尖点处有水平切线。
The unrestricted implicit equation would also include x beyond 1.
不加限制的隐式方程还会包含 x > 1 的部分。
Check yourself.
自我检测。
The period of minus 2 cos 3x is 2 pi over 3, and its shift is pi over 3.
−2cos 3x 的周期是 2π/3，平移是 π/3。
The ellipse slope at pi over 4 is minus a half.
椭圆在 π/4 处的斜率是 −1/2。
The cusp of t squared, t cubed has a horizontal tangent.
(t², t³) 的尖点有水平切线。
And a quarter parameter interval traces one arc of the astroid.
四分之一参数区间只描出星形线的一段弧。