純数学3
Pearson Edexcel · International A-Level · Pure Mathematics トピック 3 4:17 英語ナレーション・英語＋中文字幕 burning-in
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Transcript
Paper WMA13: 1 hour 30 minutes, 75 marks, with P1 and P2 assumed.
WMA13 试卷：1 小时 30 分钟，75 分，默认已学 P1 和 P2。
None of these are in the booklet: the compound and double angle formulae, the reciprocal identities, and the standard derivatives and integrals.
这些都不在公式册里：和角与二倍角公式、倒数恒等式，以及标准导数和积分。
f of x is 2x over 3x plus 1.
f(x) = 2x/(3x + 1)。
It starts at 0 and climbs towards two thirds without reaching it, so that is the range.
它从 0 开始，向 2/3 增长但达不到，所以这就是值域。
Its inverse is its reflection in the line y equals x, and the two meet on the mirror line.
反函数是它关于直线 y = x 的对称图形，两者在对称轴上相交。
Write y equals f of x and solve for x: x is y over 2 minus 3y, so that is the inverse.
令 y = f(x)，解出 x：x = y/(2 − 3y)，这就是反函数。
Setting the inverse equal to f and cross-multiplying gives x times 9x minus 3: x is 0 or a third.
令反函数等于 f 并交叉相乘，得 x(9x − 3) = 0：x = 0 或 1/3。
y equals the modulus of 2x minus 1 is a V: the part below the axis reflected up.
y = |2x − 1| 是一个 V 形：x 轴下方的部分翻折到上方。
The line meets it twice, once on each arm.
直线与它相交两次，每支一次。
Each arm is a case: 2x minus 1 equals x plus 5 gives 6, and its negative gives minus four thirds.
每一支是一种情况：2x − 1 = x + 5 得 6，取相反数得 −4/3。
The coefficients 7 and 24 make a right triangle: legs 7 and 24, hypotenuse 25.
系数 7 和 24 构成一个直角三角形：直角边 7 和 24，斜边 25。
Its angle alpha, with tan alpha equal to 24 over 7, is 1.287.
它的角 α 满足 tan α = 24/7，为 1.287。
So the whole expression is one cosine, amplitude 25.
所以整个式子是一个余弦，振幅 25。
Split 3x as 2x plus x and use the compound angle formula.
把 3x 拆成 2x + x，用和角公式。
Replace tan 2x by its double angle form, then multiply top and bottom by 1 minus t squared: 3t minus t cubed, over 1 minus 3t squared.
把 tan 2x 换成二倍角形式，再将分子分母同乘 1 − t²：(3t − t³)/(1 − 3t²)。
Plotting log y against log x gives a straight line through minus 3.5, 0 and 0, minus 2.
以 log y 对 log x 作图得到一条直线，过 (−3.5, 0) 和 (0, −2)。
Its gradient, minus four sevenths, is the power; its intercept, minus 2, is log a, so a is one hundredth.
斜率 −4/7 就是指数；截距 −2 是 log a，所以 a = 1/100。
The processor starts at 75 degrees and cools quickly towards a floor of 21, room temperature.
处理器从 75 °C 开始，迅速冷却并趋向 21 °C 的下限，即室温。
It reads 25 after 5 minutes.
5 分钟后读数为 25 °C。
The tangent marks the moment it falls at 9 degrees per minute.
切线标出它每分钟下降 9 °C 的时刻。
At time 0, 75 equals 21 plus A, so A is 54.
t = 0 时 75 = 21 + A，所以 A = 54。
At 5 minutes, e to the minus 5k is 2 over 27, so k is 0.521.
5 分钟时 e^{−5k} = 2/27，所以 k = 0.521。
Setting the derivative to minus 9 and taking logs gives T equals 2.19 minutes.
令导数等于 −9 并取对数，得 T = 2.19 分钟。
Differentiate e to the minus x squared times sine 3x.
对 e^{−x²} sin 3x 求导。
Each factor needs the chain rule: u dash is minus 2x e to the minus x squared, v dash is 3 cos 3x.
每个因子都要用链式法则：u′ = −2x e^{−x²}，v′ = 3cos 3x。
The product rule combines them; at 0 the gradient is 3.
乘积法则把它们组合起来；在 0 处斜率为 3。
Sine squared 3x cannot be integrated as it stands.
sin²3x 不能直接积分。
The double angle identity turns it into a half of 1 minus cos 6x, which integrates to x over 2 minus sine 6x over 12, plus c.
二倍角恒等式把它变成 ½(1 − cos 6x)，积分得 x/2 − sin 6x/12 + c。
The numerator 6x is exactly the derivative of 3x squared plus 4, so the integral is the log of the denominator.
分子 6x 正是 3x² + 4 的导数，所以积分是分母的对数。
The powered version works the same way, with a sixth to fix the constant.
带幂次的版本同理，用 1/6 调整常数。
Each step goes up to the curve, then across to the line y equals x.
每一步先竖直走到曲线，再水平走到直线 y = x。
The staircase spirals in towards the crossing: 0.4, 0.4367, 0.4292, 0.4307.
这个蛛网图螺旋式收敛到交点：0.4、0.4367、0.4292、0.4307。
A sign change between 0.4295 and 0.4305 confirms 0.430.
0.4295 与 0.4305 之间的变号确认根为 0.430。
Check yourself.
自我检测。
The inverse graph is the reflection in y equals x.
反函数的图像是关于 y = x 的对称图形。
R is the root of a squared plus b squared.
R = √(a² + b²)。
A power law is plotted as log y against log x.
幂函数模型用 log y 对 log x 作图。
The derivative of tan kx is k sec squared kx.
tan kx 的导数是 k sec² kx。
And a sign change proves a root only for a continuous function.
变号只有在函数连续时才能证明有根。