Newton's second law of motion F = ma = $m\cdot \dfrac{dv}{dt} $can be written in the form $F=d/dt\left( mv\right)$ in terms of the momentum mv of a particle of mass m and velocity v, and remains valid in this form even if m is not constant, as assumed so far. Suppose a spherical raindrop f...
In a certain barbarous land, two neighboring tribes have hated one another from time immemorial. Being bar barous peoples, their powers of belief are strong, and a solemn curse pronounced by the medicine man of the first tribe deranges the members of the second tribe and drives them to murder a...
给出两个来自chatgpt的参考证明方法，显然方法二更符合题意（可以直接看方法二）。 方法一： Let $y=u^n$, where $n$ is a positive integer. Expanding $\Delta y = (u + \Delta u)^n - u^n$ using the binomial theorem, we get: $\Delta y = u^n + \binom{n}{1}u^{n-1}(\Delta u) + \binom{n}{2}u^{n-2}...
我们可以使用链式法则和乘法法则来对该函数求导。具体来说，我们需要将函数表示为若干个简单函数的复合。 设 $u = 4x^3 - 9x^2$，$v = 3x - 2x^2$，则原函数可以写成： $$y = u^2 v^3$$ 接下来我们分别对 $u$ 和 $v$ 求导： $$\frac{du}{dx} = 12x^2 - 18x$$ $$\frac{dv}{dx} = 3 - 4x$$ 然后根据乘法法则和链式法则，我...
二项式定理（英语：binomial theorem），又称牛顿二项式定理，由艾萨克·牛顿于1664年、1665年间提出。该定理给出两个数之和的整数次幂诸如展开为类似项之和的恒等式。二项式定理可以推广到任意实数次幂，即广义二项式定理（以上定义来自于百度百科） 通常来说，其数学表达式可以写为：$$\left( a+b\right) ^{n}=a^{n}+n...