Average Rates of Change and Secant Lines 给定任意函数y=f(x),我们通过y的改变量\Delta y =f(x_{2})-f(x_{1})除以间隔的长度\Delta x=x_{2}-x_{1}=h来计算y在区间\left[ x_{1}, x_{2}\right]上相对于x的平均变化率。(我们使用 h 替换\Delta x来简化符号。) 几何上,f在[x1,x2]上...
The speed at which you walk is called the rate of change. Rate of change tells us how fast something is changing. In calculus, we study these rates of change. For example, if a car is moving, we can find out how fast it is going at any moment. This is called the instantaneous ...
12 feet is the length of the trough which is a constant. h is the height of the water in the trough which is changing. At a given point in time one of the bases of the trapezoid cross section is 4, but the length of the other base depends on h. In fact the 2nd base will be ...
Thomas' Calculus - Chapter2:Limits and Continuity (目录&概述) 2.1 Rates of Change and Tangents to Curves(变化率和曲线切线) 2.2 Limit of a Function and Limit Laws(函数极限和极限定律) 2.3 The Precise Definition of a Limit(极限的准确定义) 2.4 One-Sided Limits(单侧极限) 2.5 Continuity(连续性...
1.1RatesofChangeandLimits TheTangentProblemLetfbeafunctionandletP(a,f(a))beapointonthegraphoff.TofindtheslopemofthetangentlinelatP(a,f(a))onthegraphoff,wefirstchooseanothernearbypointQ(x,f(x))onthegraph(seeFigure1)andthencomputetheslopemPQofthesecantlinePQ.mpQ f(x)f(a).xa 0 ...
1.1RatesofChangeandLimits Supposeyoudrive200miles,andittakesyou4hours.Thenyouraveragespeedis:200mi4hr50mihr averagespeeddistancexelapsedtimet Ifyoulookatyourspeedometerduringthistrip,itmightread65mph.Thisisyourinstantaneousspeed.1.1RatesofChangeandLimits Arockfallsfromahighcliff.Thepositionoftherockisgiven...
Tangents, Velocities, and Other Rates of Change 正切,速度和其他速率变化 我们计算2个点的斜率, 是通过 k 值确定的 左图是对应的计算图像 简单的计算过程,可以见下图 右侧表示,Q点越接近P, 越能表示出P点的瞬时速度 (前几章将瞬时速度的时候,有提到) ...
Students completed the Precalculus Concept Assessment (PCA) and two open-ended tasks with questions about rates of change. After adjusting for prior achievement with the Iowa Algebra Aptitude Test, students from these two paths performed comparably (F = 3.54, p = .063) on the PCA. Student ...
Tangents, Velocities, and Other Rates of Change 正切,速度和其他速率变化 我们计算2个点的斜率, 是通过 k 值确定的 左图是对应的计算图像 简单的计算过程,可以见下图 右侧表示,Q点越接近P, 越能表示出P点的瞬时速度 (前几章将瞬时速度的时候,有提到) ...
Calculus is a branch of mathematics that deals with rates of change and accumulation. For example, when we want to know how fast a car is going, we can use calculus. The basic concept of calculus is the derivative. The derivative tells us how fast something is changing at a particular ...