Limits, the Foundations Of Calculus, seem so artificial and weasely: “Let xapproach0, but not get there, yet we'll act like it's there… ” Ugh. 极限是微积分的根基,但是它常用的定义又有点模糊“让x接近一个值,但是又没到那个值,我们假装它到了...bluh bluh bluh”. 而它严谨的ε和δ的...
36 Lectures 1Basic Functions of Calculus and Limits 2Differentiation Warm-up 3Integration Warm-up 4Differential Equations-Growth and Decay 5Applications of Differential Equations 6Linear Differential Equations 7Areas and Volumes 8Arc Length, Surface Area, and Work 9Moments, Centers of Mass, and ...
0 Understanding Why |x|<min{1,δ}|x|<min{1,δ} in Spivak's Solutions. 0 Spivak, Calculus, Ch 5 - Limits, Problem **23a: Why can we use the trick g(x)=f(x)g(x)f(x)g(x)=f(x)g(x)f(x)? 2 Why is limx→af′(αx)g′(αx)=limx→af′(x)g...
Suppose we're measuring a stock portfolio, not a bank account. We might want a second-by-second model of our salary and account balance. The idea is to measure at intervals short enough to get the detail we need -- a large aspect of calculus is deciding what "limit" is enough to say...
attached any limits to the integral, this is called the indefinite integral as it simply refers to any x over any interval. Suppose we were to integrate over a definite interval from a to b, then just as the constants in the polynomial functions canceled out, so will the constant in ln ...
Yet when viewed from the macroscopic perspective of the network of global actors, the more one sees the intrinsic smoothness of accelerating technological change, as we discuss in Acceleration and Punctuated Equilibrium in Technological Development. The Astonishing Lack of Limits To Computational Growth ...
In a calculus class, we should rely heavily on the intuition of the student, and we shouldn't attempt to proof things that already are intuitively obvious! (until we reach the point where we realize that obvious things are not that obvious :D ) Intuition is way more important than...
Functional Limits and Continuity Stephen Abbott Pages 111-144 The Derivative Stephen Abbott Pages 145-167 Sequences and Series of Functions Stephen Abbott Pages 169-213 The Riemann Integral Stephen Abbott Pages 215-248 Additional Topics Stephen Abbott Pages 249-303 Back Matter ...
where the integral is between the limits s1and s2 What it means in English: The intensity of radiation at wavelength λ is reduced as it travels through the atmosphere according to thetotal amount of the absorberalong the path. “exp” is e, or 2.718, to the power of t...
Ch 3. Vectors in Calculus Ch 4. Geometry and Trigonometry Ch 5. How to Use a Scientific... Ch 6. Limits Ch 7. Rate of Change Ch 8. Calculating Derivatives and Derivative Rules Using Limits to Calculate the Derivative 8:11 The Linear Properties of a Derivative 8:31 Derivatives of Tr...