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Math1931 Calculus Of One Variable (SSP) 单变量微积分 悉尼大学

The mathematical modelling of physical systems is based upon differential equations and linear algebra. This unit will introduce fundamental techniques for studying linear systems and differential equations, focusing on applications to physical systems. The topics in linear algebra to be considered include eigenvalues and eigenvectors, diagonalisation of square matrices, matrix functions, LU-decomposition, applications. The topics in optimisation include Lagrange multipliers, the method of least-squares, linear programming, applications. Finally, the topics in differential equations include matrix solutions of constant coefficient systems of ordinary differential equations, conservative systems and phase-planes of simple non-linear ordinary differential equations. Solutions of first order partial differential equations will be analysed and applied to real world problems. You will be introduced to the Mathematica computer package, and learn how to use it for analytical and numerical calculations and graphics. Mathematica will be integrated into most activities.

Content: One-dimensional calculus is a branch of mathematics that studies functions and their rates of change. It includes the concepts of differentiation and integration, studies the behaviour of functions and is used to solve problems in a wide range of fields, including physics, engineering and economics.

Math1931 Calculus Of One Variable (SSP) 单变量微积分 悉尼大学
问题 1.

Replace each of the following infinitesimals with an equivalent one:
(a) $3 \sin \alpha-5 \alpha^3$;
(b) $(1-\cos \alpha)^2+16 \alpha^3+5 \alpha^4+6 \alpha^5$.

Solution. (a) Note that the sum of two infinitesimals $\alpha$ and $\beta$ of different orders is equivalent to the summand of the lower order, since the replacement of an infinitesimal with one equivalent to it is tantamount to the rejection of an infinitesimal of a higher order.
In our example the quantity $3 \sin \alpha$ has the order of smallness 1 , $\left(-5 \alpha^3\right)$ – the order of smallness 3 , rence
$$
3 \sin \alpha+\left(-5 \alpha^3\right) \sim 3 \sin \alpha \sim 3 \alpha .
$$
(b) $(1-\cos \alpha)^2+16 \alpha^3+5 \alpha^4+6 \alpha^5=4 \sin 1^4 \frac{\alpha}{2}+16 \alpha^3+5 \alpha^4+6 \alpha^5$.
The summand $16 \alpha^3$ is of the lower order, therefore
$$
(1-\cos \alpha)^2+16 \alpha^3+5 \alpha^4+6 \alpha^5 \sim 16 \alpha^3 .
$$

Proof. Around the disease-free equilibrium point $E_1\left(\frac{\Lambda}{\mu}, 0,0,0\right)$, we investigate the eigenvalues of the following Jacobian matrix
$$
J\left(\frac{\Lambda}{\mu}, 0,0,0\right)=\left[\begin{array}{cccc}
-\mu & 0 & -\beta \bar{S} & 0 \
0 & -\phi_1 & (1-p) \beta \bar{S} & 0 \
0 & \alpha & p \beta \bar{S}-\phi_2 & 0 \
0 & \varepsilon_1 & \varepsilon_2 & -\mu
\end{array}\right] .
$$
Around the disease-free equilibrium point $E_1$, the Jacobian matrix (10) takes the following form,
$$
J\left(E_1\right)=\left[\begin{array}{cccc}
-\mu & 0 & -\beta \frac{\Lambda}{\mu} & 0 \
0 & -\phi_1 & (1-p) \beta \frac{\Lambda}{\mu} & 0 \
0 & \alpha & p \beta \frac{\Lambda}{\mu}-\phi_2 & 0 \
0 & \varepsilon_1 & \varepsilon_2 & -\mu
\end{array}\right] .
$$

问题 2.

Using the definition of continuity of a function in terms of ” $\varepsilon-\delta$ “, test the following functions for continuity:
(a) $f(x)=a x+b \quad(a \neq 0)$;
(b) $f(x)=\left{\begin{aligned} x^2 & \text { if } x \text { is rational, } \ -x^2 & \text { if } x \text { is irrational. }\end{aligned}\right.$

Solution. (a) Choose an arbitrary point $x_0$. According to the ” $\varepsilon–\delta$ ” definition it is necessary to show that for any preassigned, arbitrarily small number $\varepsilon>0$ it is possible to find a number $\delta>0$ such that at $\left|x-x_0\right|<\delta$ the inequality $\left|f(x)-f\left(x_0\right)\right|<\varepsilon$ holds true.
Consider the absolute value of the difference
$$
\left|f(x)-f\left(x_0\right)\right|=\left|(a x+b)-\left(a x_0+b\right)\right|=\left|a x+b-a x_0-b\right|=|a|\left|x-x_0\right| \text {. }
$$
Let us require that $\left|f(x)-f\left(x_0\right)\right|<\varepsilon$. This requirement will be fulfilled for all $x$ satisfying the inequality
$$
|a|\left|x-x_0\right|<\varepsilon \text { or }\left|x-x_0\right|<\varepsilon /|a| \quad(a \neq 0) .
$$
Hence, if we take $\delta \leqslant \varepsilon /|a|$, then at $\left|x-x_0\right|<\delta$ the inequality $\left|f(x)-f\left(x_0\right)\right|<\varepsilon$ is fulfilled. Continuity is thus proved for any point $x=x_0$.
(b) Choose an arbitrary point $x_0$. If $\left{x_n\right}$ is a sequence of rational numbers tending to $x_0$, then $\lim {x_n \rightarrow x_0} f\left(x_n\right)=x_0^2$. If $\left{x_n^{\prime}\right}$ is a sequence of irrational numbers tending to $x_0$, then $\lim {x_n \rightarrow x_0} f\left(x_n^{\prime}\right)=-x_0^2$. At $x_0 \neq 0$ the indicated limits are different and hence the function is discontinuous at all points $x \neq 0$.

On the other hand, let now $x=0$. Find the absolute value of the difference $|f(x)-f(0)|$ :
$$
|f(x)-f(0)|=\left| \pm x^2-0\right|=x^2 .
$$
It is obvious that $x^2<\varepsilon$ at $|x|<\sqrt{\varepsilon}$. If $\varepsilon>0$ is given, then, putting $\delta \leqslant V^{-}$and $|x-0|=|x|<\delta$, we obtain $|\Delta f(0)|=x^2<\varepsilon$. Hence, at the point $x=0$ the function is continuous. And so, the point $x=0$ is the only point at which the function is continuous. Note that the function under consideration can be expressed through the Dirichlet function (see Problem 1.14.4 (b)): $f(x)=x^2[2 \lambda(x)-1]$.

学习资料推荐

针对 Calculus Of One Variable (SSP) 推荐三本教材:

✅Real Functions in One Variable Examples of Taylor’s Formula and Limit Processes Calculus

✅Lectures on Differential Calculus of Functions of One Variable

✅ Modelling financial derivatives with Mathematica: mathematical models and benchmark algorithms

单变量微积分预测介绍:

  • 微积分也是由德国数学家和哲学家戈特弗里德-威廉-莱布尼茨独立发现或发明的。牛顿和莱布尼茨的朋友和追随者就发现的先后顺序展开了激烈的争论,每个人的朋友都声称是自己首先独立发现了微积分,而另一个人则抄袭了他的发现。如今,人们普遍认为,牛顿和莱布尼茨各自的发现或发明都是独立于对方的,两人都得到了充分的肯定。我们将在本书中使用的许多公式和定理都是牛顿首先提出的,但我们将使用的微分、导数和积分的符号却是莱布尼茨首先使用的。