Calculus Review

Chapter 0.

The requirement of our college is that the process should be written in English...

The content of this essay is very very trivial, so should you find yourself with some leisure time, you might find it mildly amusing to peruse. And what I truly concern is whether I can get full score in the test...

Chapter 1. Function

1.1 Odd function and even function

Example 1. Let \(f(x)\) be a function defined on \((-l,l)\). Prove that \(f(x)\) can be written as the sum of an even function and an odd function.

Proof:

It is a classical construction problem... And it is easy to find that:

\[g(x)=\left(f(x)+f(-x)\right) / 2 \]

\[h(x) = \left(f(x)-f(-x)\right) / 2 \]

are the functions that meet the conditions. \(\quad \square\)

1.2 The convertion between polar equation and Cartesian equation

\[\begin{cases} x = r\cos \theta \\ y = r \sin \theta \\ x ^ 2 + y ^ 2 = r ^ 2 \end{cases} \]

There are not many noticable problems.

1.3 Some inequalities

Example 2. To prove that:

\[\prod _ {k = 1} ^ {n} \dfrac{2k-1}{2k} < \dfrac{1}{\sqrt{2n+1}} \]

Proof:

It is obvious that:

\[\prod _ {k = 1} ^ {n} \dfrac{\sqrt{(2k-1)(2k+1)}}{2k} < 1 \]

Thus we obtain the original inequality. \(\quad \square\)

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Example 3. To prove that:

\[\sum _ {k = 1} ^ {n} \dfrac{1}{\sqrt{k}} < \sqrt{n} \quad (n \ge 2) \]

Proof:

\[\sum _ {k = 1} ^ {n} \dfrac{1}{\sqrt{k}} < \sum _ {k=1}^{n} \dfrac{1}{\sqrt{n}}<\sqrt{n}. \quad \square \]

Chapter 2. Limit and Continuity

2.1 Squeeze Theorem

Example 1. To prove that:

\[\lim _ {n \to \infty} n ^ {1/n} = 1 \]

Proof:

Let \(n ^ {1/n} = 1 + \alpha _ n\). We obtain:

\[n = ( 1 + \alpha _ n ) ^ n \geqslant \dfrac{n(n-1)}{2} \alpha _ n ^ 2 \quad (\forall n \ge 2), \]

which yields that \(0 \leqslant \alpha _ n \leqslant \sqrt{2/(n-1)}\).

Therefore, \(\lim _ {n \to \infty} \alpha _ n = 0\), which implies that:

\[\lim _ {n \to \infty} n ^ {1/n} = 1. \quad \square \]

2.2 Monotonic Bounded Principle

We see that if a generating sequence converges to \(A\), it will satisfy that \(A=G(A)\).

Thus, if we prove that the sequence is convergent by Mnotonic Bounded Principle, we will be able to use the property to obtain the limit.

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Example 2. \(a _ {n+1} = \sqrt{ 6 + a _ n}\), and \(a _ 1 = 10\).

Solution:

To prove monotonicity, we consider the induction on \(n\).

For the base case, we know that \(a_2 = \sqrt{6+ a _ 1 } = 4\), which gives \(a _ 2 - a _ 1 < 0\).

Suppose that $\forall n \le k : a _ n - a _ {n-1} < 0 $, from which we obtain that

\[a _ {n+1} - a_n = \sqrt{6 + a_ n} - \sqrt{6 + a_{n-1}} = \dfrac{a_n - a_ {n-1}}{\sqrt{6 + a_ n} + \sqrt{6 + a_{n-1}} } < 0. \]

By the induction, we yields \(a _ {n+1} - a_ n < 0\) is true for all \(n > 0\).

Since \(a_ n > 0\), it follows that \(\left\{ a_ n\right\}\) is convergent frome Monotonic Bounded Principle.

Without loss of generality, suppose that \(\lim _ {n \to \infty} a _ {n+1}= A\).

Since

\[\lim _ {n\to \infty} a_{n+1} = \lim _ {n\to \infty} \sqrt{6 + a _ n} = A, \]

it implies that \(A = \sqrt{6+A}\), which yields \(A = 3\).

Thus, \(\lim _ {n\to\infty} a_n = 3\). \(\quad \square\)

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Example 3. To prove:

\[\dfrac{1}{n+1} < \ln \left(1 + \dfrac{1}{n}\right)< \dfrac{1}{n} \quad (\forall n \in \mathbb{N _ +}) \]

Proof:

Prove it by the inequality:

\[\left(1 + \dfrac{1}{n}\right) ^ n < \mathrm{e} < \left( 1 + \dfrac{1}{n}\right) ^ {n+1}. \quad \square \]

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Example 4. To prove that: the sequence

\[x_ n = \sum _ { k = 1 } ^ {n} \dfrac{1}{k} - \ln n \]

is convergent. Thus, there is

\[x_ n = \gamma + \varepsilon _ n \quad \left(\lim _ {n \to \infty} \varepsilon _ n = 0\right). \]

Proof:

Prove it by the inequality in Exp. 3 and the Monotonic Bounded Principle. \(\quad \square\)

2.3 Stolz's Theorem

Theorem. If:

• \(y_{n+1} > y_n \quad (n = 1, 2, \cdots)\);
• \(\lim _ {n\to \infty} y_ n = +\infty\);
• \(\lim _ {n\to \infty} \dfrac{x_{n+1} - x _n}{y_{n+1} - y_n}\) exists,

then we have:

\[\lim _ {n\to \infty} \dfrac{ x _n}{ y_n} = \lim _ {n\to \infty} \dfrac{x_{n+1} - x _n}{y_{n+1} - y_n}. \]

2.4 Discontinuity

2.4.1 Removable Discontinuity

It is defined as \(x_0\) which satisfies that $\lim _ {x\to x_0 ^ +} f(x) = \lim _ {x\to x_0 ^ -} f(x) \ne f(x_0) $.

2.4.2 Discontinuity of the First Kind

It is also called the jump discontinuity, which satisfies that \(f(x_0 ^ +) \ne f(x_0^-)\).

2.4.3 Discontinuity of the Second Kind

Discontinuities of the second kind can be classified into oscillating discontinuities and infinite discontinuities.

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Example 5. If exactly one of \(f(x)\) and \(g(x)\) is dioscontinuous at \(x = x_ 0\), does it follow that \(f(x)+g(x)\) is discontinuous at \(x=x_0\)?

Solution:

Without loss of generality, suppose that \(f(x)\) is discontinuous and \(g(x)\) is continuous.

If \(f(x)+g(x)\) is continuous, we obtain that \(f(x) = (f(x)+g(x)) -g(x)\) is continuous, which yields a conflict.

Thus, it follows that \(f(x)+g(x)\) is discontinuous at \(x = x_0\). \(\quad \square\)

2.5 Intermediate Value Theorem

Theorem. If \(f(x)\) is continuous on \([a,b]\), and \(f(a)< \mu < f(b)\) or \(f(b) < \mu < f(a)\) is true, there will exist a point \(\xi\) which satisfies that \(f(\xi) = \mu\).

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Example 6. Suppose that \(f(x)\) is continuous on \([a,b]\), and \(f(x) \in (g(a),g(b))\) for every \(x\in [a,b]\). Prove that there exists at least one point \(\xi \in [a,b]\), \(\mathrm{s.t.}\) \(f(\xi) = g(\xi)\).

Proof:

Let \(F(x) = f(x) - g(x)\). We obtain that \(F(a) > 0\) and \(F(b) < 0\).

Thus, \(\exists \xi \in [a,b] : F(\xi) = 0\) is true from IVT. \(\quad \square\)

Chapter 3. Derivative and Differential

The former part is trivial, so I skip it.

3.5 Differential of Functions

Definition 1. The increment of the function can be expressed in the form (\(\mathrm{d}x = \Delta x\))

\[\Delta y = \sum _{i=1} ^ { n } \dfrac{A^{(i)} ( x _ 0 )}{i!} (\Delta x) ^ {i} + o ((\Delta x)^{n}) \mathrm{d}y=A'(x_0)\mathrm{d}x. \]

And \(A'(x_0)\Delta x\) is called the differential of the function at \(x_0\).

We can make use of diffeerntial calculus to do some approximate calculations:

\[f(x_0 + \Delta x) \approx \sum _ {i=0} ^ {n} f^{(i)}(x_0) (\Delta x) ^{i} \]

Chapter 4. Differential Mean Value Theorem and Its Applications

4.1 Differential Mean Value Theorem

Definition 1. Let \(f(x)\) be defined in some neighborhood \(U(x_0,\delta)\) of the point \(x_0\). If for every \(x\in U(x_0,\delta)\),

\[f(x)\leqslant f(x_0) \]

holds, then \(f(x)\) is said to have a local maximum at \(x_0\), and \(x_0\) is called a local maximum point.

(Similarly for local minimum and local minimum point...)

Definition 2. Local extremum includes both local maxima and local minima.

An extremum poitn refers to either a local maximum point or a local minimum point.

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Definition 3. Let \(f(x)\) be defined on an interval \(I\). If there exists \(x_M\in I\) such that for every \(x\in I\),

\[f(x)\leqslant f(x_M) \]

then \(f(x_M)\) is called the absolute maximum of \(f\) on \(I\), and \(x_M\) is the absolute maximum point.

(Similarly for absolute minimum and absolute minimum point...)

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Theorem 1. (Lagrange's Mean Value Theorem)

Let a function \(f(x)\) satisfy the following conditions:

• It is continuous on the closed interval \([a,b]\).
• It is differentiable on the open interval \((a,b)\).

Then, there exists at least one point \(\xi \in (a,b)\) such that:

\[f'(\xi) = \dfrac{f(b)-f(a)}{b-a}. \]

Theorem 2. (Cauchy's Mean Value Theorem)

Let two functions \(f(x)\) and \(g(x)\) satisfy the following conditions:

• They are continuous on the closed interval \([a,b]\).
• They are differentiable on the open interval \((a,b)\).
• \(\forall x\in (a,b):g'(x)\ne 0\).

Then, there exists at least one point \(\xi \in (a,b)\), such that:

\[\dfrac{f(b)-f(a)}{g(b)-g(a)} = \dfrac{f'(\xi)}{g'(\xi)}. \]

4.3 Taylor Formula

Taylor's formula for \(f\) at \(x_0\) with Peano remainder form:

\[f(x) = \sum _ {k=0} ^{n} \dfrac{f^{(k)}(x_0)}{k!}(x-x_0)^k + o((x-x_0)^k) \quad (x\to x_0) \]

Taylor's formula for \(f\) at \(x_0\) with Lagrange remainder form (\(\xi\) is between \(x_0\) and \(x\)):

\[f(x) = \sum _ {k=0} ^{n} \dfrac{f^{(k)}(x_0)}{k!}(x-x_0)^k + \dfrac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1} \]

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Some commonly used Taylor formula:

\(\displaystyle e^{x} = \sum \limits _ {k=0} ^{n} \dfrac{e^{x_0}}{k!}(x-x_0)^k+o((x-x_0)^n) \quad (x\to x_0)\)

\(\displaystyle \sin x = \sum \limits _ {k=0} ^{n} \dfrac{\sin (x_0+k\pi /2)}{k!}(x-x_0)^k+o((x-x_0)^n) \quad (x\to x_0)\)

\(\displaystyle (1+x)^{\alpha} = \sum \limits _ {k=0} ^{n} \dfrac{\alpha ^{\underline{k}}(1+x_0)^{\alpha -k}}{k!}(x-x_0)^k+o((x-x_0)^n) \quad (x\to x_0)\)

\(\displaystyle \ln (1+x) = \ln (1+x_0)-\sum \limits _ {k=1} ^{n} \dfrac{(-1)^k}{k(1+x_0)^k}(x-x_0)^k+o((x-x_0)^n) \quad (x\to x_0)\)

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Some commonly used Maclaurin Series:

\(\displaystyle \arctan x = \int _{0} ^{x} \dfrac{1}{1+t^2}\mathrm{d}t = \sum_{k=0} ^{+\infty} (-1)^{n} \dfrac{1}{2n+1}x^{2n+1}\)

\(\displaystyle \arcsin x = \int _{0} ^{x} \dfrac{1}{\sqrt{1-t^2}}\mathrm{d}t = \sum_{k=0} ^{+\infty} \dfrac{(2n)!}{4^{n} (n!)^{2}(2n+1)}x^{2n+1}\)

\(\displaystyle \arccos x = \int _{0} ^{x} \dfrac{-1}{\sqrt{1-t^2}}\mathrm{d}t =\dfrac{\pi}{2} - \sum_{k=0} ^{+\infty} \dfrac{(2n)!}{4^{n} (n!)^{2}(2n+1)}x^{2n+1}\)

4.5 Concavity, Points of Inflection, Asymptotes

Convex (or Concave Upward): \(\bigcup\) is like v in the convex.

Concave (or Concave Downward): \(\bigcap\) is like c in the concave.

Definition 4.

Convexity: \(f(x)\) is convex on \((a,b)\) if for all \(x_1,x_2\in (a,b)\) and for all \(\lambda \in [0,1]\), the folowing inequality holds:

\[f(\lambda x_1+ (1-\lambda)x_2) \leqslant \lambda f(x_1) + (1-\lambda) f(x_2) \]

(If strict inequality holds for \(x_1\ne x_2\) and \(\lambda \in (0,1)\), the function is strictly convex.)

Concavity: \(f(x)\) is concave on \((a,b)\) if for all \(x_1,x_2\in (a,b)\) and for all \(\lambda \in [0,1]\), the folowing inequality holds:

\[f(\lambda x_1+ (1-\lambda)x_2) \geqslant \lambda f(x_1) + (1-\lambda) f(x_2) \]

(If strict inequality holds for \(x_1\ne x_2\) and \(\lambda \in (0,1)\), the function is strictly concave.)

Theorem 3. For a function \(f\) that is twice differentiable on an interval \((a,b)\),

• \(f(x)\) is convex if \(\forall x\in (a,b):f''(x)> 0\).
• \(f(x)\) is concave if \(\forall x\in (a,b):f''(x)< 0\).

Definition 5. A point \((x_0,f(x_0))\) on the graph of \(f\) is called an inflection point if the cacavity changes at \(x_0\). At an inflection point, the second derivative may be zero or undefined.

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Definition 6. For a function \(f(x)\) is defined on an interval \(I\), a point \(c\in I\) is called a critical point if either of the following is true:

• \(f'(c)=0\).
• \(f'(c)\) does not exist.

Definition 7. For a function \(f(x)\) is defined on an interval \(I\), a point \(c\in I\) is called a stationary point if:

• \(f'(c)=0\).