"Calculus" 07-Application of calculus

1. Differential Application 1.1 Differential 1.1.1 monotonicity, extremum and asymptote of unary function

The derivative gives the direction of the function, which is very useful for the graphical properties of our analytic functions, and here we use the knowledge of calculus to understand the nature of the function. The effect of the first derivative on the function is the most direct one, where first-order derivative is seen. For the constant function on the interval \ (f (x) =c\), its derivative everywhere is zero, the inverse is known by the mean value theorem, the derivative is constant zero function is the constant value, therefore the function in the interval \ (f (x) =0\) the necessary and sufficient condition is \ (f ' (x) =0\). This conclusion also shows that the difference function of the same derivative function is a constant, which is useful in proving the equivalence of functions, such as proving \ (3\arccos{x}-\arccos{3x-4x^2}=\pi\).

It is easy to prove by using the median theorem that the necessary and sufficient conditions for the function of the interval to be monotonically ascending (descending) are \ (f ' (x) \geqslant 0\) (\ (f ' (x) \leqslant 0\)). When the equals sign is not true, the function is still strictly monotonically rising (falling). When a part of the equals sign is established, the contradiction can be used to know that the function is strictly monotonically rising (falling) as long as the equal sign is not in an interval.

We already know that for any function \ (f (x) \), if \ (x_0\) is the extremum point , then there is \ (f ' (X_0) =0\). Conversely, it is not necessarily (for example \ (x^3\) 0 points), so that \ (f ' (X_0) =0\) the point \ (x_0\) is generally referred to as the stationary point . If in the domain of \ (x_0\) can be guided, then the derivative of the two sides of the same number is a general stationary point, the number is the extremum point (and depending on the circumstances can be judged very large or very small). In general, if \ (f ' (x_0) =f ' (x_0) =\cdots=f^{(n-1)} (X_0) =0\), but \ (f^{(n)}\ne 0\), the Taylor formula (1) is established. It can be proved that \ (n\) is odd when \ (x_0\) is the general static point, \ (n\) is an even number, if \ (f^{(n)}>0\) then \ (x_0\) is a minimum, if \ (f^{(n)}<0\) then \ (x_0\) is the maximum point.

\[f (x) =f (x_0) +\dfrac{1}{n!} f^{(N)} (X_0) (X-X_0) ^n+o ((x-x_0) ^n) \tag{1}\]

With the above conclusions, we can draw the approximate curve of the function, as long as the 0 points and monotonicity can be found out. In addition, sometimes there will be asymptote , it is in fact one of the following three situations: (1) \ (x\to x_0\) when \ (f (x) \to\infty\), (2) \ (x\to\infty\) when \ (f (x) \to y_0\) ; (3) formula (2) has finite limit respectively. The first two respectively with \ (x=x_0\) and \ (y=y_0\) for the asymptotic line, the third with a \ (y=ax+b\) as the asymptotic line.

\[\lim\limits_{x\to\infty}{\dfrac{f (x)}{x}}=a;\quad \lim\limits_{x\to\infty}{[f (x)-ax]}=b\tag{2}\]

1.1.2 Convex function

Finally, the second derivative can be regarded as the tangent slope of the curve, then the second derivative can depict the change of the slope in the function graph. It can be imagined that when \ (f ' (x) >0\) The function curve is convex, and when \ (f ' (x) <0\) The function curve is convex. How to express such a curve strictly? It can be said that any two points on the connection curve form a straight line, the function values between the two points are on the straight side. Inspired by this, the definition of any point (3) is the function of the lower convex (convex) function , the equal sign is not immediately called the strict convex (convex) function .

\[f (tx_1+ (1-t) x_2) \leqslant (\geqslant) tf (x_1) + (1-T) f (x_2), \quad (0<t<1) \tag{3}\]

The above definition of convex function does not assume that the function can be guided, so it is not good to describe the nature of the derivative, for this reason change the nature of the secant on the observation curve. Specifically, for example, for the lower convex function, set \ (x_1<x_2<x_3\), using the definition of easy to prove (4) they can also be used as a convex function equivalent definition. The inequality on the right shows the right side of any point \ (x_0\), and \ (\dfrac{f (x)-F (x_0)}{x-x_0}\) decreases with the (x\to x_0\) monotonically, but the left-hand inequality shows that it has a lower bound, so that \ (x_0\) has a right limit (or infinite limit). It is also possible to have a left limit (or limit of infinity) (x_0\), of course, if \ (x_0\) is the endpoint, only one of them is established. When \ (x_0\) is an intra-interval point, it is obvious that \ (f (x) \) is continuous at \ (x_0\).

\[\dfrac{f (X_2)-F (x_1)}{x_2-x_1}\leqslant\dfrac{f (X_3)-F (x_2)}{x_3-x_2};\quad\dfrac{f (x_2)-F (x_1)}{x_2-x_1}\ Leqslant\dfrac{f (X_3)-F (x_1)}{x_3-x_1}\tag{4}\]

In fact, the convex function is not necessarily conductive, such as \ (v\) Glyph of \ (f (x) =|x|\) is the lower convex function, but in \ (x_0\) is not guided. When \ (f (x) \) is available, it is easy to prove that the necessary and sufficient conditions for (f (x) \) convex (convex) are, \ (f ' (x) \) monotonically Ascending (descending). The necessary and sufficient condition is also equivalent to: the curve is above any tangent of its line. If \ (f (x) \) Differentiable, it can also be proved that \ (f (x) \) convex (upper convex) the necessary and sufficient conditions are \ (f "(x) \geqslant 0\) (\ (f" (x) \leqslant 0\). These are more intuitive and prove to be simple, please self-argument.

The above conclusion shows that if \ (f ' (x) \) is continuous and \ (f ' (X_0) =0\), and in the domain \ (f ' (x) \ne 0\), Visible \ (f (x) \) is the upper and lower convex functions on the left and right sides of \ (x_0\), so the curve is in the field of \ (x_0\) respectively Both sides of the \ (x_0\) tangent. More generally, if \ (f (x) \) is x_0\, and the points of the left and right realms fall on both sides of the tangent, the x_0\ is called the inflection point of \ (f (x) \).

Formula (3) The convex function of the differentiable guide is further extended, set \ (\sum\limits_{k=1}^n{p_k}=1, (p_k>0) \), and remember \ (x=\sum\limits_{k=1}^n{x_k}\). The following convex function \ (f (x) \) can have a formula (5), the \ (n\) sub-multiplication (p_k\) and the addition of the formula (6) is established. This conclusion of the convex function can be used to easily prove some inequalities, such as (f (x) =\ln{x},\,p_k=\dfrac{1}{n}\), can prove \ (\prod x_k\leqslant \dfrac{1}{n}\sum x_k\).

\[f (X_k) =f (x) +f ' (x) (X_k-x) +\frac{1}{2}f "(\xi) (x-x) ^2\geqslant f (x) +f ' (x) (x_k-x) \tag{5}\]

\[\SUM\LIMITS_{K=1}^N{P_KF (X_k)}\geqslant f\left (\sum\limits_{k=1}^n{p_kx_k}\right) \tag{6}\]

1.2 Differential 1.2.1 tangent and normal plane of multi-element function

Now, using the method of differential to review the point line surface of space, please review the basic content of analytic space. The simplest expression of a spatial curve is the parametric equation \ (x=x (t), y=y (t), z=z (t) \), first order continuous derivative \ ((x ' (t), y ' (t), Z ' (t)) \) determines the tangent vector of the curve at \ ((x (t), Y (t), Z (t)) \) \ (\vec{t}\), the curve of tangent continuous change is called smooth curve . The curve may also be represented as the intersection of two surfaces ((7) left), and the Tangent vector ((1,y ' _x (x), Z ' _x (x)) is obtained by the conclusion of the implicit function of the vector-valued function (7) right.

\[\left\{\begin{matrix}f (x, Y, z) =0\\g (x, Y, z) =0\end{matrix}\right.\quad\rightarrow\quad \vec{t}=\left (\,\dfrac{\ Partial (F,g)}{\partial (y,z)},\:\dfrac{\partial (f,g)}{\partial (z,x)},\:\dfrac{\partial (f,g)}{\partial (z,y)}\,\ right) \tag{7}\]

Now look at the surface of the space \ (f (x, Y, z) =0\), if \ (f ' _x,f ' _y,f_z\) is contiguous, it is called a smooth surface . The x_0,y_0,z_0 curve of the surface is examined by the surface equation (F (x (t), Y (t), Z (t)) =0\), and the derivative formula (8) is derived from the surface's micro-accessibility curve. This formula indicates that the tangent of all curves at \ ((X_0,Y_0,Z_0) \) is on the same plane, which is called the plane of the surface at point \ (( x_0,y_0,z_0) \), and its normal vector is \ ((F ' _x,f ' _y,f ' _ z), as shown in. The surface may also be represented by the parametric equation to the left of the formula (9), which can be used to determine the implicit function \ (U (x, y), V (x, y) \). The surface expression is obtained by taking the third formula, and the normal vector (9) Right is then calculated after the normals.

\[f ' _x (x_0,y_0,z_0) x ' _t (t_0) +f ' _y (x_0,y_0,z_0) y ' _t (t_0) +f ' _z (x_0,y_0,z_0) z ' _t (t_0) =0\tag{8}\]

\[\left\{\begin{matrix}x=x (u,v) \\y=y (u,v) \\z=z (u,v) \end{matrix}\right.\quad\rightarrow\quad \vec{T}=\left (\,\ Dfrac{\partial (y,z)}{\partial (u,v)},\:\dfrac{\partial (z,x)}{\partial (u,v)},\:\dfrac{\partial (x, y)}{\partial (U, V)}\,\right) \tag{9}\]

1.2.2 Curvature

I wonder if you have noticed that the second derivative of the planar curve, while representing the rate of change of the slope, does not reflect the curvature of the curve because the slope is not proportional to the angle. To accurately measure the degree of curvature of a curve, we must examine the rate of change in the angle itself, specifically, at a certain point \ (m_0\) tangent angle (\alpha\) compared to length \ (s\) rate of change. If the limit of formula (10) exists, it is called \ (k\) is the curvature of point \ (m_0\), and \ (\dfrac{1}{k}\) is called the radius of curvature .

\[k=\left|\dfrac{\text{d}\alpha}{\text{d}s}\right|=\left|\lim\limits_{\vardelta s\to 0}\dfrac{\varDelta\alpha}{\ Vardelta S}\right|\tag{10}\]

If the curve is denoted by the parametric equation \ (x (t), Y (t) \), first there is \ (\text{d}s=\sqrt{{x ' _t}^2+{y ' _t}^2}\text{d}t\), then by \ (\alpha=\arctan{\dfrac{y ' _t}{x ' _t}}\) is also easy to get (\text{d}\alpha\), thus prone to curvature of the expression (11), the latter is the coordinate equation \ (y=y (x) \) results. for polar equation \ (r=r (\theta) \), it can be written as parametric equation \ (x=r (\theta) \cos{\theta},y=r (\theta) \sin{\theta}\), with the formula (11) can be obtained (12). In particular, for the Circle \ (r=r_0\), it is easy to know its radius of curvature is \ (r_0\).

\[k=\dfrac{|x ' _ty ' _{t^2}-x ' _{t^2}y ' _t|} {({x ' _t}^2+{y ' _t}^2) ^{\frac{3}{2}}}=\dfrac{|y ' _{x^2}|} {(1+{y ' _x}^2) ^{\frac{3}{2}}}\tag{11}\]

\[r=r (\theta) \quad\rightarrow\quad k=\dfrac{|r^2+2{r ' _{\theta}}^2-rr ' _{\theta^2}|} {(R^2+{r ' _{\theta}}^2) ^{\frac{3}{2}}}\tag{12}\]

1.2.3 Extremum

Similar to the conclusion of a unary function, the function of the partial derivative (f (x_1,\cdots,x_n) \), if \ (f ' _{x_i} (x_{01},\cdots,x_{0n}) =0\) is established, then \ (\vec{x}_0= (x_{01},\ CDOTS,X_{0N}) \) is called a static point of \ (f\). When is a stationary point a extremum point ? Then assume that \ (f\) has a sequential second-order partial differential, using the Taylor formula (13). If you remember the symmetric matrix \ (Q=\{a_{ij}=f ' _{x_ix_j} (\VEC{X}_0) \}\), the value of the formula (13) depends on \ (q\) the two type of \ (\vardelta\vec{x}\). Two-positive definite (negative), then the stationary point is the maximum point (minimum), otherwise it is uncertain.

\[\vardelta f (\vec{x}_0) =f (\vec{x})-F (\vec{x}_0) =\dfrac{1}{2} (\vardelta x_1\dfrac{\partial}{\partial x_1}+\cdots+ \vardelta x_n\dfrac{\partial}{\partial X_n}) ^2f (\vec{x}_0+\theta\vardelta\vec{x}) \tag{13}\]

The above extrema assume that variables can change in one field, but there are often limitations in practical problems. For example known \ (G_i (\vec{x}) =0,\, (i=1,\cdots,m) \), Ask \ (F (\vec{x}) \) The extremum, such a problem is called the conditional extremum . In fact, if the local \ (\dfrac{\partial (g_1,\cdots,g_m)}{\partial (x_1,\cdots,x_m)}\ne 0\), then according to \ (g_i=0\) can get \ (x_1,\cdots,x_m\) about \ (x_{m+1},\cdots,x_n\), bring them into \ (F (\vec{x}) \) to turn the problem into an unconditional extremum problem.

However, many times, such implicit functions cannot be written directly, or the result will destroy the original symmetry, thus making the computation complicated. We already have a \ (m\) equation \ (g_i=0\), now we need to find \ ((n-m) \) A "good" equation. We still consider the problem of \ (x_{m+1},\cdots,x_n\) as the argument, the extremum of \ (f\) First has \ (\TEXT{D}F=\SUM\LIMITS_{I=1}^NF ' _{x_i}\,\text{d}x_i=0\), notice where \ (\text{d}x_i,\, (i=1,\cdots,m) \) is a function. If we can make the equation only the independent variable \ (x_{m+1},\cdots,x_n\) differential, then the differential coefficients are \ (0\), this gives the other \ (n-m\) equation.

You can also g_i\ \ (\text{d}g_i=\sum\limits_{i=1}^n{(g_i)} ' _{x_i}\,\text{d}x_i=0\), because \ (\dfrac{\partial (G_1,\cdots , g_m)}{\partial (x_1,\cdots,x_m)}\ne 0\), you can select the parameter \ (\lambda_i,\, (i=1,\cdots,m) \) so that \ (\text{d}f+\sum\limits_{i=1}^m The coefficients for \ (\text{d}x_i,\, (i=1,\cdots,m) \) are \ (0\) in {\lambda_i\text{d}g_i}\). At this time \ (\text{d}x_i,\, (i=m+1,\cdots,n) \) The coefficients must be zero, they are to find the \ (n-m\) equation.

Now to summarize the equations that need to be solved, for the convenience of discussion, the \ (\lambda_i\) also as the unknown, and remember \ (\varphi\) as the formula (14) left. The original \ (m\) equation \ (g_i=0\) is actually \ (\varphi ' _{\lambda_j}=0\), (\lambda_i\) the \ (m\) equation is actually \ (\varphi ' _{x_i}=0,\, (i=1,\cdots,m \), and the last \ (n-m\) equation is \ (\varphi ' _{x_i}=0,\, (i=m+1,\cdots,n) \). This method is called the Lagrange multiplier method , and the formula (14) is more convenient for memory. However, it is important to note that we are only looking for "static point", and we need to determine whether it is an extremum according to the actual situation.

\[\varphi (\vec{x},\vec{\lambda}) =f (\vec{x}) +\sum\limits_{i=1}^m{\lambda_ig_i (\vec{x})}\quad\Rightarrow\quad\ Varphi ' _{x_i}=0\;\wedge\;\varphi ' _{\lambda_j}=0\tag{14}\]

2. Application of Integral 2.1 integral 2.1.1 of a unary function--plane area, volume

Before we set the definite integral as a definition of area, we now look at the rationality of this definition and the more extensive application of definite integrals. First, we give a visual definition of the area of the plane graph, and for polygons, they can always be divided into several triangles. For the general plane graphics \ (p\), we can always construct two polygons \ (b,a\), \ (b\) surround \ (p\) and \ (a\) surrounded by \ (p\), obviously \ (b\) area is not less than \ (a\) area. All satisfies the condition \ (a\) The area has the upper bound \ (s_*\), all satisfies the condition \ (b\) The area has the next certainty boundary \ (s^*\), when \ (s_*=s^*\) is called \ (p\) to calculate the product , and \ (s=s_*=s^*\) is known as \ (p\) area .

for arbitrary graphs \ (p\), it is easy to prove that the necessary and sufficient condition of its integrable is that there are polygon sequences \ (\{a_i\},\{b_i\}\), and their area limits are the same. This condition is good for the definition of definite integral, so it is reasonable to define the area with definite integral for integrable function. For complex graphics (the domain is defined as \ ([a,b]\)), the line length (x=x_0\) truncated is \ (g (x) \) (continuous), the graphic area is (15) left. If \ (x\) is the parametric equation of \ (t\) and \ (x ' (T) \) is contiguous, it is also available on the right side of the formula (15).

\[S_P=\INT_A^BG (x) \,\text{d}x=\int_{\alpha}^{\beta}g (x (t)) x ' (t) \,\text{d}t,\quad (x (\alpha) =a,a (\beta) =b) \tag{ 15}\]

The above defined area method can actually be generalized, if the required amount \ (q\) is continuous on \ ([a,b]\), it is divided into several parts, each part uses an approximate value of the integral (f (x_i) \vardelta x_i\) instead. Then it proves that the error part tends to \ (0\), so the amount is equal to the definite integral \ (\INT_A^BF (x) \,\text{d}x\). For each specific problem, it is necessary and sometimes difficult to prove that the error part tends to be zero, and the argument to the F (x) \) is essential. But the following conclusion, I only intend to give a rough description, specific proof please refer to the textbook.

Some graphs are more convenient with polar coordinates, and the sector (R=r (\theta) \) defined on \ ([\alpha,\beta]\) can prove to be of the formula (16). Similarly, the volume can be defined with a multipatch approximation, which is more convenient to use with multiple prism calculations. If stereoscopic \ (v\) is defined on \ ([a,b]\), and \ (x\) has a cross-sectional area of \ (S (x) \), it can be proved that its volume is \ (\int_a^bs (x) \,\text{d}x\).

\[r=r (\theta) \quad\rightarrow\quad s_p=\dfrac{1}{2}\int_{\alpha}^{\beta}r^2 (\theta) \,\text{d}{\theta}\tag{16}\]

2.1.2 length, rotational surface, curve quality

Now we discuss a curve segment in the plane \ (l\), which is given by the parametric equation \ (x (t), Y (t), t\in[p,q]\). Take a number of points online, connect them sequentially with line segments, and define the length of \ (l\) with the limit of the sum of the lengths of these segments. If the curve segment itself does not intersect and is not closed, it can be proved that the length of the formula (17). When the curve is connected to the end, it can be split into two-segment calculations, which is easy to prove that the formula is still valid. If \ (s\) is considered as a function of \ (t\), \ (s ' (t) =x ' ^2 (t) +y ' ^2 (t) \geqslant 0\) and continuous, thus \ (T^{-1} (s) \) exists. \ (x,y\) can be seen as a function of \ (s\), by the \ (\text{d}s^2=\text{d}x^2+\text{d}y^2\) known formula (18) is established.

\[s_l=\int_p^q\sqrt{x ' ^2 (t) +y ' ^2 (t)}\,\text{d}t=\int_a^b\sqrt{1+y ' ^2 (x)}\,\text{d}x=\int_{\alpha}^{\beta}\sqrt {R ' ^2 (\theta) +r^2 (\theta)}\,\text{d}\theta\tag{17}\]

\[(\dfrac{\text{d}x}{\text{d}s}) ^2+ (\dfrac{\text{d}y}{\text{d}s}) ^2=1\tag{18}\]

For the area of the general surface, a general method is given later, where only the area of a particular surface is discussed. For a curve defined on \ ([a,b]\) \ (y (x) \), rotate it around the \ (x\) axis for one week, and the path of the curve forms the rotational surface \ (\sigma\). You can use the rotational surface of the segmented segment on the curve (the side of the round table) as the area of the rotated surface , knowing that the area of the rotated surface of each segment is \ (\pi (y_i+y_{i+1}) d_i\) (\ (d_i\) is the line section), which proves that the rotational surface area is \ (2\pi\ int_0^ly\,\text{d}s\) (\ (l\) is the curve length), the collation of the formula (19) is established.

\[s_{\sigma}=2\pi\int_p^qy (t) \sqrt{x ' ^2 (t) +y ' ^2 (t)}\,\text{d}t=2\pi\int_a^by (x) \sqrt{1+y ' ^2 (x)}\,\text{d}x\tag {19}\]

More generally, the density of each point of the curve \ (l\) is \ (f (x, y) \) (and possibly other meaning), so what is the quality of the curve? The same way, the curve is divided into small segments \ (\vardelta s\), with the mass and the limit of all segments as the definition of \ (l\) weight . This limit is also known as the first curve integral , which is recorded as \ (\int_lf (x, y) \,\text{d}s\). A similar length analysis can be expressed in relation to the \ (t\) parameter equation (x,y,s\), and the first curve integral is converted to one-dimensional integral (equation (20)).

\[\INT_LF (x, y) \,\text{d}s=\int_p^qf (× (t), Y (t)) \sqrt{x ' ^2 (t) +y ' ^2 (t)}\,\text{d}t\tag{20}\]

2.2 Integral 2.2.1 Plane area and volume of multi-element function

The integral is the integral of the area (volume), so the integral function is set to \ (1\) to find the plane area and volume (21), and then the repeated integral or the replacement method can be used to obtain the re-integral.

\[s=\iint_d\text{d}x\,\text{d}y;\quad V=\iiint_{\omega}\text{d}x\,\text{d}y\,\text{d}z\tag{21}\]

2.2.2 Surface area, surface quality

Now look at the area of general space surface \ (\gamma\), first introduce a basic conclusion: Set plane \ (\pi_1,\pi_2\) between the angle is \ (\theta\), it is easy to prove that \ (\pi_1\) any graphics in \ (\pi_2\) The area of the vertical projection on is the \ (\cos{\theta}\) of the original graphic. To define the surface area , we split the surface \ (f (x, y) \) into multiple small regions \ (\gamma_1,\cdots,\gamma_n\), and the vertical projection of each region on the \ (xy\) plane is \ (d_i\). For each zone \ (\gamma_i\), it is possible to approximate the normal plane of the \xi_i,\eta_i\ (t_i\) with any point above it, assuming that \ (f (x, y) \) has a continuous partial derivative.

The area of the set \ (t_i,d_i\) is \ (\vardelta\tau_i,\vardelta\sigma_i\), because the normal vector of \ (t_i\) is \ ((f_x (\xi_i,\eta_i), f_y (\xi_i,\eta_i),- 1) \), so \ (t_i,d_i\) The angle of the satisfaction formula (22) left. So the approximate area of the surface, as shown in the formula (22), is actually an integral on \ (d\) and therefore a heavy integral of the surface area of the formula (23). If \ (x,y,z\) is given by parameter \ (u,v\), re-computes the re-integral of the formula (24).

\[\cos{\theta}=\dfrac{1}{\sqrt{1+f_x^2 (\xi,\eta) +f_y^2 (\xi,\eta)}};\quad\sum_{i=1}^n\vardelta\tau_i=\sum_{i=1} ^n\dfrac{\vardelta\sigma_i}{\cos{\theta_i}}\tag{22}\]

\[s=\iint_d\sqrt{1+f_x^2 (x, y) +f_y^2 (x, y)}\,\text{d}x\,\text{d}y\tag{23}\]

\[s=\iint_{d '}\sqrt{a^2+b^2+c^2}\,\text{d}u\,\text{d}v,\quad (a=\frac{\partial (y,z)}{\partial (u,v)},\; B=\frac{\partial (z,x)}{\partial (u,v)},\; C=\frac{\partial (x, y)}{\partial (u,v)}) \tag{24}\]

Similar to the first curve integral, if the density on the surface \ (\gamma\) is \ (f (x, y, z) \) (or other meaning), the surface quality is called the first shape area . This integral is recorded as \ (\iint_{\gamma}f (x, Y, z) \,\text{d}\tau\), the surface area above is actually \ (\iint_{\gamma}\text{d}\tau\). The differential \ (\text{d}\tau\) can be expanded to convert the first surface area into double integrals (such as formula (25), or as a parametric \ (u,v\).

\[\iint_{\gamma}f (x, Y, z) \,\text{d}\tau=\iint_df (x, y) \sqrt{1+z_x^2+z_y^2}\,\text{d}x\,\text{d}y\tag{25}\ ]

"Calculus" 07-Application of calculus