目录

指数函数及e

指数增长

复数指数

练习

1. expgui

2. 计算e

3 五角星绘制

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指数函数及e

（1）的比值总是常数

（2）的导数为其自身。（根据比值=1推导出e的值）

%% Plot a^t and its approximate derivativea = 2;t = 0:.01:2;h = .00001;y = 2.^t;ydot = (2.^(t+h) - 2.^t)/h;plot(t,[y; ydot])legend('y','dy/dx')

%% Compute eformat longformat compacth = 1;while h > 2*epsh = h/2;e = (1 + h)^(1/h);disp([h e])end

对指数函数进行二项式展开，求和式的每一项都通过前一项简单计算得出。该循环在r==s时结束，即两个相连的部分浮点数相同时结束。

%% Experimental version of exp(t)t = rands = 1;term = 1;n = 0;r = 0;while r ~= sr = s;n = n + 1;term = (t/n)*term;%求和式的每一项都通过前一项简单计算得出s = s + term;endexp_of_t = s

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指数增长

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年利、月利、连续利息

%% Compound interestfprintf('             t        yearly       monthly     continuous\n')format bankr = 0.05;y0 = 1000;for t = 0:20y1 = (1+r)^t*y0;y2 = (1+r/12)^(12*t)*y0;y3 = exp(r*t)*y0;disp([t y1 y2 y3])end

贷款清还

%% Payments for a car loany0 = 20000r = .10h = 1/12n = 36p = (1+r*h)^n/((1+r*h)^n-1)*r*h*y0

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复数指数

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绘制八边形

%% Complex exponentialtheta = (1:2:17)'*pi/8z = exp(i*theta)p = plot(z);set(p,'linewidth',4,'color','red')axis square off

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练习

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1. expgui

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2. 计算e

%% 计算e
%在h很小时，10的负多少次方不能由浮点数很好表示，将会发生偏差
clear
format long
format compact
h=1
while h>1.e-15h=h/10; e=(1+h)^(1/h); disp([h e])
end%% 计算e
clear
format long
format compact
h=1
while h>1.e-15h=h/10; e=(1+h)^(1/(1+h-1)); disp([h e])
end

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3 五角星绘制

%% 利用复数指数绘制五角星theta = (0:3:15)'*(2*pi/5)z = exp(i*theta)p = plot(z);set(p,'linewidth',4,'color','red')axis square off