CalculusEasyMade_CH10

A

1

a, b, x=symbols('a b x', real=True)
y=b*(exp(a*x)-exp(-a*x))
y
$b \left(e^{a x} - e^{- a x}\right)$

dydx=y.diff(x)
dydx
$b \left(a e^{a x} + a e^{- a x}\right)$

3

f,n,t=symbols('f, n t', real=True)
f1=log(n*t)
f1
log(nt)

eq=exp(f)-exp(f1)
deq=idiff(eq, f, t)
deq
ne^-f

deq.subs(f, f1)
$\frac{1}{t}$

f1.diff(t)
$\frac{1}{t}$

5

w, p, v, n=symbols('w p v n', real=True)
w1=p*v**n
w1
pvⁿ

eq=log(w)-log(p*v**n)
deq=idiff(eq, w, v)
deq
$\frac{nw}{v}$

deq.subs(w, w1)
$\frac{npv^n}{v}$

w1.diff(v)
$\frac{npv^n}{v}$

7

y, x=symbols('y x', real=True)
y1=3*exp(-x/(x-1))
y1
$3 e^{- \frac{x}{x - 1}}$

eq=log(y)-log(y1)
deq=idiff(eq, y, x)
deq
$\frac{y}{\left(x - 1\right)^{2}}$

deq.subs(y, y1)
$\frac{3 e^{- \frac{x}{x - 1}}}{\left(x - 1\right)^{2}}$

simplify(y1.diff(x))
$\frac{3 e^{- \frac{x}{x - 1}}}{\left(x - 1\right)^{2}}$

9

y, x=symbols('y x', real=True)
y1=log(x**a+a)
y1
log(a+ax)

eq=exp(y)-exp(y1)
deq=idiff(eq, y, x)
deq
$a x^{a - 1} e^{- y}$

deq.subs(y, y1)
$\frac{a x^{a - 1}}{a + x^{a}}$

y1.diff(x)
$\frac{a x^{a - 1}}{a + x^{a}}$

11

y, x=symbols('y x', real=True)
y1=log(x+3)/(x+3)
y1
$\frac{\log{\left(x + 3 \right)}}{x + 3}$

eq=exp(y)-exp(y1)
deq=idiff(eq, y, x)
deq
$\frac{\left(1 - \log{\left(x + 3 \right)}\right) e^{- y + \frac{\log{\left(x + 3 \right)}}{x + 3}}}{\left(x + 3\right)^{2}}$

deq.subs(y, y1)
$\frac{1 - \log{\left(x + 3 \right)}}{\left(x + 3\right)^{2}}$

y1.diff(x)
$\frac{1 - \log{\left(x + 3 \right)}}{\left(x + 3\right)^{2}}$

B

1

a, y=symbols('a y', real=True)
s=a*y**2*log(1/y)
s
$a y^{2} \log{\left(\frac{1}{y} \right)}$

dsdy=s.diff(y)
dsdy
$2 a y \log{\left(\frac{1}{y} \right)} - a y$

ds2dy=s.diff(y, 2)
ds2dy
$a \left(2 \log{\left(\frac{1}{y} \right)} - 3\right)$

sol=solve(dsdy, y)
sol
[exp(-1/2)]

위 결과에 의하면 극점은 $e^{-\frac{1}{2}}이며 이 지점에서의 2차이미분계수는 음수입니다. 그러므로 위 극점은 극대를 나타냅니다. 극대값은 다음 계산과 같습니다.

ds2dy.subs(y, sol[0])
-2a

s.subs(y, sol[0])
$\frac{a}{2e}$

2

a, x=symbols('a x', real=True)
y=x**3-log(x)
y
x3−log(x)
dydx=y.diff(x)
dydx
$3x^2−\frac{1}{x}$
sol=solve(dydx, x)
sol
[3**(2/3)/3]
dy2dx=y.diff(x, 2)
dy2dx.subs(x, sol[0])
$3⋅3^{\frac{2}{3}}$
minimum=y.subs(x, sol[0])
minimum.evalf(3)
0.7

3_a

x=symbols('x', real=True)
y=x**x
y
x^x

dydx=diff(y, x)
dydx
x^x(log(x)+1)

sol=solve(dydx, x)
sol
[exp(-1)]

y.diff(x, 2).subs(x, sol[0])
$\frac{e}{e^{e^{-1}}}$

print(f"극소값: {N(y.subs(x, sol[0]), 5)}")
극소값: 0.69220

3_b

x=symbols('x', real=True)
y=x**(1/x)
y
$x^{\frac{1}{x}}$

sol=solve(y.diff(x), x)
sol
[E]

y.diff(x, 2).subs(x, sol[0])
$- \frac{e^{e^{-1}}}{e^{3}}$

print(f"극대값: {N(y.subs(x, sol[0]), 5)}")
극대값: 1.4447

3_c

x=symbols('x', real=True)
y=x*a**(1/x)
y
$a^{\frac{1}{x}} x$

sol=solve(y.diff(x), x)
sol
[log(a)]

y.diff(x, 2).subs(x, sol[0])
$\frac{e}{log(a)}$

print(f"극소값: {N(y.subs(x, sol[0]), 5)}")
극소값: 2.7183*log(a)