Python Part 2 - Calculus I
Instructions
- This page focuses on Calculus I examples, but it is useful for learning Python even if you are not taking Calculus.
- For each video, you can see the problem being solved, as well as the Python code used.
- You can also open the link to the Python Notebook file. You can click File>Download to download the notebook file to run on your own computer.
- Each video has a link to the full video page. These pages include links to related videos covering the same topics.
Learning Objectives
- Using list comprehension to estimate limits
- Finding and evaluating derivatives
- Calculating definite and indefinite integrals
- Word problems
- Simplifying difference quotients
- Plotting functions, derivatives, and tangent lines
- Patterns in higher derivatives
- Implicit and Parametric plots and tangent lines
- Finding absolute maxima and minima
- Finding intervals of increase and decrease, along with concavity
- Approximating definite integrals with Riemann Sums
Exercises
1.
Python Code:
from sympy import *
x=symbols('x')
f=(sqrt(x)-1)/(x**3-1)
xval=[0.9,0.99,0.999,1.1,1.01,1.001,1.0001]
yval=[f.subs(x,i) for i in xval]
print(yval)
print('The limit appears to be about 1.6666')
matplotlib notebook
plot(f,(x,0,2))L=limit(f,x,1)print('The limit is',L)
L=limit(f,x,1,'-')
print('The left hand limit is',L)
L=limit(f,x,1,'+')
print('The right hand limit is',L)
Open Python Notebook File
from sympy import *
x=symbols('x')
f=(sqrt(x)-1)/(x**3-1)
xval=[0.9,0.99,0.999,1.1,1.01,1.001,1.0001]
yval=[f.subs(x,i) for i in xval]
print(yval)
print('The limit appears to be about 1.6666')
matplotlib notebook
plot(f,(x,0,2))L=limit(f,x,1)print('The limit is',L)
L=limit(f,x,1,'-')
print('The left hand limit is',L)
L=limit(f,x,1,'+')
print('The right hand limit is',L)
Open Python Notebook File
2.
Python Code:
from sympy import *
mu,m,g,x,h=symbols('mu m g x h')
F=mu*m*g*sqrt(x**2+h**2)/(x+mu*h)
dF=F.diff(x)print('The general derivative is',dF)
dFat10=F.subs({m:18,h:10,mu:0.55,g:9.8,x:10})
print('When x=10, the rate of change in t is',dFat10.evalf() ,'Newtons/meter')
dFat40=F.subs({m:18,h:10,mu:0.55,g:9.8,x:40})
print('When x=40, the rate of change in t is', dFat40.evalf(), 'Newtons/meter')
Open Python Notebook File
from sympy import *
mu,m,g,x,h=symbols('mu m g x h')
F=mu*m*g*sqrt(x**2+h**2)/(x+mu*h)
dF=F.diff(x)print('The general derivative is',dF)
dFat10=F.subs({m:18,h:10,mu:0.55,g:9.8,x:10})
print('When x=10, the rate of change in t is',dFat10.evalf() ,'Newtons/meter')
dFat40=F.subs({m:18,h:10,mu:0.55,g:9.8,x:40})
print('When x=40, the rate of change in t is', dFat40.evalf(), 'Newtons/meter')
Open Python Notebook File
3.
Python Code:
from sympy import *
V,omega,t=symbols('V omega t', positive=True)
v=V*cos(omega*t)
v_int=(v**2).integrate(t)
#You can also use: v_int=integrate(v**2,t)
print(v_int)
vdef=integrate(v**2,(t,0,2*pi/omega))
print('The integral from 0 to 2pi/omega is', vdef)
rms=sqrt(omega/(2*pi)*vdef)
print('The root mean square value of the voltage is', rms)
Open Python Notebook File
from sympy import *
V,omega,t=symbols('V omega t', positive=True)
v=V*cos(omega*t)
v_int=(v**2).integrate(t)
#You can also use: v_int=integrate(v**2,t)
print(v_int)
vdef=integrate(v**2,(t,0,2*pi/omega))
print('The integral from 0 to 2pi/omega is', vdef)
rms=sqrt(omega/(2*pi)*vdef)
print('The root mean square value of the voltage is', rms)
Open Python Notebook File
4.
from sympy import *
# Step 1
r,h=symbols('r h')
V=pi*r**2*h+2/3*pi*r**3
S=2*pi*r*h+2*pi*r**2
# Step 2 command: solve V-4=0
hval=solve(V-4,h)
print('the height solution is',hval)
# Step 3 command: subs
Sofr=S.subs(h,hval[0])
print(Sofr)
matplotlib notebook
# Step 4
plot(Sofr,(r,0,10),ylim=[0,100])
Open Python Notebook File
# Step 1
r,h=symbols('r h')
V=pi*r**2*h+2/3*pi*r**3
S=2*pi*r*h+2*pi*r**2
# Step 2 command: solve V-4=0
hval=solve(V-4,h)
print('the height solution is',hval)
# Step 3 command: subs
Sofr=S.subs(h,hval[0])
print(Sofr)
matplotlib notebook
# Step 4
plot(Sofr,(r,0,10),ylim=[0,100])
Open Python Notebook File
5.
from sympy import *
# Use limit definition to find derivative of f(x)=1/x^3
h,x=symbols('h x')
f=1/x**3
#Step 1 command: subs
diffqt=(f.subs(x,x+h)-f)/h
print('The difference quotient is',diffqt)
#Step 2 command: factor, expand, or simplify
print(diffqt.expand().simplify())
#Step 3 command: subs
print(diffqt.expand().simplify().subs(h,0))
# Use limit definition to find derivative of f(x)=1/x^3
h,x=symbols('h x')
f=1/x**3
#Step 1 command: subs
diffqt=(f.subs(x,x+h)-f)/h
print('The difference quotient is',diffqt)
#Step 2 command: factor, expand, or simplify
print(diffqt.expand().simplify())
#Step 3 command: subs
print(diffqt.expand().simplify().subs(h,0))
6.
Python Code:
from sympy import *
x=symbols('x')
f=exp((x-75)**2/200)/(10*sqrt(2*pi))
y0=f.subs(x,80)
df=diff(f,x)
m=df.subs(x,80)
tanline=y0+m*(x-80)
print('The equation of the tangent line is y=',tanline.evalf())
matplotlib notebook
plot((f,(x,60,100)),(tanline,(x,60,100)))
Open Python Notebook File
7.
Python Code:
from sympy import *
x=symbols('x')
y=(6*x**2-4*x**3)/(2*sqrt(x**3*(2-x)))
Python Code:
matplotlib notebook
plot(y,(x,0,2),ylim=[-10,10])
dy=diff(y,x).simplify()
print(dy)
horiz=nsolve(dy,x,0.75)
print('The graph has a horizontal tangent line at x=',horiz)
Open Python Notebook File
8.
Python Code:
from sympy import *
# List Comprehension
x=symbols('x')
f=1/x
# 1st 6 derivatives IN ONE LINE OF CODE!
fderivs=[diff(f,x,i) for i in range(1,7)]
#range(a,b)-> list of integers from a INCLUSIVE to b EXCLUSIVE
print(fderivs)
# exponent is 1 bigger than the derivative we took: x^(n+1)
# signs alternate (-1)^n or (-1)^(n+1). n=1 is negative, so (-1)^n
#factorial numbers in the numerator: n!
print('The formula for the nth derivative of f is (-1)^n*n!/x^(n+1)')
Open Python Notebook File
from sympy import *
# List Comprehension
x=symbols('x')
f=1/x
# 1st 6 derivatives IN ONE LINE OF CODE!
fderivs=[diff(f,x,i) for i in range(1,7)]
#range(a,b)-> list of integers from a INCLUSIVE to b EXCLUSIVE
print(fderivs)
# exponent is 1 bigger than the derivative we took: x^(n+1)
# signs alternate (-1)^n or (-1)^(n+1). n=1 is negative, so (-1)^n
#factorial numbers in the numerator: n!
print('The formula for the nth derivative of f is (-1)^n*n!/x^(n+1)')
Open Python Notebook File
9.
Python Code:
from sympy import *
x,y=symbols('x y')
eqn=x**3+y**3-3*x*y
matplotlib notebook
peqn=plot_implicit(eqn,(x,-3,3),(y,-3,3))
dydx=idiff(eqn,y,x)
print('dy/dx=',dydx)
m=dydx.subs({x:2/3,y:4/3})
print('The slope of the tangent line is',m)
tanline=4/3+m*(x-2/3)
print('The equation of the tangent line is y=',tanline)
peqn.extend(plot(tanline,(x,-3,3)))
peqn.show()
Open Python Notebook File
10.
Python Code:
from sympy import *
x,t=symbols('x t')
xt=t**3-3*t**2
yt=t**3-3*t
matplotlib notebook
pcurve=plot_parametric(xt,yt,(t,-2,3))
dydx=diff(yt,t)/diff(xt,t)
m=dydx.subs(t,-2)
x0=xt.subs(t,-2)
y0=yt.subs(t,-2)
tanline=y0+m*(x-x0)
print('The equation of the tangent line is y=',tanline)
pcurve.extend(plot(tanline,(x,-20,0)))
pcurve.show()
Open Python Notebook File
from sympy import *
x,t=symbols('x t')
xt=t**3-3*t**2
yt=t**3-3*t
matplotlib notebook
pcurve=plot_parametric(xt,yt,(t,-2,3))
dydx=diff(yt,t)/diff(xt,t)
m=dydx.subs(t,-2)
x0=xt.subs(t,-2)
y0=yt.subs(t,-2)
tanline=y0+m*(x-x0)
print('The equation of the tangent line is y=',tanline)
pcurve.extend(plot(tanline,(x,-20,0)))
pcurve.show()
Open Python Notebook File
11.
Python Code:
from sympy import *
x,t=symbols('x t')
xt=cos(t)**3
yt=sin(t)**3
eq1=xt-1/27
# To get exact answers, use the Rational command: eq1=xt-Rational(1,27)
eq2=yt-16*sqrt(2)/27
# To get exact answers, use the Rational command: eq2=yt-Rational(16,27)*sqrt(2)
tval=solve([eq1,eq2],t)
print(tval)
dydx=diff(yt,t)/diff(xt,t)
print('dy/dx=',dydx)
m=dydx.subs(t,tval[0][0])
print(m)
tanline=16*sqrt(2)/27+m*(x-1/27)
print('The equation of the tangent line is y=',tanline.evalf())
matplotlib notebook
pcurve=plot_parametric(xt,yt,(t,0,pi/2))
pcurve.extend(plot(tanline,(x,0,1)))
pcurve.show()
Open Python Notebook File
from sympy import *
x,t=symbols('x t')
xt=cos(t)**3
yt=sin(t)**3
eq1=xt-1/27
# To get exact answers, use the Rational command: eq1=xt-Rational(1,27)
eq2=yt-16*sqrt(2)/27
# To get exact answers, use the Rational command: eq2=yt-Rational(16,27)*sqrt(2)
tval=solve([eq1,eq2],t)
print(tval)
dydx=diff(yt,t)/diff(xt,t)
print('dy/dx=',dydx)
m=dydx.subs(t,tval[0][0])
print(m)
tanline=16*sqrt(2)/27+m*(x-1/27)
print('The equation of the tangent line is y=',tanline.evalf())
matplotlib notebook
pcurve=plot_parametric(xt,yt,(t,0,pi/2))
pcurve.extend(plot(tanline,(x,0,1)))
pcurve.show()
Open Python Notebook File
12.
Python Code:
from sympy import *
T=symbols('T')
D=.0694*T**2-10.505*T+399.05
dD=diff(D,T)
critvals=solve(dD,T)
print(critvals)
candidates=[53,critvals[0],107]
yvals=[D.subs(T,i) for i in candidates]
print(yvals)
print('The minimum distance is', yvals[1],'degrees when T is about 75.68 degrees.')
Open Python Notebook File
from sympy import *
T=symbols('T')
D=.0694*T**2-10.505*T+399.05
dD=diff(D,T)
critvals=solve(dD,T)
print(critvals)
candidates=[53,critvals[0],107]
yvals=[D.subs(T,i) for i in candidates]
print(yvals)
print('The minimum distance is', yvals[1],'degrees when T is about 75.68 degrees.')
Open Python Notebook File
13.
Python Code:
Analyze the graph of y=2x^7-21x^6-36x^5+3x^4-38x^3+24x^2
from sympy import *
x=symbols('x')
y=2*x**7-21*x**6-36*x**5+3*x**4-38*x**3+24*x**2
matplotlib notebook
plot(y,(x,-10,10),ylim=[-10,10])
dy=diff(y,x)
# To find concavity, repeat this process but use dy=diff(y,x,2) to get the second derivative
cvals=solve(dy,x)
cvalsfloat=[i.evalf() for i in cvals]
print(cvalsfloat)
matplotlib notebook
plot(dy,(x,-2,11),ylim=[-10,10])
print('The function is increasing from(-∞,-1.625),(0,0.35),(10,253,∞)')
print('and decreasing from (-1.625,0) and (0.35,10.253)')
Open Python Notebook File
Analyze the graph of y=2x^7-21x^6-36x^5+3x^4-38x^3+24x^2
from sympy import *
x=symbols('x')
y=2*x**7-21*x**6-36*x**5+3*x**4-38*x**3+24*x**2
matplotlib notebook
plot(y,(x,-10,10),ylim=[-10,10])
dy=diff(y,x)
# To find concavity, repeat this process but use dy=diff(y,x,2) to get the second derivative
cvals=solve(dy,x)
cvalsfloat=[i.evalf() for i in cvals]
print(cvalsfloat)
matplotlib notebook
plot(dy,(x,-2,11),ylim=[-10,10])
print('The function is increasing from(-∞,-1.625),(0,0.35),(10,253,∞)')
print('and decreasing from (-1.625,0) and (0.35,10.253)')
Open Python Notebook File
14.
Python Code:
from sympy import *
import numpy as np
x=symbols('x')
f=exp(-x**2)
a=0
b=2
n=50
dx=(b-a)/n
xi=np.arange(a,b,dx)
yi=[f.subs(x,i) for i in xi]
Approx=np.sum(yi)*dx
print('The area is approximately', Approx)
Open Python Notebook File
from sympy import *
import numpy as np
x=symbols('x')
f=exp(-x**2)
a=0
b=2
n=50
dx=(b-a)/n
xi=np.arange(a,b,dx)
yi=[f.subs(x,i) for i in xi]
Approx=np.sum(yi)*dx
print('The area is approximately', Approx)
Open Python Notebook File