Tutorial 2: Basic Functions & Derivatives

Housekeeping: Importing required libraries

Sympy: SymPy is a Python library for symbolic mathematics. We will use this library to solve all the tutorials.

import sympy as sy
from sympy.plotting import plot,plot_implicit
from sympy.abc import c,x,y                       # variable x,y
from sympy.solvers import solve                   # Solving equations
from sympy import sin,cos
from sympy import diff, idiff                     # Solving differentiation
sy.init_printing(use_latex=True)                  # Show it in natural display

Q1: Find \(y'\) for \((x-y)^2=x+y-1\) by using implicit differentiation.

eqn = (x-y)**2-x-y+1
idiff(eqn,y,x)

Q2: Implicit differentiation to find an equation of the tangent line to the curve \(5x^2+10xy^2+5y=20\) at point \((2,2)\).

eqn = 5*x**2+10*x*y**2+5*y-20
yp = idiff(eqn,y,x)
display('Differential equation, y\':',yp)
m = yp.subs({x:2,y:2})
print('At point (2,2), the gradient is: ', m)
interception = solve(sy.Eq(2,m*2+c),c)[0]
display('The equation is: ',sy.Eq(y,m*x+interception))

"Differential equation, y':"

At point (2,2), the gradient is:  -12/17

'The equation is: '

Q3: \(y\cos{x}=1+\sin{(xy)}\). Find \(\frac{dy}{dx}\) by implicit differentiation.

eqn = y*cos(x)-1-sin(x*y)
idiff(eqn,y,x)

Q4: Find \(\frac{dy}{dx}\) by implicit differentiation \(4\cos{x}\sin{y}=1\).

eqn = 4*cos(x)*sin(y)-1
idiff(eqn,y,x)

Q5: An open box is to be made from cutting a square from each corner of a 12 in by 12 in piece of metal and then folding up the sides. What size square should be cut from each corner to produce a box of maximum volume?

eqn = 4*x**3-48*x**2+144*x
a=solve(diff(eqn))
print('Possible value of x: ',a)
print('When a = {}, V = {}'.format(a[0],eqn.subs(x,a[0])))
print('When a = {}, V = {}'.format(a[1],eqn.subs(x,a[1])))

Possible value of x:  [2, 6]
When a = 2, V = 128
When a = 6, V = 0

Q6: We are going to fence in a rectangular field. If we look at the field from above the cost of the vertical sides are RM 10/ft, the cost of the bottom is RM2/ft and the cost of the top is RM 7 /ft. If we have RM 700 determind the dimensions of the field that will maximize the enclosed area.

x=sy.symbols('x')
area = x*(35-(9/20)*x)
ans_x = solve(diff(area))[0]
ans_y = 35-9/20*ans_x
(ans_x,ans_y)