Essential Math for Data Science

Appendix B. Exercise Answers

Chapter 1

1. 62.6738 is rational because it has a finite number of decimal places, and therefore can be expressed as a fraction 626738 / 10000.

2. ${10}^7{10}^{-5}={10}^{7+-5}={10}^2=100$

3. ${81}^{\frac{1}{2}}$ = $\sqrt{(}81)=9$

4. ${25}^{\frac{3}{2}}={\left({25}^{1/2}\right)}^3=5^3=125$

5. The resulting amount would be $1,161.47. The Python script is as follows:

   from math import exp
   
   p = 1000
   r = .05
   t = 3
   n = 12
   
   a = p * (1 + (r/n))**(n * t)
   
   print(a) # prints 1161.4722313334678

6. The resulting amount would be $1161.83. The Python script is as follows:

   from math import exp
   
   p = 1000 # principal, starting amount
   r = .05 # interest rate, by year
   t = 3.0 # time, number of years
   
   a = p * exp(r*t)
   
   print(a) # prints 1161.834242728283

7. The derivative calculates to 6x, which would make the slope at x = 3 to be 18. The SymPy code is as follows:

   from sympy import *
   
   # Declare 'x' to SymPy
   x = symbols('x')
   
   # Now just use Python syntax to declare function
   f = 3*x**2 + 1
   
   # Calculate the derivative of the function
   dx_f = diff(f)
   print(dx_f) # prints 6*x
   print(dx_f.subs(x,3)) # 18

8. The area under the curve between 0 and 2 is 10. The SymPy code is as follows:

   from sympy import *
   
   # Declare 'x' to SymPy
   x = symbols('x')
   
   # Now just use Python syntax to declare function
   f = 3*x**2 + 1
   
   # Calculate the integral of the function with respect to x
   # for the area between x = 0 and 2
   area = integrate(f, (x, 0, 2))
   
   print(area) # prints 10

Chapter 2

1. 0.3 × 0.4 = 0.12; refer to “Joint Probabilities”