# Ivana Diaz
# 2/22/2017
# Lab 2 Assignment

# Question 1
f(x)=(3*x^2+1)*e^(5*x*sin(x))

# Part a
f(pi)

f(3.14)

f(1.71)

f_prime(x)=diff(f(x), x)

# Part b

f_prime(pi)

f_prime(3.14)

f_prime(1.71)

# Question 2
g(x)=ln(x^2+1)

N(f(5+0.0001)-f(5))/0.0001

N(f(5-0.0001)-f(5))/-0.0001

# We can estimate that f'(5)= about 7.928
# Part B
g_prime(x)=diff(g(x),x)

g_prime(5)

g_prime(5.0)

g_prime(3.14)

# Question 3
var("x y")

implicit_plot((x^2+y^2-4*x)^2==16*(x^2+y^2),(x,-1,8),(y,-6,6))

#Question 4

var("x,y")

implicit_plot(((4*(x^2+y^2-x)^3)==27*(x^2+y^2)^2),(x, -1,4),(y,-5, 5))