CALCULUS I & II
TUTORIAL 3: FUNCTIONS
Problem 1: For f( x) = 2x2+ 5x+3 and g(x) = 4x+1 find the following
a) (f+g)(x)
b) (f-g)(x)
c) (f.g)(x)
d) (f/g)(x)
e) f0g(x)
Problem 2: The number N of cars produced at a certain factory in 1 day after t hours of
operation is given by N(t) = l00t- 5t2, 0≤t≤10. If the cost C (in dollars) of producing N cars is
C(N) = 15,000 + 8,000N, find the cost C as a function of the time t of operation of the factory.
Problem 3: Find the inverse of the following functions.
a) f(x) = 2x-3
b) f(x) = x3-1
c) f(x) = x2-1
Graph f, f-1 , and y = x on the same coordinate axes.
Problem 4: The price p, in dollars, of a Honda Civic DX Sedan that is x years old is given by
p(x) = 16,630(0.90)x
a) How much does a 3-year-old Civic DX Sedan cost?
b) How much does a 9-year-old Civic DX Sedan cost?
Problem 5: When you drive an Ace Rental compact car x kilometers in a day, the company
charge f(x) dollars, where
if
0  x  100
30
f ( x)  
if
30  0.07( x  100)
x  100
Describe Ace Rental’s pricing policy in plain English. (Be sure to interpret the constants 30, 0.07, and
100 that appear in the pricing formula)
Problem 6: For the following demand and supply functions of a product, state the economically
sensible ranges of price and quantity for which they are defined. Draw the market diagram for
this product. What are the equilibrium price and quantity?
QD = 16 – 2p
QS = -4 + 3p
Problem 7: Consider the following demand and supply functions for a product.
q = 500 -10p and q = -100+5p
a) Find the inverse demand function and the inverse supply function.
b) Draw the market diagram for this product.
c) Find the equilibrium price and quantity.
FMT - 2012
Page 1
CALCULUS I & II
TUTORIAL 4: SEQUENCES, SERIES, LIMITS
Problem 1: Write down the first five terms of the following sequences
1 𝑛−1
1
{ };{
} ; { 𝑛}
𝑛
𝑛
2
Problem 2: Determine the convergence or divergence of the following sequences
1 𝑛−1
1
{ };{
} ; { 𝑛}
𝑛
𝑛
2
Problem 3: Compute the following limits
1) lim
𝑛2 −2𝑛+3
𝑛→∞ 2𝑛2 −1
−2𝑛+3
2) lim
𝑛→∞ 2𝑛2 −1
3) lim (√𝑛 + 25 − √𝑛)
𝑛→∞
Problem 4: Determine the convergence or divergence of the following series.
2
1)∑∞
𝑛=1 5𝑛−1
1
2) ∑∞
𝑛=1 𝑛3𝑛
1
3) ∑∞
𝑛=1 3𝑛
Problem 5: Determine the sum of the following geometric series, when they are convergent.
1
1
1
1)1 + 6 + 62 + 63+….
1
1
1
2)1 + 23 + 26 + 29 +….
1
1
1
32
34
3)32 − 34 + 36 - ….
36
4)1 + 6 + 62 + 63 +….
Problem 6: 29(577)
Problem 7: 33(577)
FMT - 2012
Page 2