1 Problems
1.1 (⋆)
How is the number e defined?
1.2 (⋆)
Find the exponential function f (x) = Cax whose graph is given as below.
1.3 (⋆)
If f (x) = 5x , show that
f (x + h) − f (x) 5h − 1
= 5x ( )
h h
1.4 (⋆)
Suppose you are offered a job that lasts one month. Which of the following methods
of payment do you prefer?
1. One million dollars at the end of the month.
2. One cent on the first day of the month, two cents on the second day, four cents
on the third day, and, in general, 2n−1 cents on the nth day.
1.5 (⋆)
Under ideal conditions a certain bacteria population is known to double every three
hours. Suppose that there are initially 100 bacteria.
(a) What is the size of the population after 15 hours?
(b) What is the size of the population after t hours?
(c) Estimate the size of the population after 20 hours.
Calculus (I) Sections 1.4 and 1.5 1
1.6
A bacterial culture starts with 500 bacteria and doubles in size every half hour.
(a) How many bacteria are there after 3 hours?
(b) How many bacteria are there after t hours?
(c) How many bacteria are there after 40 minutes?
1.7 (⋆)
1. What is a function?
2. What is a one-to-one function?
1.8 (⋆)
If f (x) = x5 + x3 + x, find f −1 (3) and f (f −1 (2)).
1.9 (⋆)
In the theory of relativity, the mass of a particle with speed v is
m0
m = f (v) = q
2
1 − vc2
where m0 is the rest mass of the particle and is the speed of light in a vacuum. Find
the inverse function of and explain its meaning.
1.10 (⋆)
y = ln(x + 3)
1. Find the domain of the function.
2. Find a formula for the inverse of the function.
1.11
Find the exact value of each expression.
1. (⋆) e−2 ln 5
10
2. ln(ln ee )
1.12
When a camera flash goes off, the batteries immediately begin to recharge the flash’s
capacitor, which stores electric charge given by
t
Q(t) = Q0 (1 − e− a )
(The maximum charge capacity is Q0 and t is measured in seconds.)
1. Find the inverse of this function and explain its meaning.
2. How long does it take to recharge the capacitor to 90% of capacity if a = 2 ?
Calculus (I) Sections 1.4 and 1.5 2
1.13
Find the exact value of each expression.
1. sec−1 2
2. arctan 1
3. sin−1 ( √12 )
4. sin−1 (sin( 7π
3
))
5. (⋆) sin(2 sin−1 ( 35 ))
1.14 True or False
Determine whether the statement is true or false. If it is true, explain why. If it is
false, explain why or give an example that disproves the statement.
1. (⋆) If f (s) = f (t), then s = t.
2. (⋆) If x1 < x2 and f is a decreasing function, then f (x1 ) > f (x2 ).
3. (⋆) If f is one-to-one, then f −1 (x) = 1
f (x)
.
ln x
4. If x > 0 and a > 1, then ln a
= ln xa .
5. tan−1 (−1) = 3π
4
sin−1 x
6. tan−1 x = cos−1 x
2 Problems Plus
x
(i) Find the number of solutions of the equation sin x = 100
.
(ii) Write Fibonacci sequence Fn+2 = Fn+1 + Fn as an iteration of a function. (Hint:
f : R2 → R2 ).
(iii) Find the exact formula for Fn .
(iv) Consider the quadratic map fµ (x) = µx(a − x), a > 0. Define the iteration as
follows: xn+1 = fµ (xn ). 0 ≤ x0 ≤ 1. Let xn be the size of population at the
time n. Explain the meaning of the parameters µ and a.
Calculus (I) Sections 1.4 and 1.5 3
