M129:
Applied Calculus
Q−1: [4+1 Marks] let 
a) Find the zeros
of g(x) and h(x).
b) Find the
points of intersection, if any, of the graphs of y = f (x) and
y = -4
.
Q−2: [2+2+1 Marks] Let 
.
.
a)
Find f (g(x)) and describe
its domain.
b) Find g
(f (x)), and describe its domain
c) Describe the
domains of f (x) and
g(x).
Q−3: [2+3
Marks] Find the slope of the tangent
line to the following curves at the designated points:
a) 
b)
Q−4: [2+3
Marks] Use
the definition of the derivative to find f ′(x) if
a) 
.
.
b)
Q−5: [5×1 Marks] Let
a)
Find
f ′(x) and f ′′(x).
b)
Find
the intervals on which f (x) is increasing or decreasing.
c)
Find
the local maximum and minimum of f (x), if any.
d)
Find
the intervals on which the graph of f (x) is concave up or
concave down.
e)
Find
the points of inflection, if any.
Q−6: [5 Marks] A
window is being built and the bottom is a rectangle and the top is a
semicircle. If there is 12 m
of framing materials what must the dimensions of the window be to let in the most light?
Q−7: [2+3
Marks]
a)
Differentiate the
function 
b)
Use the logarithmic
differentiation to differentiate the function
Q−8: [5×1
Marks] Let P(t) be the
population (in millions) of a certain city t years after 1950, and suppose
that P(t) satisfies the differential equation P′(t)
= 0.017 P(t), P(0)= 2560
a)
Find the formula for P(t).
b)
What was the initial
population (the population in 1950) ?
c)
What is the growth
constant?
d)
What was the population
in 1960?
e)
Estimate the population
in the year 2020.
