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Calculus Worksheet 1
a) Lim
1. Find the following limits, if they
exist:
2x  5
b) Lim
x x  4
x 2  3x  4
c) Lim 2
x 3x  2 x  5
3x  5
d) Lim 2
x x  2 x
2  3x
e) Lim
3
x 4  x
4x 2  5
f) Lim
x 6x  3
4x 3  3x 2  2x
g) Lim
x0
x
h) Lim (6-3x)
x2
x 2  16
x4
x3
2
x3 x  9
5x 2 h  3h
j) Lim
h0
h
(a  h) 2  a 2
k) Lim
h0
h
tan 2x
l) Lim
x0
2x
sin 3x
m) Lim
x0
x
4
n) Lim
x2 x  2
i)
Lim
x2
Exercise 3.1, p51
2. Differentiate the following
a) y = 3x3 – 6x + 4
b)
c)
d)
h) y =
4
y = x5
5
i) y =
y = 6x1/3
j) y =
y = 7x-3/4
3x 2  4x  5
n) y =
x
4
55 x 3
(2x  1) 2
o) y =
2x
x
1
x
3
k) y = 5
x
e) y = 24
p) y =
q) y =
4
x3
7
m) y = 4
x
f) y = - 3 x 5
l) y =
g) y = 6 4 x 3
x 2  7x  12
x
x 3  2x  4
3
x2
3. Differentiate
a) y =
x (3x2 + 2x – 4)
b) y =
1
x
(2x + 3)
4.
a) If x =
2
y 3 , find
dx
dy
Mount Albert Grammar School
J.McHardy
b) If A =  r2, find
dA
dr
and
dr
dA
Mathematics with Calculus- calculus worksheet 1
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c) If g : y  ( y + 1)( y - 1), find
gx 
d) If f (x) = ( x +
1
f x 
5. If 2x3 – 21x2 + 72x – 6y = 0, show that
x
)2 find
dx
1
=
(x  3)(x  4)
dy
6. Given g : x  x3 +2x2 -3x + 4 find g(2), g(2), and g (2)
7. Given f : x  1 + 2x + 5x2 + 4x3 where x  R. Find the solutions of
a)
f ' (x) = 0
8. Find
d (3y )
a)
dy
b) f (x) = 0
c) (x) = 0
d) f iv (x) = 0
d (3x 3 )
b)
dx
1
d( )
u
c)
du
d)
d( v)
dv
Exercise 3.3, p60
Delta, Exercise 6.5, p73 Nos 1 – 5, 10 – 12, 14 - 18
9. Differentiate using first principles
a) y = 3x2 – 4
b) y =
x5
10. What feature of the graphs y = 4x3 + 6 and y = 5 + 4x3 means that they have the
same derived function.
11. What function would have the derived function given by
(x  h)2  3(x  h)  x 2  3x
Lim
?
h0
h
Exercise 3.2, p58
Delta, Exercise 6.5, p74, Nos 6 – 9, 13
Mount Albert Grammar School
J.McHardy
Mathematics with Calculus- calculus worksheet 1
Page 2