MESH Workshop for Mathematics 1A Class Test 3
Question 1
If possible, differentiate with respect to x:
(a) y = 4 sin
√
x.
(b) y = x3 sec2 x.
[
]
tan x
(c) y = 1+tan
x .
(d) y = ln
[
x(x2 +1)
√
sin x
(e) y = etan
(f) y =
−1 x
]
.
+ tan−1 (ex ).
ln(4−x2 )
√
.
x2 −9
Question 2
x+1
Consider the function f (x) = x2 +3x+2
.
(a) Find f ′ (x).
(b) Does f ′ (−1) exist? (Give reasons.)
Question 3
A table of values for f, g, f ′ and g ′ is given.
(
)
Find the values of h′ (1), p′ (1) and H ′ (1) where h, p and H are defined by h(x) = f (g(x)), p(x) = g x2 + 1
and H(x) = f (x)g(x).
x
1
2
3
f (x)
3
7
3
g(x)
3
5
9
f ′ (x)
4
4
2
g ′ (x)
6
9
6
Question 4
Let x and y be such that
(1)
sin(xy) + x = 0.
(a)
dy
Use implicit differentiation to find dx
.
(b)
dy
Find all values of x and y for which dx
is not defined.
Mathematics Education Support Hub (MESH)
www.westernsydney.edu.au/mesh
1
−1
(c)
Solve equation (1) to show that y = sin x(−x) .
(d)
dy
Differentiate y = sin x(−x) explicitly to find dx
.
(e)
Are the derivatives found in (a) and (d) the same?
−1
Question 5
Use logarithmic differentiation to find
dy
when
dx
(a) y = (1 − 5x)cos(x)
2
(b) y = xe(−x )
Question 6
(a) If f (x) is a differentiable function with f (1) = 1 satisfying
xf (x) + f (x2 ) = 2.
Find f ′ (1).
(b) If possible, differentiate y =
(
x+sin x
x−sin x
)2/3
with respect to x.
Question 7
Let f be a function for which:
(1) f (x + y) = f (x) + f (y) + x2 y + xy 2 , for all real numbers x and y
(2)
f (x)
=1
x→0 x
lim
Find f ′ (x).
Mathematics Education Support Hub (MESH)
www.westernsydney.edu.au/mesh
2