The University of Zambia
Department of Mathematics & Statistics
MAT2110-Engineering Mathematics I
Tutorial sheet 2
1. Find y 0 , y 00 and y 000 for each of the following:
a) y = (3 − 2x)7 .
b) y =
6
.
(x−1)2
c) y =
x−1
.
x+1
d) y = tan x.
e) y =
sin x
.
x
2. Calculate enough derivatives to enable you guess the general formula
for f (n) (x).
a) f (x) = x1 .
b) f (x) =
1
.
2−x
c) f (x) =
√
x.
d) f (x) =
1
.
a+bx
e) f (x) = cos(ax).
3. If y = sec(kx), show that y 00 = k 2 y(2y 2 − 1).
4. Use mathematical induction to prove that the nth derivative of the
function y = tan x is of the form Pn+1 (tan x) where Pn+1 is a polynomial
of degree n + 1.
5. If f and g are twice differentiable functions, show that
(f g)00 = f 00 g + 2f 0 g 0 + f g 00 .
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6. Find dy/dx in terms of x and y in the following:
a) x3 y 3 = 2x − y.
b)
x−y
x+y
=
x2
y
+ 1.
√
c) x x + y = 8 − xy.
7. Find the equation of the tangent to the given curve at the given point.
a) x2 y 3 − x3 y 2 = 12 at (−1, 2).
b) x + 2y + 1 =
y2
x−1
at (2, −1).
c) x sin(xy − y 2 ) = x2 − 1 at (1, 1).
√
d) 2x + y − 2 sin(xy) = π2 at ( π4 , 1).
8. Find y 00 in terms of x and y in the following:
a) xy = x + y.
b) x2 + 4y 2 = 4.
c) x3 − 3xy + y 3 = 1.
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