UNIVERSITY OF NAIROBI
Department of Mathematics
CALCULUS REVISION QUESTIONS
Isaiah 1:19
“The only way to do MATHEMATICS is by DOING MATHEMATICS!”
************************Enjoy as many problems as possible!**********************
SET ONE
Question 1 Total for Question 1: 54 marks
(a) Evaluate the following integrals:
∫ x
i. √1+5x 2 dx.
(4 marks)
∫
ii. (2x3 + x)(x4 + x2 + 1)49 dx. (4 marks)
∫
iii. sec2 4x tan 4xdx.
(4 marks)
∫ √
iv. (sec2 x) tan3 xdx. (4 marks)
(b) Find the 100-th derivative of
p(x) = (x + x5 + x7 )10 (1 + x2 )11 (x3 + x5 + x7 ).
(4 marks)
(c) Determine the slope of the tangent line to the graph of x2 + 4y 2 = 4.
( marks)
(d) What are y (100) (x) and y (101) (x) for y = cosh x?
(4 marks)
(e) State the formal definition of a limit of a function f (x) about a point x = a.
(2 marks)
,For any inquiry, send an email to:
(f) Show that the circles
C1 : x2 + y 2 − 12x − 6y + 25 = 0
and
C2 : x2 + y 2 + 2x + y − 10 = 0
are tangent to each other at the point (2,1).
(7 marks)
Question 2 Total for Question 2: 50 marks
(a) Evaluate the following integrals:
∫
i. tan2 θ sec4 θdθ
(5 marks)
∫ 3 5 2
ii. (x + x) (3x + 1)dx (5 marks)
(b) Find a general formula for f (n) (x).
i. f (x) = x−2
(4 marks)
−1
ii. f (x) = (x + 2) (4 marks)
(c) What is the slope of the tangent line to y = 4x at x = 0?
(3 marks)
(d) Give an ε − δ proof of the fact that lim x2 = 100.
x→10
(5 marks)
Question 3 Total for Question 3: 107 marks
(a) Use formal definition of limit to prove the following:
i. lim 2x + 8 = 14 (4 marks)
x→3
ii. lim x2 = 9 (4 marks)
x→3
(b) Find dy
dx
given that y 3 + y 2 − 5y − x2 = −4.
(4 marks)
(c) Give an ε − δ proof of the fact that lim x2 = 100.
x→10
(5 marks)
(d) The small arches have the shape of parabolas. The first is given by f (x) = 1−x2
for −1 ≤ x ≤ 1 and the second by g(x) = 4 − (x − 4)2 for 2 ≤ x ≤ 6. A board is
placed on top of these arches so it rests o both. What is the slope of the board?
HINT: Find the tangent line to y = f (x) that intersects y = g(x) in exactly
one point.
(10 marks)
Page 2 of 50
NOTE: In order to get free access to ALL the solutions for these problems and many
more other Mathematical work; subscribe to the Youtube Channel (Mathematics Can
Smile) and you will be directed to the solutions playlist.
,For any inquiry, send an email to:
Question 4 Total for Question 4: 76 marks
(a) Calculate the higher derivative:
i. f ′′ (θ), f (θ) = θ sin θ
(5 marks)
′′ 2
ii. f (t), cos t (5 marks)
(b) Find lim f (x+h)−f (x)
h
when f (x) = x3 + 3x2 − 3x + 9.
h→0
(4 marks)
f (x+h)−f (x)
√
(c) Find lim h
when f (x) = 2x.
h→0
(4 marks)
(d) The conical watering pail has a grid of holes. Water flows out through the holes
at a rate of kA m3 /min, where k is a constant and A is the surface area
√ of the
part of the cone in contact with the water. This surface area is Aπr h2 + r2
and the volume is V = 31 πr2 h. Calculate the rate dh
dt
at which the water level
changes at h = 0.3 m, assuming that k = 0.25 m. (6 marks)
(e) If f is an odd function and f (0) is defined, must f (0) = 0? (2 marks)
(f) If f (x) = x2 + kx + 1 for all x and f is an even function, find k. (2 marks)
(g) If f (x) = x − kx + 2x for all x and f is an odd function, find k.
3 2
(2 marks)
Page 3 of 50
NOTE: In order to get free access to ALL the solutions for these problems and many
more other Mathematical work; subscribe to the Youtube Channel (Mathematics Can
Smile) and you will be directed to the solutions playlist.
, For any inquiry, send an email to:
Question 5 Total for Question 5: 182 marks
(a) Evaluate the following integrals:
∫√
i. 2x + 1dx
(4 marks)
∫ 2
ii. sec (5x + 1)5dx (4 marks)
(b) Find lim f (x+h)−f (x)
h
when f (x) = 4x2 − x.
h→0
(4 marks)
(c) Determine the slope of the graph of 3(x2 + y 2 )2 = 100xy at the point (3, 1).
(5 marks)
(d) Find the area of the region between the graphs of the functions
f (x) = x2 − 4x + 10 g(x) = 4x − x2 , 1 ≤ x ≤ 3.
(10 marks)
(e) Find the domain and range of the following functions
i. h(x) = 4 − x2 (3 marks)
√
ii. G(x) = −2 x (3 marks)
√
iii. U (x) = x2 − 4 (3 marks)
√
iv. H(x) = 4 − x2 (3 marks)
v. V (x) =| x − 1 | (3 marks)
vi. f (x) = [2x] =the greatest integer ≤ 2x. (3 marks)
vii. h(x) = x1 (3 marks)
1
viii. F (x) = x−1 (3 marks)
ix. H(x) = − 12 x3 (3 marks)
x2 −4
x. G(x) = x+2
(3 marks)
|x|
xi. f (x) = x
(3 marks)
xii. g(x) = √ 1 (3 marks)
1−x2
(f) In the following problems, evaluate f (5t + 2) for the given function f.
i. f (x) = x + 4 (3 marks)
ii. f (x) = 2x + 5 (3 marks)
iii. f (y) = y − 9 (3 marks)
iv. f (w) = y 2 + 6 (3 marks)
v. f (a) = 5x + 10 (3 marks)
vi. f (x) = x2 + 3x − 9 (3 marks)
(g) In the following problems, evaluate f ogoh, f ohog, gof oh, gohof, hof og and hogof
for the given functions f, g and h.
i. f (x) = 3x2 + 3 g(x) = cos x h(x) = tan x
(6 marks)
ii. f (x) = e 2x
g(x) = − cos(θx) h(x) = tan 5x (6 marks)
2
iii. f (x) = x + 3x + 3 g(x) = sin(3x) h(x) = tan x (6 marks)
iv. f (x) = x − 4
2
g(x) = 3x + cos x 2
h(x) = 1 − tan x (6 marks)
Page 4 of 50
NOTE: In order to get free access to ALL the solutions for these problems and many
more other Mathematical work; subscribe to the Youtube Channel (Mathematics Can
Smile) and you will be directed to the solutions playlist.
Department of Mathematics
CALCULUS REVISION QUESTIONS
Isaiah 1:19
“The only way to do MATHEMATICS is by DOING MATHEMATICS!”
************************Enjoy as many problems as possible!**********************
SET ONE
Question 1 Total for Question 1: 54 marks
(a) Evaluate the following integrals:
∫ x
i. √1+5x 2 dx.
(4 marks)
∫
ii. (2x3 + x)(x4 + x2 + 1)49 dx. (4 marks)
∫
iii. sec2 4x tan 4xdx.
(4 marks)
∫ √
iv. (sec2 x) tan3 xdx. (4 marks)
(b) Find the 100-th derivative of
p(x) = (x + x5 + x7 )10 (1 + x2 )11 (x3 + x5 + x7 ).
(4 marks)
(c) Determine the slope of the tangent line to the graph of x2 + 4y 2 = 4.
( marks)
(d) What are y (100) (x) and y (101) (x) for y = cosh x?
(4 marks)
(e) State the formal definition of a limit of a function f (x) about a point x = a.
(2 marks)
,For any inquiry, send an email to:
(f) Show that the circles
C1 : x2 + y 2 − 12x − 6y + 25 = 0
and
C2 : x2 + y 2 + 2x + y − 10 = 0
are tangent to each other at the point (2,1).
(7 marks)
Question 2 Total for Question 2: 50 marks
(a) Evaluate the following integrals:
∫
i. tan2 θ sec4 θdθ
(5 marks)
∫ 3 5 2
ii. (x + x) (3x + 1)dx (5 marks)
(b) Find a general formula for f (n) (x).
i. f (x) = x−2
(4 marks)
−1
ii. f (x) = (x + 2) (4 marks)
(c) What is the slope of the tangent line to y = 4x at x = 0?
(3 marks)
(d) Give an ε − δ proof of the fact that lim x2 = 100.
x→10
(5 marks)
Question 3 Total for Question 3: 107 marks
(a) Use formal definition of limit to prove the following:
i. lim 2x + 8 = 14 (4 marks)
x→3
ii. lim x2 = 9 (4 marks)
x→3
(b) Find dy
dx
given that y 3 + y 2 − 5y − x2 = −4.
(4 marks)
(c) Give an ε − δ proof of the fact that lim x2 = 100.
x→10
(5 marks)
(d) The small arches have the shape of parabolas. The first is given by f (x) = 1−x2
for −1 ≤ x ≤ 1 and the second by g(x) = 4 − (x − 4)2 for 2 ≤ x ≤ 6. A board is
placed on top of these arches so it rests o both. What is the slope of the board?
HINT: Find the tangent line to y = f (x) that intersects y = g(x) in exactly
one point.
(10 marks)
Page 2 of 50
NOTE: In order to get free access to ALL the solutions for these problems and many
more other Mathematical work; subscribe to the Youtube Channel (Mathematics Can
Smile) and you will be directed to the solutions playlist.
,For any inquiry, send an email to:
Question 4 Total for Question 4: 76 marks
(a) Calculate the higher derivative:
i. f ′′ (θ), f (θ) = θ sin θ
(5 marks)
′′ 2
ii. f (t), cos t (5 marks)
(b) Find lim f (x+h)−f (x)
h
when f (x) = x3 + 3x2 − 3x + 9.
h→0
(4 marks)
f (x+h)−f (x)
√
(c) Find lim h
when f (x) = 2x.
h→0
(4 marks)
(d) The conical watering pail has a grid of holes. Water flows out through the holes
at a rate of kA m3 /min, where k is a constant and A is the surface area
√ of the
part of the cone in contact with the water. This surface area is Aπr h2 + r2
and the volume is V = 31 πr2 h. Calculate the rate dh
dt
at which the water level
changes at h = 0.3 m, assuming that k = 0.25 m. (6 marks)
(e) If f is an odd function and f (0) is defined, must f (0) = 0? (2 marks)
(f) If f (x) = x2 + kx + 1 for all x and f is an even function, find k. (2 marks)
(g) If f (x) = x − kx + 2x for all x and f is an odd function, find k.
3 2
(2 marks)
Page 3 of 50
NOTE: In order to get free access to ALL the solutions for these problems and many
more other Mathematical work; subscribe to the Youtube Channel (Mathematics Can
Smile) and you will be directed to the solutions playlist.
, For any inquiry, send an email to:
Question 5 Total for Question 5: 182 marks
(a) Evaluate the following integrals:
∫√
i. 2x + 1dx
(4 marks)
∫ 2
ii. sec (5x + 1)5dx (4 marks)
(b) Find lim f (x+h)−f (x)
h
when f (x) = 4x2 − x.
h→0
(4 marks)
(c) Determine the slope of the graph of 3(x2 + y 2 )2 = 100xy at the point (3, 1).
(5 marks)
(d) Find the area of the region between the graphs of the functions
f (x) = x2 − 4x + 10 g(x) = 4x − x2 , 1 ≤ x ≤ 3.
(10 marks)
(e) Find the domain and range of the following functions
i. h(x) = 4 − x2 (3 marks)
√
ii. G(x) = −2 x (3 marks)
√
iii. U (x) = x2 − 4 (3 marks)
√
iv. H(x) = 4 − x2 (3 marks)
v. V (x) =| x − 1 | (3 marks)
vi. f (x) = [2x] =the greatest integer ≤ 2x. (3 marks)
vii. h(x) = x1 (3 marks)
1
viii. F (x) = x−1 (3 marks)
ix. H(x) = − 12 x3 (3 marks)
x2 −4
x. G(x) = x+2
(3 marks)
|x|
xi. f (x) = x
(3 marks)
xii. g(x) = √ 1 (3 marks)
1−x2
(f) In the following problems, evaluate f (5t + 2) for the given function f.
i. f (x) = x + 4 (3 marks)
ii. f (x) = 2x + 5 (3 marks)
iii. f (y) = y − 9 (3 marks)
iv. f (w) = y 2 + 6 (3 marks)
v. f (a) = 5x + 10 (3 marks)
vi. f (x) = x2 + 3x − 9 (3 marks)
(g) In the following problems, evaluate f ogoh, f ohog, gof oh, gohof, hof og and hogof
for the given functions f, g and h.
i. f (x) = 3x2 + 3 g(x) = cos x h(x) = tan x
(6 marks)
ii. f (x) = e 2x
g(x) = − cos(θx) h(x) = tan 5x (6 marks)
2
iii. f (x) = x + 3x + 3 g(x) = sin(3x) h(x) = tan x (6 marks)
iv. f (x) = x − 4
2
g(x) = 3x + cos x 2
h(x) = 1 − tan x (6 marks)
Page 4 of 50
NOTE: In order to get free access to ALL the solutions for these problems and many
more other Mathematical work; subscribe to the Youtube Channel (Mathematics Can
Smile) and you will be directed to the solutions playlist.
