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MATH275 Practice Sheets
Calculus for Engineers and Scientists (University of Calgary)
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MATH 275 Practice Sheets
FALL 2021
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Practice Sheet 1
1. Let h x
x2
2. Let g x
x 3 . Determine g
f x
2
f
3 f x . Determine h
2x
3 2
3
2 given that f
f2
2
2 3 , and
3
2, g 4
3
7, f
f
f
2
f x3
6. Let f x
20x 1 , find f x and simplify your answer.
2x 1 10
10. Let y
x2
f 4
h
h
h
f 1
f 1
h
0
9. Determine lim
x
. Determine g 2 , given that f 8
f 4
8. Determine lim
3 , and g 4
f x
x
2
14
f 2
if f x
2
6 3 x2
1 , and f 8
4 h 2
.
2h 4 h
x5
if f x
3
8 . Compute
3. Determine
1
5. Let g x
h
6
9 , and f
2, f 1
1
7. Find f 4 if
1
4.
g
4. Let f be functions such that f 1
a
4, and f
1
60.
1.
3. Let f and g be functions such that f
Determine f
1 given that f
14
10x 3
1
3x .
dy
at x
dx
3.
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2.
1.
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ANSWERS
1. 240
36 , b 27
4. a
7.
1
16
2. 1
3.
2
5.
6.
360x
2x 1 11
144
8.
25
9.
1
4
10. 9
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Practice Sheet 2
1. Find an equation of the line tangent to the curve f x
the point on the curve where x
at the point on the curve where x
y
7 3 2x
5 10
17 at
2.
31x 3
2. Find an equation of the tangent line to the curve y
12x 2 36x
4x 3
16
2.
2
x
3. Find all points on the curve y
4x
5x where the normal line is parallel to the line
4 .
11
1x
3
4. Find equations of all tangent lines to the curve y
2x 2
10x
7 which pass through
the point 2, 3 ( Not on the curve).
x4 1
1 x
4
2
x 2x 3
2x 2 3x 1
5. Given f x
b
6. Let f x
2x 3
16
x
ax 2 1
x
1
x
1 Is f differentiable at x
x
1
x
2
x
2
1?
Determine constant real numbers a and b so that the function f is differentiable
at x
7. Let y
2. What is f
x2
x
2?
2x 2 . Find y " and simplify!
1 2
8. Find all values of the constant real number k such that y
x 2 y " 3xy
32y
9. Determine lim
f" 1
2
f" 1
h
4
10. Evaluate lim
0.
h
0
h
x
0, x
3x 5
x 2
x k satisfies the equation
1
if f x
x5
10x 3
1
.
Hint : Interpret limit as derivative of some function f at c.
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ANSWERS
1. y
8x
3. 1, 7 , and
2. y
1, 7
5. f is not differentiable at x
7.
9.
x
0
18
1 4
1.
9x
4
4. y
34x
65 , and y
2x
6. a
5, b
3 , and f
2
8. k
8 , or 4.
10. 3
4
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4
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Practice Sheet 3
2 sin x
dy
if y
dx
1. Find
9 tan x 7 sec x tan x
tan x
25
cos x 3 .
2. Find f " x for f x
cos 3x 2 satisfies the equation xy" y
3. Verify that y
4. Find an equation of the tangent line to the curve y
the curve where x
1
1
5. Let f x
36x 3 y
0.
tan
at the point on
x2
2.
cos 2x
cos 2x
with 0
x
2
, find f " x .
Hint : First , simplify f using Double Angle Identities.
6. Given y
sin 2x
2 sin x
2x
1, 0
x
. Determine the x
point on the curve where the tangent line is horizontal.
7. In each case find f x :
a
tan 2
f x
sec 2
x
10
x
x2
cos 4 x
b f x
sin 4 x
Hint : First , simplify f using suitable Trigonometric Identities.
f2 1
8. Let g x
tan 3x
for some differentiable function f.
Determine g 0 given that f 1
9. Let y
tan sin x cos x
10. Let f x
tan x
. Find
1 tan 3 x
3
2 and f 1
5.
dy
.
dx
3. Show that f x
sec 4 x .
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coordinate of each
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ANSWERS
2 sin x
1. y
6x sin x 3
2. f " x
4. y
7 sec x tan x
2
25 csc 2 x
9x 4 cos x 3
1
x
2 sec 2 x tan x
5. f " x
, 2 .
3
6. Two points with x
coordinates
20
x 21
b f x
2 sin 2x
cos x cos x
cos x
7. a f x
8. g 0
9. y
2
60
sec 2 sin x cos x
x sin x
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Practice Sheet 4
1. Find the equation of the tangent and normal lines to the graph of the relation :
x2
y2
5xy
16y at the point P 3, 4 on its graph.
1
2. Assuming that the relation y 3
function of x. Find
2xy 2
5x 3
1
x
2 defines y as a differentiable
y
dy
.
dx
y2
3. Given the relation x
2y Assuming relation defines y as a function of x ,
find y" as a function of y only.
4. a A cube has a volume V 1000 cubic centimeters. Determine approximately by
how many cubic centimeters should V increase if the side length x of the cube is
to be increased by 1%
b Determine by approximately what percentage the volume of a spherical balloon
increase if its surface area increased by 4%.
5. The pressure difference P between the ends of an arteriole is related to its radius r
by the equation P
k , where k is a constant. Use differentials to determine by
r4
approximately what percentage the pressure difference P increase , if the radius
of an arteriole decreases by 10%.
6. Which of the following is an antiderivative of f x
i F1 x
csc 4 x
ii F 2 x
cot 4 x
iii F 4 x
csc 4 x
2 cot 2 x
4 csc 2 x cot 3 x ?
9
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x2
7. Which of the following is an antiderivative of f x
2
1 x 1 3 x 2 3x 90
3
1x x 1 2 7x 1
3
3
1 x 3 2x 61
3
i F1 x
ii F 2 x
iii F 3 x
8. Evaluate :
i
cos 2 4x dx
ii
sin 2x
dx
cos 2 2x
iii
12 sin 3x cos 3x dx
iv
cot 2 x dx
9. Find the function g with the properties :
g" x
24 sin 12x
15 cos 3x
1 , and g 0
3
2, g 0
10. Find the function f such that
f 1
9 , and lim
h
0
f x
h
h
f x
1
2 x
2x
4.
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ANSWERS
1. Equation of tangent line : y
Equation of normal line : x
2. y
15x 2
2y 2
3y 2
4xy
103x
103y
305 , and
415
0.
1 x 1/2
2
2y 2
1
3. y
y
1 3
4. a Volume of Cube increases by 30 cubic centimeters.
b Volume of cube increases by 6 %.
5. The Pressure P increases by 40 %.
6. F 2 x & F 3 x are antiderivatives ; but F 1 x is not.
7. F 1 x & F 3 x are antiderivatives ; but F 2 x is not.
8. i
1 sin 8x
8
1 x
2
iii
cos 6x
C.
C.
9. g x
1 sin 12x
6
10. f x
x
x2
5 cos 3x
3
4x
ii 1 sec 2x
2
iv
cot x
x2
7x
C.
x
C.
2.
13.
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Practice Sheet 5
1. Evaluate each of the following limits :
10
2
3x
1
a lim
x2 4
x 2
3
2. Find
lim
b
x
64x 6
1
64x 4
3x 14
2x 2
17
x3
x
9 has an inverse on R
3. Assuming the function f x
If f
1
x
1
2
x
i
x
x
1
lim
ii If f
1
2x
7, x
1 has an inverse.
2x
3, x
R has an inverse g.
2 , find a.
a
x2
4. Assuming the function f x
Find an expression for y
f
1
x.
x5
5. Assuming the function f x
b , find b.
9
Find an equation of the line tangent to graph of g at the point 0, 1 on its graph.
6. Assume that the function f has an inverse g. Find d
dx
given that f 4
7. Express y
log x
8. Express y
log
8, f 4
2 log x
6
6 , and f
1 log x 2
2
1
x 1 cot 8 2x
1 4x 30 x 4x
8
gx
at x
8
15.
1 as a Single Logarithm.
as a Sum or Difference of Logarithms.
9. Solve for x :
a 7 2x 1
1
b 4x
2
1
8
5/2
c 2 log 4 x
4
log 4 x
1
2
10. Simplify Completely:
a log 1/ x x 3 , x
0 , x
1
b log 9 27
x
c 5 6 log 5 x , x
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ANSWERS
1. a
2.
4
3.
i a
3
10
b
1
2
1.
ii b
0.
1
4. y
f
5. y
1x
7
6.
x
6.
1.
1
24
7. y
log
xx
1 2
x2
1
1 log x
6
8. y
9.
1
x
1
2
a x
10. a
6
1
8 log |cot 2x |
b x
b
3x
2
30 log |1
4x|
4x log |x|
1
c x
2
c
1
x6
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Practice Sheet 6
1. Find the slope of the line tangent to the curve x 2
y2ex y
2 at the point 1, 1
x
1 6 2x 3 4 5 2x
at the
4x 7 2 13 6x 5
on the curve.
2. Find an equation of the line tangent to the curve y
point 2, 1 on the curve.
dy
if y
dx
3. Find
log e x 1
x x
, x
0.
4. Find exact values of
a tan tan 1 100
b
c cos 2 cot 1 7
d sin cos 1 x
e tan tan 1 100
f tan 1 tan 100
tan 1 x 2
g
cos
h
e ln cos
1
1 , x
, 1
cos sec 1 2
, x
0, 1
1,
1/2
5. Find Exact values of
i cos 1 cos 53
5
ii tan 1 tan
17
3
iii
sin 1 sin 53
5
6. Find the derivative of each of the following functions. Simplify your answers.
i f x
2 x arcsin
x
3 3/2 sin 1 2x
ii y
3 2x
iii y
x 4 arccos x 2
iv f x
x tan
vi h x
e ln cos
1 x
2 1
x
x 2 arctan x 2
1 ln 1
2
3
4
2x 3/2
3 4
2x
1 x2 1
2
x4
1 arcsin x 2
2
v gx
e cos
x4
7
1 x
1 x
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7. Let f x
2 arcsin x 1 , find f 1 .
8. Find the exact values of each of the following ;
i cos 1
1
2
ii sin 1 1
iii
csc 1 2
This Problem is for Student to do at Home.
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ANSWERS
1. Slope m
3
2. Equation : 34x
dy
dx
3.
67
y
0
1
1
x
4. a 100
b
1
2
e 100
f
0
3
5
ii
5. i
1 sin 1
x
6. i f x
dy
dx
9 2x
iii
dy
dx
4x 3 cos 1 x 2
iv f x
x tan
3 sin 1
iii
3
2x
ln x
1 x2
1 x
g
3
4
d
1
|x|
h
1
x
3
2
2
5
2x tan 1 x 2
x
ii
c
3
tan 1 x
x
1
e cos x
1 x2
v g x
1
vi h x
cos
7. f 1
8. i
1
x
2
x2
1
ln 2
3
4
ii
2
iii
6
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Practice Sheet 7
1. Find the exact values of
a cosh
ln 3
b csch 2
c tanh 1 2
3
d cosh 1 3
2. If sinh ln x
12 , find the exact values of
5
ii tanh ln x
i x
3. If tanh
sech
3
, find the exact values of sinh
2
, and coth
, cosh
, csch
,
.
4. Find the derivatives of each of the following functions. Simplify your answer.
a f x
x sinh 1 x
b g x
tanh 1 sin x
1
x2 .
tan 1 sinh x .
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ANSWERS
1. a
5
3
2
b
e
2
e
2
or
2e 2
1 e4
c
1 ln 5
2
d
ln 3
8
312
313
2. i 25
ii
3. sinh
3 , cosh
4. a f x
sinh 1 x
2 , csch
b g x
1 , sech
3
sec x
1 , and coth
2
sech x
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2
3
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Practice Sheet 8
1. A child is running at a constant rate of 1 ft / sec towards a street upright light post 18
feet high above the ground. If the shadow of the child on ground is decreasing at the
rate of 0. 2 ft / sec, how tall is the child?
2. The current in an electric system is decreasing at a rate of 5 amperes / s while the
voltage remains constant at 9 volts. Find the rate of change of the resistance R when
the current is 18 amperes.
The voltage V in volts , the resistance R in ohmms, and the current I in amperes are
related by the formula V
I R.
3. Newton’s universal law of gravitation states that the bodied of masses m 1 and m 2 at a
distance r unites apart exert a force of attraction F on each other given by the formula
G m1m2
F
, where G is the Universal Gravitational Constant.
r2
Suppose when a comet is 600 million kilometer from the earth and receding at 12000 km /s ,
the force between the comet and the earth is 3000 N (Newtons).How fast is the force of
attraction changing at that instant?
4. A lighthouse which makes 0. 2 revolution /s is located on an island 200 m across from a
straight shoreline.How fast is the light beam moving along the shore at the instant it makes
an angle of 30 0 with the shore?
5. The power P (watts) in an electric circuit is related to the circuit’s resistance R (ohms)
and the current I (amperes) by the equation P
R I 2.
If the power of a circuit is a constant 100 watts and the resistance R is decreasing at
the rate of 3 omhs / min , at what rate is the current I changing at the instant the current
is 2 amperes?
6. Coffee is poured at a uniform rate of 20 cm 3 /s into a cup whose inside is shaped like a
truncated cone. If the upper and lower radii of the cup are 4 cm and 2 cm and the height
of the cup is 6 cm , how fast will the coffee level be rising when the coffee is halfway up?
Hint : Extend the cup downward to form a cone.
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ANSWERS
1. The Child is 3 feet tall ( about 91 cm)
2. The resistance is increasing at the rate of
5 ohms/s
36
3. The force of attraction is decreasing at the rate of 0. 12 N/s
4. 320 m/s
5. dI
dt
3 amperes/ min
25
6. Coffee level is rising at the rate of 20 cm/s
9
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Practice Sheet 9
1. Evaluate each of the following limits :
2x
3x
lim 3e x 2 2e
e
1
x 0
a
0
4
lim x
e
lim
f
x2
x2
1
x
1
2
x2
7
4
0
x
10
2
x
lim 2
d
3
x
tan 1 x
x3
x
0
x
2x 3 7x
ln x
1
x
lim
b
ln cos x
ln cos 3x
c lim
x
1
2. Evaluate each of the following limits :
1
x
lim
a
0
x
1
ex
0
x
lim 2x sinh 4x
lim x
1
x
1
sin x
lim e x tan 4e x
f
x
g
x
e 1/x
e
lim
d
x
e
1
x
lim ln x 1/x
c
lim
b
1
x
2 arctan x
lim cos x sec 3x
h
x
x
2
3. Evaluate each of the following limits :
a
lim
x
c
2x
5x 2
2
1 x
lim 1
b
1
lim
x
4
1
x2
1
ln x
x
1
2 ln x
3
1 x
d
lim x
1
ln x 3 1
x
Parts c & d are for students to do at home.
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ANSWERS
1. a
3
b
1
3
c
1
9
d
ln 2
e
17
f
1
3
2. a
1
2
b
c
0
d
0
e
8
f
4
g 2
3. a
e 16
c
e
1
1
3
h
b e2
6
d
3
e
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Practice Sheet 10
1. Determine :
x ln x dx
a
b
2x sin 2 x dx
c
e 2x cosh 3x dx
b
6x 3 e x dx
b
e
2. Determine :
3
6x 2 e x dx
a
2
3. Evaluate :
3
a
e x dx
3
x
dx
4. Determine :
a
cosh 1 2x dx
cot 1 3x dx
b
5. Evaluate :
1 2 cos 3x dx.
9 3x
a
b
ln 2 x dx
6. In each case , complete the square :
a 4x 2
c
20x
26
b
x2
6x
1
d x
x2
25x 2
6x
5
7. Evaluate :
a
9x
d
2
1
24x
x 1 dx
2x x 2
dx
2x
b
7
x
e
2
e 2x
5
dx
4x
13
ex
10e x
11
2x
c
dx , x
4x
0.
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2
7
dx
4x 3
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ANSWERS
1. a
2 x 3/2 ln x
3
c 1 ex
2
2. a
2e x
1 e 5x
5
3
3. a 3e x/3
4 x 3/2
9
C
C
b 3 x2
C
b 3 x 2/3
1 4x 2
2
1
5. a 3 3x
1 2 sin 3x
6 3x
b x ln 2 x
2x ln x
2x
6. a
2x
5 2
c
5x
3
5
1 cos 3x
6 sin 3x
4x
13
c
1 ln 4x 2
4
4x
d
2x
x2
C
e
x
cosh 1 e
5
6
2
1 cos 2x
4
C
2x 1/3
2 ex
C
b 4
x
3 2
1
4
x
1
2
d
2
C
9 sinh 1 x
2
3 tan 1
2
3
C
3
2x
1
2
1/3
1 ln 1
6
C
16
25
7. a 1 sin 1 3x 4
3
3
1 ex
b x cot 1 3x
C
1
2
1 x sin 2x
2
C
4. a x cosh 1 2x
b 2 x2
1 x2
2
b
C
C
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C
9x 2
C
C
lOMoARcPSD|16110024
Practice Sheet 11
1. Evaluate :
x2
dx
1 x 2 3/2
a
1
x2 x2
b
dx
4
2. Find :
x2 1
dx , x
x
a
c
x
1.
16
dx
x 2 16
2
x2
4
b
x
dx
x2
d
x
2
dx
1 2
3. Determine :
x 3 6x 2
x 2 3x
a
2x dx
2
b
20
dx
4x 2 20x
b
2x 3 tan 1 x dx
4. Determine
a
x3
x4
108
18x 2
81
dx
5. Determine whether the improper integral converges or diverges
using a suitable limit :
0
a
c
1
0
d
dx
1
1
e 3x
e 2x
dx
b
d
dx
x sinh 3x
dx
d
4
d
dx
1
2
e 3x
e 2x
d x tan
x
dx
dx
dx
6. In each case , determine whether the improper integral converges or diverges using
an appropriate limit :
2
a
1
5
1
dx
1 x
ln
d
1
4
x
2
x
2
b
c
3
f
0
0
dx
0
tan 2 x dx
e
x 2 e x dx
1
dx
9x 2 1
1
e 1/x dx
2
1 x
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ANSWERS
x
1. a
2. a
c
3.
1
x2
x2
1
x2
16
sec 1 x
a
1 x2
2
b
1 tan 1 x
x2
b
ln|x
C
C
3x
4x
2 ln
b
2
4
x
d 1 tan 1 x
2
12 ln|x
1
x
2|
3
x
3 ln|x
tan 1 x
1 tan 1 x 2
4
2
3|
x2
4
b
C
x
4. a ln|x|
5.
sin 1 x
3
ln|x
1|
C
x2
4
x
1
x2
x2
C
C
C
C
1 ln x 2
2
4x
20
3|
3
C
x
3
C
a The Improper Integral Diverges to
b The Improper Integral Converges to 1
6
c The Improper Integral Converges to 3
sinh 3
d The Improper Integral Converges to
4
5
4
6. a The Improper Integral Converges to
c The Improper Integral Converges to
e The Improper Integral Diverges to
6
b The Improper Integral Diverges to
d The Improper Integral Converges to 1
4
f The Improper Integral Diverges to
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Practice Sheet 12
1. Find vertical and horizontal asymptotes of
4x 2 1
,
x2 x
x 2 6x 9 ,
x 2 3x
1 x
,
f x
3
x
3
0
x
0
x
2. Find the horizontal asymptotes of
x 6 4x 1
2 x2
x2 x 1
x 9
3
f x
,
x
3
,
x
3
3. Find the horizontal asymptotes of
e x
x2 1
x2 4
a gx
0
,
0
x
2 tan 1 x
x
b f x
, x
log x 2 2x
3
, x
0
x
0
,
4. Find the vertical asymptotes of
f x
x2
x4
1
5. Let g x
x3
4x 2
g x
x2 x
2x
3
8x
8 e x 1 . The first order derivative of g is given by
1 ex 1.
Determine open intervals where g is increasing or decreasing , and the x & y
coordinates of local Maxima or local Minima of g.
6. Find absolute extrema of f x
10
4x
9 on the closed interval 1, 3 .
x
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x3
7. Let f x
7x 2
30 e x 1 . The first and second order derivatives of f
22x
x3
are given by f x
4x 2
8 ex 1 , f " x
8x
x3
x2 ex 1.
Determine intervals of concavity as well as points of inflection.
8. Use the second derivative test to find local maxima and local minima of
1 x2
2
3x
ln x
3.
9. Given f x
x
1 4 4
x
f x
a Show that f x
1 3 3
5x
x , f" x
20 x
1 2 2
x.
b Determine the x and y
coordinates of local maxima and minima of f.
c Determine the x and y
coordinates of all points of inflection of f.
10. Find the vertical asymptotes of f x
11. Let f x
f" x
84
25
x
x
12
5
3
x
2
3
3
5
6x
x 2 4x
x 3 2x 2
21 .
15x
3 75 . The second order derivative of f is given by
.
Determine open intervals where f is concave up or down and points of inflection.
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ANSWERS
1. Vertical Asymptote x
0 , Horizontal Asymptote y
2. " Left " Horizontal Asymptote y
1 , " Right " Horizontal Asymptote y
3. a " Left " Horizontal Asymptote y
1
0 , " Right " Horizontal Asymptote y
b " Left " Horizontal Asymptote y
4. Vertical Asymptote x
4
1
2 , " Right " Horizontal Asymptote y
1
1
5. The function g is increasing on 1,
, and is decreasing on
g has only a local Minimum at the point x, y
,0
0, 1 .
1, 3 .
6. Absolute Maximum value is 25, and Absolute Minimum value is 22.
7. The function f is concave up on 1,
, and is concave down on
f has only one inflection point at x, y
0, 1 .
1, 14 .
8. The function f has only a local Minimum at the point x, y
9. b Local Minimum at the point
,0
2, 4 .
1, 0 and local Maximum at the point 3, 256 .
c Only one inflection point at 2, 162 .
10. Vertical Asymptotes x
0 , and x
11. The function f is concave up on
f has inflection points at x, y
5
, 3
2,
3, 0 , and x, y
, and is concave down on
2, 5 .
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3, 2 .
lOMoARcPSD|16110024
Practice SAheet 13
1. In each case , find the Linearization ( Linear Approximation) for the function f about
the specified value of c.
a
f x
5
3x
5 , c
c
f x
ln 3
2x , c
2
1
b
f x
tan x , c
d
f x
tan 1 x , c
4
1
Note : Taylor Polynomial of degree one for the function f about c is referred to
as the Linearization ( or Linear Approximation) of f about c.
2. In each case find the Taylor’s Polynomial of degree n about c
a f x
10 x , n
3
b
f x
ln 2x
3
x
, n 2
d f x
1 x2
2 3x
Note : Taylor Polynomial of degree n for the function f about c
c
0 for the given function f
, n
, n
f x
3
4
0 is also referred to
as Maclaurin’s Polynomial of degree n.
3. In each case , write out Taylor formula with Remainder for the given function
and the specified values of c and n.
a fx
ln 3x
b fx
3
c f x
ln 3
d
f x
x
1 , c
0 , and n
3
c
1 , and n
2
2x , c
1 , and n
1
2 ,
x , c
1, n
2
4. Refer to problem (3) :
i Use Taylor formula of part a to estimate the value of ln 1. 3 .
ii Use Taylor formula of part b to estimate the value of
3
1. 18 .
iii Use Taylor formula of part c to estimate the value of ln 1. 2 .
iv Use Taylor formula of part d to estimate the value of 1. 12 .
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5. The linear approximation of f x
the value of cos 1
cos 1 x about x
0. 5 is used to estimate
0. 6 . Is the estimated value smaller or greater than the exact value?
6. The linear approximation of f x
tan 1 x about x
1 is used to estimate the value of
tan 1 1. 2 . Is the estimated value smaller or greater than the exact value?
Note : You may either use Concavity Test or sketch the graph of f x
together with its tangent line at x
1.
7. Find the first three non-zero terms of Taylor’s Series of the function f x
about c
tan 1 x
x
x
1
3
1.
8. Find the first five terms of Maclaurin’s Series of the function f x
e 2x 3
9. Find the Centre , and Radius of Convergence of each of the following the Power Series :
3 nxn
n 1
a
n
b
0
n! x
n
2n x 4 n
n!
c
n
d
d
0
n
1
3 2n
4n n
3 n
0
x
1 n
10. In each case find the centre and radius of convergence of the Power Series:
a The Power series
n
0 converges for all real numbers.
an x
c n , an
0 converges only at x
an x
c n converges only on the open interval
3.
2
0
c The Power series
n
3 n , an
0
b The Power series
n
an x
0
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4, 16
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ANSWERS
1
3 x
5
c Lx
2x
1
2. a P 3 x
1
ln 10
x
1!
1. a L x
2
b Lx
ln 3
2x
3
c P2 x
1x
2
3 x2
4
d P4 x
1
1 x2
2
1
ln 2 10 2
x
2!
3
x
2
2 x2
9
2x
9 x2
2
3x
1 x
3
1
81
x4
4 3s 1 4
9x 3
1 x
9
1
0 and x.
1 2
2x
1
3
2
x
2s 2
ln 1. 3
1 3
1 and x.
1 and x.
4. i
1 x
8
x
1 2
for some real number s between c
1
2 8/3
1 and x.
1
x
16s 5/2
1
1 x
2
5
81 s
1 2
x
1
1 x4
8
for some real number s between c
d
1 x
2
8 x3
81
for some real number s between c
c ln 3
4
4
ln 3 10 3
x
3!
for some real number s between c
b
2x
d Lx
b P3 x
3. a ln 3x
1
1 3
0. 264
ii
3
1. 18
1. 0564
iii
ln 1. 2
0. 2
iv
1. 12
1. 0582
5. The Exact Value of cos 1
0. 6 is Greater than the Estimated Value.
6. The Exact Value of tan 1 1. 2 is Smaller than the Estimated Value.
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x
x
1
3
1 x
2
1
1 x
4
1 2
1 x
8
8. e 2x
3
e 3 1
2x
2x 2
4 x3
3
2 x4 . . . .
3
7.
1 3 ....
,
x
a Centre of Convergence : c
0 ; Radius of Convergence : R
1
3
b Centre of Convergence : c
3 ; Radius of Convergence : R
0
c Centre of Convergence : c
4 ; Radius of Convergence : R
d Centre of Convergence : c
1 ; Radius of Convergence : R
10. a Centre of Convergence : c
3 ; Radius of Convergence : R
9.
b Centre of Convergence : c
c Centre of Convergence : c
4
9
3 ; Radius of Convergence : R
2
6 ; Radius of Convergence : R
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0
10