Assignments - Chu Hai College

DEPARTMENT OF COMPUTER SCIENCE
CALCULUS I – MAT103A
Due Dates for Thursday Group Only
Assignments
Assignment 0-4
Assignment 5-7
Due date
23 Oct 2014
18 Dec 2013
Assignment 0- Functions, Limit and Continuity
If f(x)=x+5 and g(x) = x2-3, find g(f(x))
If f(x)=x, g(x)=x/4, and h(x)=4x-8, find h(f(g(x)))
Q1.
(a)
(b)
Q2.
Graph the functions
1 − x ,
(a)
f ( x) = 
2 − x,
Q3.
0 ≤ x ≤1
1< x ≤ 2
1 / x, x < 0
g ( x) = 
 x, 0 ≤ x
(b)
Complete the following table.
(i)
g(x)
x-7
(ii)
(iii)
x+2
?
(iv)
(v)
(vi)
x /( x − 1)
?
1/ x
f(x)
x
3x
x−5
x /( x − 1)
1+ 1 / x
?
f(g(x))
?
?
x2 − 5
?
x
x
 x 2 − 1, - 1 ≤ x < 0

0 < x <1
 2 x,
Q4. Given the following function
f ( x) =  1,
x =1
 - 2x + 4, 1 < x < 2

0,
2<x <3
(a)
(b)
Graph the above function f(x).
Does f(-1) exist? Does Lim+ f ( x ) exist? Does Lim+ f ( x ) =f(-1)?
x → −1
(c)
x → −1
Is f(x) continuous at x=-1? Does f(1) exist? Does Lim f ( x ) exist?
x→1
Q5. Refer to Q4, answer the function questions
(a) Does Lim f ( x ) =f(1)? Is f(x) continuous at x=1?
x→1
(b)
(c)
Does f(x) defined at x=2? Is f(x) continiuous at x=2
At what values of x is f continuous? What value should be assigned to f(2) to make the
extended function continuous at x=2? To what new value should f(1) be changed to
remove the discontinuity?
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DEPARTMENT OF COMPUTER SCIENCE
CALCULUS I – MAT103A
Assignment 1 – Derivatives
Q1. Find the derivative of f(x) from first principle by using the equation of f’(x), examine the
derivative at x=-4/3, where the function is discontinuous.
f ( x + h )− f ( x )
2x − 3
f ' ( x ) = Lim
f(x)=
,
h
→
0
3x + 4
h
Q2. Find the derivative of (i) f ( x ) = ( x 2 + 4 ) 2 ( 2 x 3 − 1 ) 3 (ii) f ( x ) = x 2 / 4 − x 2
Q3.
Q4.
Given f(x)=2/(1-x), find df/dx, d2f/dx2 and d3f/dx3 and hence suggest a solution for dnf/dxn .
Also proof you solution by mathematical induction.
Find (i) dy/dx, d2y/dx2 for x2-xy+y2=3 (ii) dy/dx for x2y-xy2 + x2+ y2=0
Q5. A weight W is attached to a rope 15m long which passes over a pully at P, 6m above the
ground. The other end of the rope is attached to a truck at a point A, 0.5m above the ground as
shown in Fig. 5.1. If the truck moves off at the rate of 3ms-1. How fast is the weight rising
when it is 2m above the ground ?
Fig. 5.1
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DEPARTMENT OF COMPUTER SCIENCE
CALCULUS I – MAT103A
Assignment 2 – Derivatives
Q1. Find the first derivative dy/dx of the function y =
3
( 5 x + sin 2 x ) 3 / 2
2
Q2. Find d2y/dx2 by implicit differentiation:
2
(b) y 2 = 1 −
(a) x 2 + y 2 = 1
x
Q3. Find the points on the curve y = 2 x 3 − 3 x 2 − 12 x + 20 where the tangents is
(a) perpendicular to the line y = 1 − x / 24
(b) parallel to the line y = 2 − 12 x
Q4. Find the limit for the equation
Lim
θ →( π / 2 ) −
Q5.
4 tan 2 θ + tan θ + 1
tan 2 θ + 5
Water drains from the conical tank (x=4) shown in Fig. Q5 at the rate 5 ft3/min
(a) What is the relation between the variable h and r in Fig. Q5 ?
(b) How fast is the water level dropping when h = 6 ft ?
Fig. Q5
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DEPARTMENT OF COMPUTER SCIENCE
CALCULUS I – MAT103A
Assignment 3 – Application of Derivatives
Q1.
Find the limit of
(i)
Lim
x →∞
Q2.
x2 + 2
3x 2 + 4
(ii)
Lim
x→∞
x2 − x +7
6 x3
Find the critcal points, point of inflection and asymptotes of the function y = f ( x ) =
x 3 +1
,
x
sketch the graph of the function y=f(x).
Q3.
Find the linearization of
(i)
(ii)
y = tan x at x = -π/4
f ( x ) = 1 + x + sin x − 0.5 at x = 0
Q4.
To find the height of a tree, you measure the angle from the ground to the treetop from a
point 100ft away from the base. The best figure you can get with the equipment at hand is
300±10. About how much error could the tolerance of ±10 create in the calculated height?
Remember to work in radian.
Q5. The pipe in Fig. Q5 consists of two section AB and BC. The perdendicular distance between
point A and DC is 10m, point C is 25m away from point D. If the section AB is made of
material which costs $40,00 per metre and the section BC is made of materials which costs
$20,00 per metre. What value of x gives the least expensive connection?
Fig. Q5
-4-
DEPARTMENT OF COMPUTER SCIENCE
CALCULUS I – MAT103A
Assignment 4 – Application of Derivatives
Q1. Find df/dx and d2f/dx2 for the function f(x) = sin x cos 3x.
Q2. Given the function y = f ( x ) = −2 x 3 + 12 x 2 − 18 x + 4 . Consider the critical points, point of
inflection and the variation of df/dx and d2f/dx2 in the range 0 < x < 4, graph the curve y = f(x).
Q3. Starting with an initial estimates of x1=1.3, find the root of the equation x 3 + 2 x − 5 = 0 by
using Newton’s method.
Q4. Write a formula that estimates the change occurs in the volume of a right circular cone (Fig. Q4)
when the radius changes from r0 to r0+dr and the height does not change.
Q5. Find the dimension of the right cylinder of maximum volume V which can be inscribed in a
sphere of radius Ro as shown in Fig. Q5.
Fig. Q4
Fig. Q5
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DEPARTMENT OF COMPUTER SCIENCE
CALCULUS I – MAT103A
Assignment 5 – Integration
Q1. (a)
By using the substitution u = 2+tan x, find the indefinite integral of ∫
3
18 tan 2 x sec 2 x
(2 + tan 3 x) 2
dx .
By using the substitution u = 1+sin2(x-1), find the indefinite integral of
2
∫ 1 + sin ( x − 1) sin( x − 1) cos(x − 1)dx .
Q2. Find the indefinite integral of
( 2r − 1) cos 3( 2r − 1) 2 + 6
(a)
∫
(b)
∫
dr
3( 2r − 1) 2 + 6
sin θ
dθ
3
θ cos θ
Q3. Find the area of the region between the y=x and the x-axis on the interval[0,b] by using the
b
n
Riemann sums ∫ f ( x)dx = Lim ∑ f (c k )∆x
a
P →0 k =1
Q4. Using the concept of area under the curve, find the following definite integral.
1
0
(a)
2
∫ 16 − x dx
−4
(b)
2
∫ (1 + 1 − x ) dx
−1
Q5. A nose cone of a rocket is a paraboloid (Fig. Q5) obtained by revolving the curve y=(x)1/2, 0 ≤ x ≤ 5,
about the x-axis, where x is measured in feet. Estimate the volume of the cone by finding the sum
S5 of the volumes of the cylinders. Is the result is a under or over estimate of the actual volume.
Fig. Q5
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DEPARTMENT OF COMPUTER SCIENCE
CALCULUS I – MAT103A
Assignment 6 – Integration
b
2
Q1. What values of a and b maximize the value of ∫ ( x − x )dx .
a
b
4
2
Q2. What values of a and b minimize the value of ∫ ( x − 2 x )dx
a
Q3. Find the following definite integrals:
1
(a)
2
∫ t t + 1dt
0
2
(b)
∫
−1
tdt
2t 2 + 8
Q4. Show that the area under a parabolic arch is two-thirds the base times the height
(a) Use an integral to find the area under the arch y = 6 − x − x 2 , - 3 ≤ x ≤ 2
(b) Find the height of the arch.
(c) Show that the area is two-thirds the base b times the height h.
(d) Sketch the parabolic arch y = h − ( 4h / b 2 ) x 2 , - b/2 ≤ x ≤ b / 2 , assuming that h and b are
positive. Then use integration to find the area of the region enclosed between the arch and
the x-axis.
Q5. The marginal cost of printing a poster when x posters have been printed is
dc
1
=
dx 2 x
Find
dollars.
(a) c(100)-c(1), the cost of printing posters 2-100.
(b) c(400)-c(100), the cost of printing posters 101-400.
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DEPARTMENT OF COMPUTER SCIENCE
CALCULUS I – MAT103A
Assignment 7 – Integration
Q1. Evaluate the following definite integrals:
1
5
4
∫ t + 2t (5t + 2)dt
(a)
0
1
∫
(b)
0
y2 + 4y − 4
dy
y 3 + 6 y 2 − 12 y + 9
Q2. Find the following definite integrals:
π
3 x
2 x
(a) ∫ tan sec dx
4
4
2π / 3
π /2
π /2
4
∫ 15 sin 3x cos 3xdx
(b)
(c) ∫
0
−π / 2
3 sin x cos x
dx
1 + 3 sin 2 x
Q3. Suppose that a company’s marginal revenue from the manufacture and sale of egg beaters if
dr
2
= 2−
dx
( x + 1) 2
where r is measured in thousand of dollars and x in thousand units. How much money should
the company expect from a production run of x=3 thousand egg beaters? To find out integrate
the marginal revenue from x=0 to x=3.
Q4.
(a)
Show that the average value of v (t ) = Vmax sin 120πt over a period is zero.
(b)
(c)
Find the average value of v(t ) = Vmax sin120πt over a half-period.
If an electric stove is rated at 240 volts rms, what is the peak value of the allowable
voltages?
1 / 60
(d)
Show that
∫ (V
max
) 2 sin 2 120πtdt = (Vmax ) 2 / 120 .
0
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DEPARTMENT OF COMPUTER SCIENCE
CALCULUS I – MAT103A
Assignment 8 – Applications of Integration
Q1.
Find the area of the shaded area in Fig. Q1a and Fig Q1b.
Fig. Q1a
Q2.
Fig. Q1b
Find the volume of the solid generated by revolving the shaded region in Fig. Q2a and Fig.
Q2b.
Fig. Q2a About the y-axis
Fig. Q2b About the x-axis
Q3.
Find the volumes of solid generated by revolving the region bounded by the line and curves
as shown as follows about the x axis.
(a)
y = x 2 + 1, y = − x + 3
(b)
y = 4 − x2 , y = 2 − x
Q4.
(a)
Find the length of the curve 4√2 / 3 x3/2-1, 0≦x≦1.
Q5.
(a)
(b)
Find the area between the curve y = sec2x and y=sin x from 0 to π/4.
Find the area enclosed by y = 2-x2 and y = -x .
Q6.
(a)
Find the center of mass of a thin plate of constant density δ covering the region
bounded above by the parabola y = 4-x2 and below by the axis
Find the center of mass of (a) if the density d is given by δ =2x2
(b)
Q7.
Find the center of mass of a wire of constant density δ shaped like a semicircle of radius a.
Q8.
(a) Find
∂f
∂f
and
at (1,1) from first principle if f ( x, y ) = 2 x 2 + 5 xy + 3 y + 1 .
∂x
∂y
2y
∂f
∂f
and
if f ( x, y ) =
.
(b) Find
∂y
y + cos x
∂x
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