Calculus I
Problem 1:
(6 points)
Sunday, 22nd November, 2015
MATB110: HW#9
(1 part, 6 points).
Suppose that f and h are integrable and that
9
Z
9
Z
Z
f (x) dx = 1,
f (x) dx = 5,
1
9
h(x) dx = 4.
7
7
Use the rules of integration to find
(a)
9
Z
2f (x) dx
1
Calculus I
MATB110: HW#6
Tuesday, 17th November, 2015
Problem 1: Existence of root (1 part, 1 point).
(1 point)
Show that the equation
x3 + ex = 0
has exactly one real root.
Solution:
Let f (x) = x3 + ex . Then f (1) = 1 + 1/e < 0 and f (0) = 1 > 0. Since
Calculus I
Tuesday, 17th November, 2015
MATB110: HW#10
Due date: Monday, 23rd November, 2015
Problem 1:
(2 points)
(1 part, 2 points).
Sketch the region bounded by the curve
1
y = x,
2
x + 8 = y2 ;
then find its area.
Solution:
Solving
y 2 8 = x = 2y = y
Calculus I
Friday, 9th October, 2015
MATB110: HW#4
Problem 1: Optimization in volume of cylinder (1 part, 2 points).
(2 points) Find the shape of the cylinder of maximal volume that can beinscribed in a sphere of
radius R. Show that the ratio of the heigh
Calculus I
Thursday, 26th November, 2015
MATB110: HW#11
Due date: Friday, 30th November, 2015
Problem 1:
(1 part, 2 points).
(2 points) In the following questions, given the function f (x) is non-negative and continuous for
x 1. When the region lying unde
Calculus I
MATB110: HW#7
Problem 1:
(2 points)
by
Tuesday, 17th November, 2015
(1 part, 2 points).
Calculate dy/dx and d2 y/dx2 , assuming that y is defined implicitly as a function of x
sin2 x + cos2 y = 1.
Solution:
sin2 x + cos2 y = 1 :
where
d2 y
dx2
Calculus I
MATB110: HW#5
Tuesday, 17th November, 2015
Problem 1: Tangent line again (1 part, 2 points).
(2 points) Use implicit differentiation to find an equation of the line tangent to the curve xy =
6e2x3y at the given point (3, 2).
Solution:
10
dy
12e
Calculus I
Tuesday, 17th November, 2015
MATB110: HW#8
Due date: Wednesday, 11th November, 2015
Problem 1:
(1 point)
(1 part, 1 point).
Show that the obviously different functions
F1 (x) =
1
1x
and
F2 (x) =
x
1x
are both antiderivatives of f (x) = 1/(1 x)2