Math 101, Fall 2009

Calculus and Analytic Geometry

Quiz 2

Exercise 1. Compute the following integrals.

Z 4
1
(1) 6 points. x 3 dx.
Z ⇡4
(2) 7 points. | cos x|dx.
0
Z ⇡
2 cos x
(3) 7 points. dx.
0 (1 + 3 sin x)2

Exercise 2. 10 points. Show that the function f (x) = x3 + x2 + 2x 1 has only one root on ( 1, 1).

Exercise 3. The purpose of this exercise is to show that sin x  x on [0, 1). Define the (differentiable)
function f (x) = x sin x.
(1) 5 points. What is the sign of f 0 on [0, 1) ? Explain you answer.
(2) 5 points. Deduce from (1) that f admits a minimum at 0.
(3) 2 points. Deduce from (2) that sin x  x on [0, 1).

Z x
Exercise 4. Consider the (differentiable) function f (x) = (1 t)(t 2)2 dt.
0
(1) 5 points. Compute its derivative f 0 .
(2) 10 points. Study the zeros and the sign of f 0 .
(3) 3 points. Is 2 a local extremum ?
(4) 10 points. Compute f 00 and study the zeros and the sign of f 00 .
(5) 10 points. Sketch the graph of f .

Exercise 5. We want to find the rectangle with largest area, inscribed into a circle of radius 1. We write
x and y for the length of the sides of the rectangle.
(1) 5 points. What is the domain for x ? p
(2) 5 points. Show that the area of the rectangle is given A(x) = x 4 x2 .
(3) 10 points. find the rectangle with largest area inscribed into the circle of radius 1

1
Math 101 Exam 2
Dr. H. Yamani Summer 2014

Do each of the following problems. Show all your work.

No work shown. No credit.

Problem 1. (answer on page 1 of the booklet) (12 pts)

Solve the initial value problem
2
y !! = , y !(1) = 1, y (1) = 1
x3

Problem 2. (answer on page 2 of the booklet) (6 pts each)

Using the Fundamental theorem of Calculus to evaluate the following:
& x2 # 2x
d $ ! d &$ 1 #
a. $ cos t dt !
∫ b. ∫ dt !
dx $ 1 dx $ 2 !
% 1+ 3x 2 2 + t
!
% " "

Problem 3. (answer on page 3 of the booklet) (12 pts)

Find the area of the region bounded between x = 1 − y and x =1 − y .

Problem 4. (answer on page 4 of the booklet) (12 pts)

Consider the following function f ( x ) = x 3 + 2 x 2 − x on the interval [-1,2] find all
values of “ c ” in this interval whose existence is guaranteed by the theorem.

Problem 5. (answer on page 5 of the booklet) (12 pts)

Find the critical points of the function. f (x ) = −2 x 3 + 6 x 2 −3 . Identify the intervals on
which the given function is increasing and decreasing, concave upward and concave
downward and sketch.

Problem 6. (answer on page 6 of the booklet) (7 pts each)

Evaluate the integrals:
4 4
dx 1 x+5
a. ∫ x − 3dx b. c.
∫ 2 x (1 + x ) 3 ∫ dx d. ∫ dx
0 1
x 2 + 18 x + 81 (x + 1)2

Problem 7. (answer on page 7 of the booklet) (12 pts each)

Consider f (x ) = x 3
Use upper rectangle sum(right endpoint) with four rectangles with equal widths to
estimate the area between the graph of f (x ) = x 3 and the x-axis on the interval [0, 1].
Also find the exact area between f (x ) = x3 and the x-axis on the interval [0, 1]. Compare
your answers.
 AMERICAN(UNIVERSITY(OF(BEIRUT(

Exam(2(–(MATH(101(

I. True(or(False((2(points(each)"
Answer"by"true"when"it"is"always"true"and"by"false"otherwise.""
!
1) ! ! = ! 2! + 3!!!!!!!!! ≥ 1 "is"differentiable"at"1.""
! + 2! − 5!!!!!!!!!! < 1!!!!!!!!!!
□"True""""""""""""""""""""""""""""""""""""""""""""""""""□False"
"
!
2) If"! ! 1 = 2"and"!! 1 = 3,"then" ! ! 1 = 6."
□"True""""""""""""""""""""""""""""""""""""""""""""""""""□False"
"
!! !
! !
3) If"! 0 = 2,"! 0 = 1, !(0) = 3"and"! 0 = 0,"then" 0 = 3"
!
□"True""""""""""""""""""""""""""""""""""""""""""""""""""□False"
"
!
4) If"" = ! ,"then"!"is"not"differentiable"at"0."
!
□"True""""""""""""""""""""""""""""""""""""""""""""""""""□False"
"
! ! !! !
II. Let"! ! = "defined"on"ℝ ∖ ± !."
!!! ! !! !
1) Use"the"long"division"to"write""
!" + !
"
! ! = !" + ! +
−2! ! + 1
2) Does"the"graph"of"!"has"an"oblique"asymptote"at"±!∞?"If"yes,"find"its"equation."
"
!
III. Given"the"function"! = ! ! = !!!!"defined"for"! ≠ 2.""
! !!! !! !
1. Apply"the"formula"! ! ! = lim!→! "to"show"that"the"derivative"of"
!
!
the"function"at"every"point"of"its"domain"is"! ! ! = − ."
!!! !
2. Find"the"equation"of"the"tangent"line"to"the"graph"of"the"given"function"at"! = 1."

1"
"
"

3. For"what"values"of"!"is"the"slope"of"the"graph"equal"to"−4."

"

IV. Find"the"derivatives"of"the"following"functions:"(Don’t"simplify"the"final"result)"
!
1. ! ! = −2! ! + + ! ! − 2"
!!
!!
2. ! ! = ! 2! + 1 "
!
3. ! ! = ! "
! ! !!! ! !!!
4. ! ! = ! ! − ! ! + 2 ! tan 5!!"
"
5. ! ! = sec ! ! − 1 ! "
"
V. Let"!"be"the"curve"defined"by"the"equation"
! = ! + sin(! + !) − 1"
!" ! !
1. Find" "at"the"point" , "using"the"implicit"differentiation."
!" ! !
! !
2. Find"the"equation"of"the"normal"line"to"!"at"the"point" , "
! !
"
! !!!!
VI. Given"! ! = ! + 1 ! − 2 "defined"on"ℝ,!and""! ! ! =!
! ."
! !!! !

1) Find"the"critical"points"of"!."
2) Find"the"open"intervals"on"which"!(!)"is"increasing"and"the"open"intervals"
on"which"!(!)"is"decreasing."Show"your"work."
3) Does"!"has"the"following?"If"yes"give"their"coordinates."
P Local"maxima?"Local"minima?"

"

VII. Does"! ! = 2! ! − 9! ! + 12! + 4"has"an"absolute"maximum"and"

minimum"on" 0,3 ?"Justify"your"answer"and"if"yes"find"the"extreme"values"
of"!."
"

2"
"
 MATH 101: Calculus and Analytic Geometry I

Fall 2016-2017, Quiz 2, Duration: 60 min.

Exercise 1. Explain your answers in detail to ensure full credit. The use of L’Hospital’s Rule is
not allowed and will result in no points given.
p
x4 1
(a) (5 points) Find lim .
x! 1 x + 1
sin x
(b) (5 points) Find lim .
x!+1 x

Exercise 2. The purpose of this exercise is to show that 1 cos x  x on ( 1, 0]. Define the
differentiable function f (x) = x + cos x.
(a) (6 points) What is the sign of f 0 on ( 1, 0] ? Explain you answer.
(b) (5 points) Deduce from (a) that f admits an absolute minimum at 0.
(c) (2 points) Find f (0).
(d) (2 points) Deduce from (b) and (c) that 1 cos x  x on ( 1, 0].

Exercise 3.
(a) (10 points) Use the definition of derivative to
8 find the derivative of f (x) = x
2 1.
< ax + 1 if x < 1
(b) Consider the function f defined by f (x) =
: 2
x 1 if x 1
i. (5 points) Find the value a for which the function f is continuous on R.
ii. (5 points) Using the definition of derivative, show that f (for the value of a you have found previ-
ously) is not differentiable at 1.

Exercise 4. Let C be the curve defined by the equation x sin(xy) + x2 + y 2 = 1.

dy
(a) (10 points) Find y 0 (that is ) at the point (1, 0).
dx
(b) (5 points) Find the equation of the line that is tangent to the curve C at the point (1, 0).

2
Exercise 5. Consider the function f (x) = .
x2 1
(a) (8 points) Find the horizontal and vertical
✓ ◆asymptotes of f .
1 0
(b) (3 points) Recall the formula for the .
v
(c) (4 points) Compute the derivative f 0 of f .
(d) (5 points) Find the critical points of f .
(e) (5 points) Study the sign of f 0 and complete the given table.
x

f’

(f) (5 points) Find the local maxima and minima of f .

(g) (5 points) Sketch the graph of f .
(h) (3 points) Does f admit an absolute maximum or minimum?
(i) (2 points) Does f admit an absolute maximum or minimum on ( 1, 1)?
1
 MATH 101: Calculus and Analytic Geometry I

Fall 2017-2018, Quiz 2, Duration: 60 min.

Exercise 1. Explain your answers in detail to ensure full credit. The use of L’Hospital’s Rule is
not allowed and will result in no points given.
p p
(a) (6 points) Find lim x+2 x 1.
x!+1
2
(x3 + 2x2 + x) 3
(b) (6 points) Find lim .
x!+1 (2x2 + 1)
1 cos x
(c) (5 points) Find lim .
x!+1 x

Exercise 2. (10 points) Use the definition of derivative to find the derivative of
f (x) = x2 3x + 5.

Exercise 3. (10 points) Consider the function f (x) = x4 + 2x2 1. Find the absolute (= global)
maximum and minimum of f on [-1,2].

⇡
Exercise 4. Let C be the curve defined by the equation y cos(x + y) xy = x.
2
(a) (10 points) Differentiate implicitly both sides of the equation.
dy
(b) (3 points) Find y 0 (that is ) at the point (⇡/2, 0).
dx
(c) (5 points) Find the equation of the line that is tangent to the curve C at the point (⇡/2, 0).

x+1
Exercise 5. Consider the function f (x) = .
x2 3x
(a) (10 points) Find the horizontal and vertical asymptotes of f .
(b) (10 points) Compute the derivative f 0 of f .
(c) (8 points) Study the sign of f 0 and complete the given table. (hint: x2 + 2x 3 = (x 1)(x + 3))
x

f’

(d) (5 points) Find the local maxima and minima of f .

(e) (7 points) Sketch the graph of f .
(f) (5 points) Does f admit an absolute (= global) maximum or minimum?