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A midterm exam in mathematics with questions related to functions, limits, and asymptotes. The exam includes exercises on determining whether a relation is a function, evaluating limits, and finding vertical and horizontal asymptotes. The exam also includes an exercise on finding the values of x for which the derivative of a given function is zero. The document can be useful as study notes or exam preparation for a mathematics course covering these topics.
Typology: Exams
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a. S = {(4,7), (5,8), (6,9), (7,10), (8,11)}
b. S = {(x,y) such that y=โ xโ ; x ฯต R}
c. y = x
2
d. โyโ = x
e. y =
2 x
x + 2
lim
xโ โ
x โ 2
2 x
2
โ x + 1
x โ 2
2 x
2
โ x + 1
x
x
2
x
x
x
2 x
2
x
2
x
x
2
x
2
x
x
x
2
x
x
x
2
x
โ
x
x
2
โ
2
x
5 + 2 x
2
2 โ x โ x
2
Vertical asymptote
2 โ x โ x
2
( 2 + x ) ( 1 โ x )= 0
2 + x = 0 1 โ x = 0
x =โ 2 โ x =โ 1
โ x
x = 1
x =โ 2 , 1
Horizontal asymptote
5 + 2 x
2
2 โ x โ x
2
x
2
x
2
dy =
2 dx + 6 dx
( 1 โ 2 x โ 2 dx )( 1 โ 2 x )
dy =
8 dx
( 1 โ 2 x โ 2 dx )( 1 โ 2 x )
Step 3:
dy =
8 dx
( 1 โ 2 x โ 2 dx )( 1 โ 2 x )
dx
dy
dx
( 1 โ 2 x โ 2 dx )( 1 โ 2 x )
Step 4:
lim
dx โ 0
( 1 โ 2 x โ 2 dx )( 1 โ 2 x )
lim
dx โ 0
( 1 โ 2 x โ 2 ( 0 ))( 1 โ 2 x )
lim
dx โ 0
( 1 โ 2 x )( 1 โ 2 x )
( 1 โ 2 x )
2
4 x + 3
and verify your answer using the
differentiation formula.
y =
4 x + 3
Step 1:
y + dy =
โ
4 ( x + dx )+ 3
y + dy =
4 x + 4 dx + 3
Step 2:
y + dy โ y =
4 x + 4 dx + 3 โ
4 x + 3
Step 3:
dy =
4 x + 4 dx + 3 โ
4 x + 3 โ
dx
dy
dx
4 x + 4 dx + 3 โ
4 x + 3
dx
dy
dx
dx
4 x + 4 dx + 3 +
4 x + 3
dy
dx
4 x + 4 dx + 3 โ( 4 x + 3 )
dy
dx
4 x + 4 dx + 3 โ 4 x โ 3
dx (
4 x + 4 dx + 3 +
4 x + 3 )
dy
dx
4 x + 4 dx + 3 โ 4 x โ 3
dy
dx
4 dx
dx (
4 x + 4 dx + 3 +
4 x + 3 )
dy
dx
4 x + 4 dx + 3 +
4 x + 3
Step 4:
lim
dx โ 0
4 x + 4 dx + 3 +
4 x + 3
lim
dx โ 0
โ
lim
dx โ 0
4 x + 3 +
4 x + 3
4 x + 3
4 x + 3
4 x + 3
4 x + 3
4 x + 3
4 x + 3
Check using the Derivative of a Radical
Note: d
dx
u =
du
dx
u
d
dx
d
dx
( 4 x + 3 )
4 x + 3
