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Review Problems for Quiz 2 - Calculus I | MA 140, Quizzes of Calculus

Material Type: Quiz; Class: Calculus I; Subject: Mathematics; University: Millikin University; Term: Spring 2001;

Typology: Quizzes

Pre 2010

Uploaded on 08/04/2009

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MA140-01
2/8/09 Major Quiz 2 Review Problems
1.) Find
dx
dy
for each of the following.
(a) y = 3x
3
5x
2
+ 4x +13 (b) y = 3(x
5
+ 2x
4
) (c)
2
3
3
5
x
x
y+=
(d)
54
23
x
x
y
=
(e)
2
1
2
1
2
1
2
= xxy (f)
1
44
2
+
=
x
xx
y
(g) y = (4x
2
1)(2x
3
+ x + 5) (h)
1
1
3
3
+
=
x
x
y (i) y = (2x
4
3x)(5x
2
2x + 5)
(j)
6
3
2
+
=
x
x
y (k) y =
5
2
13
+
x
x (l) y =
2
3
4
5
4
3
5
3
3
2
x
x
x
x+
2.)
Use the definition of derivative to find f'(x) for f(x) = 3x
2
+ 5x 1.
3.)
Prove the following theorem.
"If
f
and
g
are functions and if k is the function defined by
( ) ( ) ( )
k x f x g x
then if
'( )
f x
and
'( )
g x
exist,
'( ) '( ) '( )
k x f x g x
= +
."
4.)
Find an equation of each of the lines through the point (1, 2) that is tangent to the curve
y = 2x
2
+ 8.
5.)
Find the values of a and b such that f is differentiable at 2 if f(x) =
<+
212
2
2
xifx
xifbax
.
6.)
Find an equation of the tangent line and normal line to the curve y = 3x
4
12x at the
point (1, 9).
7.)
Use the definition of derivative to find f'(x) for f(x) = 2x
2
3x + 1.
8.)
Find
'( )
f x
for each of the following.
(a) f(x) = 5x
3
+ 2x
2
3x (b) f(x) = (3x + 5)(2x
2
3x +1) (c) f(x) =
7
3
32
5
+
x
x.
(d) f(x) =
24
53
x
x
(e) f(x) =
4
2
13
2
+
x
x
9.)
Use the definition of derivative to find
'
f
for
2
( ) 3 5
f x x
=
.
10.)
Use the Intermediate Value Theorem to determine if
2
( ) 29
f x x
=
has a zero in the interval
[5,8].
11.)
Find the derivative of
2
–9 + + 7
( ) x x
f x x
=
.
12.)
Using the position function 7
( ) – 9
s t t
=, find the acceleration function.
13.)
Find the third derivative of
5
2
( ) 3 + 7
f x x x
x
=.
pf3

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2/8/09 Major Quiz 2 Review Problems

1.) Find dx

dy for each of the following.

(a) y = 3x

3 − 5x

2

  • 4x +13 (b) y = −3(x

5

  • 2x

4 ) (c) 2

3 3

(^5) x

x y = +

(d) 4 5

x x

y = − (e) 2

1 2

1

y = xx (f) 1

2

x

x x y

(g) y = (4x

2 − 1)(2x

3

  • x + 5) (h) 1

3

3

x

x y (i) y = (2x

4 − 3x)(5x

2 − 2x + 5)

(j) 6

3

2

x

x y (k) y = 2 5

x

x (l) y = 2

3

4

5

4

x

x

x

x − + −

2.) Use the definition of derivative to find f'(x) for f(x) = 3x

2

  • 5x − 1.

3.) Prove the following theorem.

"If f and g are functions and if k is the function defined by k ( x ) = f ( x ) + g ( x )

then if f '( x )and g '( x )exist, k '( x ) = f '( x ) + g '( x )."

4.) Find an equation of each of the lines through the point (1, 2) that is tangent to the curve

y = 2x

2

5.) Find the values of a and b such that f is differentiable at 2 if f(x) =



2 x if x

ax b if x .

6.) Find an equation of the tangent line and normal line to the curve y = 3x

4 − 12x at the

point (1, −9).

7.) Use the definition of derivative to find f'(x) for f(x) = 2x

2 − 3x + 1.

8.) Find f '( x )for each of the following.

(a) f(x) = 5x

3

  • 2x

2 − 3x (b) f(x) = (3x + 5)(2x

2 − 3x +1) (c) f(x) = 3 7

5

x

x .

(d) f(x) = 4 2

x x

(e) f(x) = 2 4

2

x

x

9.) Use the definition of derivative to find f 'for

2

f ( x ) = 3 x − 5.

10.) Use the Intermediate Value Theorem to determine if

2 f ( ) x = x − 29 has a zero in the interval

[5,8].

11.) Find the derivative of

2 –9 + + 7 ( )

x x f x x

12.) Using the position function

s t ( ) – 9 t

= , find the acceleration function.

13.) Find the third derivative of

f ( ) x 3 x + 7 xx

2/8/09 Major Quiz 2 Review Problems

14.) Below is a graph of f ( ) x. Sketch a graph of f ′( ) x.

3

0

lim h

h h

h

equals f ′( ) a for some function f ( ) x and some constant a.

Determine what f ( ) x and the constant a could be.

16.) Use the Intermediate Value Theorem to determine if

3 f ( ) x = x + x − 37 has a zero in the

interval [3, 9].

17.) Using the position function

s t ( ) 4 tt

= , find the velocity function.

18.) Below is a graph of f ( ) x. Sketch a graph of f ′( ) x.