Math 5a
Solutions to Self-Quiz for Section 2.4
1. The average rate of change of f (x) between x = 2 and x = 5 is
12
1
f (5) f (2)
=
= .
5 (2)
7
7
2. Let f (x) =
x
.
x+1
(a) The average rate of change of f (x) between x = 2 and x = 4 is
4 2
2
f (4) f (2)
1
Math 5a
1. (Section 2.6) Let f (x) =
Self-Quiz for Sections 2.6 and 2.7
x+1
1
and g(x) =
. Find the following, and simplify:
x1
x2
(a) (g f )(3)
(d) (f g)(x)
f
(0)
g
(f) (f f )(1)
(b) (f g)(x)
(e) (g f )(x)
(c)
2. (Section 2.6) In each of the following, a
Math 5a
Solutions to Self-Quiz for Sections 2.2 and 2.3
1. (a) (i) f (0) does not exist
(ii) f (1) = 1
(iii) f (2) = 1
2
(b) f (x) = 0 when x = 3.
(c) The domain of f (x) is [3, 0) [1, 3].
(d) The range of f (x) is [1, 0] [1, 3).
2. (a) f (2) = 2 and f (5
Math 5a
Self-Quiz for Section 2.1
1. Let f (x) = 3x2 + x 1.
(a) Find the following:
(i) f (5)
(ii) f (1)
(iii) f (0)
(iv) f ( 1 )
2
(v) f (a h)
(vi) f (4t)
(b) For what value(s) of x is f (x) = 0?
2. Let f (x) =
x
.
(x 1)2
(a) Find the following, if they
Math 5a
Solutions to Self-Quiz for Section 1.11
1. We know that y = kx. Since y = 10 when x =
1
1
, we get 10 = k
k = 30. So
3
3
y = 30x.
To nd out what y equals when x = 4, we set x = 4 and solve for y: y = 30 4 y =
120.
2. We know that T =
So T =
k
k
Math 5a
Self-Quiz for Section 1.11
1. Suppose that y varies directly with x, and that when x =
1
, y = 10. What value does
3
y have when x = 4?
2. Suppose that T is inversely proportional to the sum of x and y. Suppose that T = 4
when x + y = 5. What does
y = 0.5x + 2.5
Math 5a
!1
Self-Quiz for Sections 1.8 and 1.10
1. (Section 1.8) Find the x- and y-intercepts (if they exist) of the following:
2
2
a. y =
x +5
x2 7
3
b. y = x3 49x
2. (Section 1.10) Find the equation of the line graphed below.
4
3. (Section
Math 5a
Solutions to Self-Quiz for Section 1.5
1. (a) x2 + x = 1 x2 x + 1 = 0. Using the quadratic formula, we get x =
+
1 1 4
1 3
=
. Since 3 is not a real number, the equation has
2
2
no real solutions.
(b) (x2 + 9)(x2 + 3x 4) = 0 x2 + 9 = 0 or x2 + 3x
Math 5a
Solutions to Self-Quiz for Sections 1.1 and 1.2
1. (Section 1.1)
a. ( 3 , +)
2
b. (2, ]
c. (, 0)
2. (Section 1.1)
(a) x > 0
(b) t < 4
(c) a
(d) 5 < x <
1
3
3. (Section 1.1)
(a) A B = [4, 7]; A C = (1, 2]; A B = (1, +)
(b) Since B and C have no el
Math 5a
Self-Quiz for Sections 1.3 and 1.4
1. (Section 1.3) Consider the polynomial
x5
3
+ 10x2 x7 + 11. Find the following:
4
2
(a) The degree of the polynomial
(b) The coecient of the leading term of the polynomial
(c) The coecient of the term of degree
Math 5a
Solutions to Self-Quiz for Sections 1.8 and 1.10
1. (Section 1.8)
(a) For the y-intercept, set x = 0, getting y =
5
. For the x-intercept, set y = 0
7
x2 + 5
= 0 x2 + 5 = 0. This equation has no real solutions,
x2 7
so there has no x-intercepts.
Math 5a
1. (Section 1.3)
Solutions to Self-Quiz for Sections 1.3 and 1.4
x5
3
+ 10x2 x7 + 11
4
2
(a) The degree of the polynomial is 7.
(b) The coecient of the leading term of the polynomial is
(c) The coecient of the term of degree 5 is
3
2
1
4
(d) The
Math 5a
Self-Quiz for Section 1.5
1. Solve the following equations:
a. x2 + x = 1
b. (x2 + 9)(x2 + 3x 4) = 0
5
c. (2x3 3 x2 )(x 1 ) = 0
2
2. Find the number of real solutions of the equation 4x2 + 3 = 7x.
3. For what value of k does the equation kt2 + 5t
Math 5a
Self-Quiz for Sections 2.2 and 2.3
1. Let f (x) be the function shown below. Use it to answer the following questions.
1
y = $ $ ! ( x + 2 ) 2 + 2% 1 if x < 0 %
" 2
#
#
y = ( ! 0.25 ( x ! 5 ) 2 + 1 if x > 5 )
1.5
2
y = $
( x ! 2 ) ! 1.5 if 0 < x <
1
!2
Math 5a
Self-Quiz for Section 2.4
2
1. Let f (x) be the function shown below. Use it to answer the following questions. Find
the average rate of change of f (x) between x = 2 and x = 5.
2. Let f (x) =
x
.
x+1
(a) Find the average rate of change of f
2. (a) 150
(b) 420
(c)
5
=
radians
180
6
7 Solutions to Self-Quiz on Trigonometry, Part I
=
radians
180
3
1. An angle with positive measure that is coterminal with is = 135 + 360 = 225 .
with 7
315An angle= negative measure that is coterminal with is t
Math 5a
Solutions to Self-Quiz on Section 4.5
1. (a) First rewrite ln x + ln(x 4) = ln(2x 5) so that there is a single logarithm on
each side:
ln(x2 4x) = ln(2x 5).
Then exponentiate both sides:
eln(x
2 4x)
= eln(2x5) x2 4x = 2x5 x2 6x+5 = 0 (x5)(x1) = 0
Math 5a
Self-Quiz on Sections 4.3 and 4.4
1. (Section 4.3) Evaluate the following (if they exist):
1
a. log2 32
b. log
10
e. log5 (5)
d. log 1 27
3
1
g. ln e
h. ln 4
e
ln 6
j. ln 0
k. e
c. log3
3
f. ln e2
i. ln 3 e
l. eln()
2. (Section 4.3) Find the domai
Math 5a
Solutions to Self-Quiz on Modeling with Functions
1. Let w represent the width of the garden and l the length, and let a represent the area.
The area of the garden is a = w l. We need to write a as a function of w alone.
The perimeter of the recta
Math 5a
Self-Quiz on Sections 4.1 and 4.2
1. Solve the following equations.
a. 10 2x 7x 2x + x2 2x = 0
b.
xex 3ex
=0
x2 16
2. Solve the following inequality: x2 ex 2xex 0
3. Let h(x) = 3ex
2 +5
.
(a) Find two functions f (x) and g(x) such that h(x) = (f g
Self-Quiz for Trigonometry, Part II
1. Find the following:
a. sin 150
d. tan 450
b. cos 315
e. cos(120 )
c. sec 180
f. csc(225 )
2. Find the following (all the angles are measured in radians):
a. cos
3
4
d. cot
7
6
11
f. sin
6
b. tan()
5
3
c. csc
e. cos
Math 5a
Self-Quiz on Modeling with Functions
Note: you do not need to simplify your answers.
1. A rectangular garden has a perimeter of 48 meters. Express the area of the garden as
a function of its width.
2. A pleasure boat spends the night anchored 8 mi
Self-Quiz for Trigonometry, Part I
1. Find two angles, one with positive measure and one with negative measure, that are
coterminal with the angle = 135 .
2. Write each of the following angles using radian measure:
a. 150
b. 420
3. In the right triangle
3
Math 5a
Self-Quiz on Section 4.5
1. Solve the following equations.
a. ln x + ln(x 4) = ln(2x 5)
b. ln(x 2) = ln 8 ln x
c. ln(2x2 ) ln(x2 + 1) = 0
d. e2x + ex 12 = 0
2. Solve the following inequalities.
a. x ln x + 2x > 0
b. x2 3x2 ln x 0
3. Find the domai
Math 5a
Solutions to Self-Quiz on Sections 4.1 and 4.2
1. (a) 10 2x 7x 2x + x2 2x = 0. Factor out 2x , getting
2x (10 7x + x2 ) = 0 2x (5 x)(2 x) = 0.
Then 2x = 0, 5 x = 0 or 2 x = 0. The rst equation has no solutions, since
2anything is always positive.
4
Math 5a
3
Solutions to Self-Quiz for Section 2.5
5
1. (a) y = f (x 2) +1
1
2
!5
3
(b) y = 2f (x) 11
!
1
1
3
(c) y = f (x) + 22
OVER
2. The graph of
2
f (x) =
1 (x + 1)3 ,
2
if x 1
if 1 < x < 1
x 1 2 if x 1
3x
2
1
,
2
is shown below. 1
!
3. The formula