Exercises Calculus
Appendices A, B and D
A1
A2
Rewrite without absolute value: a. |√5 - 5|
b. |x + 1|
c. |x2 + 1|
Solve the inequality and give the solution set as an interval:
a. 2x + 7 > 3
b. 1 – x ≤ 2
c. 2x + 1 < 5x - 8 d. 4x < 2x + 1 ≤ 3x + 2
e. x3 - x2 ≤ 0
f.
<4
A3
A4
A5
A6
Solve: a. |2x| = 3
b. |x + 5| ≥ 2
Solve x for negative constants a, b and c: ax + b < c
Solve, if possible by factorizing: a. x2 - 10 x + 25 = 0
b. x2 = 8x + 9
c. 9x2 + 9x + 1 = 0 d. x2 = 3x -10
If a ball is kicked upward from the top of a building 128 ft high with initial velocity 16 ft/s, then the height h above the
ground t seconds later will be h = 128 + 16t – 16 t2 . During what time interval will the ball be at least 32 ft above ground?
B1
B2
B3
Find the distance between the points (1, 1) and (4, 5)
Find the slope of the line through the points P(-3, 3) and Q (-1, -6)
Find the equation of the line if a. Through (2,-3) and slope 6
b. the x-intercept = 1 and y-intercept = -3
c. Through (-1, -2), perpendicular to the line 2x + 5y =8
D1
D2
D3
D4
Find sin θ, cos θ and tan θ for θ = a. 3π/4
b. 9π/2
c. 5π/6
Find cos θ and tan θ if sin θ = 3/5, 0 < θ < ½π
If in a right angle triangle the hypotenuse has length 10 and θ = 30 0, compute the length of the adjacent side
Find all values of x in [0, 2π] satisfying:
a. 2cos x - 1 = 0
b. 2 sin2 x = 1
c. sin x < ½ d. sin(½x) = ½√3
Chapter 1
1.1
Find the domain of the function
1.2
Find the domain and the range and sketch the graph of the
x + 2 if x ≤ -1
b. f(x)= { 2
function
a.
x
if x > -1
Determine whether f is even, odd or neither. If f is even or odd use symmetry to sketch its
graph
a. f(x) = x -2
b. f(x) = x2 + x
c. f(x) = x3 - x
Classify each function as a power function, root function, polynomial (state its degree),
rational function, algebraic function, trigonometric function, exponential or logarithmic
function:
1.3
1.4
a.
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
b.
a.
b.
c.
d.
e. r(x) = tan 2x f. s(x) = log10 x
Suppose the graph of f is given (e.g. f(x) = x2 ).Write equations for the graph that are
obtained from the graph y = f(x) as follows:
a. Shift 3 units upward
b. Shift 3 units downward
c. Shift 3 units to the right
d. Shift 3 units to the left.
e. Reflect about the x-axis.
f. Reflect about the y-axis
g. Stretch vertically by a factor of 3
h. Shrink vertically by a factor 3.
Given is the graph y = x2 . Sketch it and also
a. y = f(x + 2) b. y = f(x) + 2
c. y = 2f(x)
d. y = ½ f(x) + 3
Graph each function by starting with the graph of a “ standard function” and then applying
an appropriate transformation: a. y = -1/x b. y = cos (x/2) c.
d. y = 1 + 2x – x2
Find f+g, f-g, fg and f/g if f(x)= x3 + x and g(x) = 3x2 – 1
Find
,
,
and
and state its domain if a. f(x)= 2x2 - x and g(x) = 3x + 2
b.
and g(x) = x2
c.
and g(x) = x3 + x
Find
and state its domain if
,
and h(x) = x – 1
Write an equation that defines the exponential function with base a > 0. State the domain
and range if a  1. Sketch the general shape of the exponential function for the cases:
(1) a > 1
(2) a = 1
(3) 0 < a < 1
Use the general shapes of 1.11 to give a rough sketch of
a.
b.
c.
Find the exponential function
whose graph is through the points (1,6) and
(3, 24)
Test your basic math skills
a
1.
b
c
0.9
Simplify
2.
d
90
2-1
Simplify
3.
(a + 3)2 – (a+2) 2 =
4.
2a + 5
10a + 25
5
1
-26
-10
22
6
x ≤ 1
x ≤ -4
x ≤ -4 or x ≥ 1
-4 ≤ x ≤ 1
-13/3
3/23
1
-1
,
,
,
5.
6.
10.
If a = 1 and b = -2,
then -(a2 b) 3 – 2 - (a b2) 2 =
Solve 2x2 + 6x -9 ≥ -x2 - 3x +
3
3x + 5y =
Solve y from
8
{
-x + 6y = 5
Solve cos (2x) = ½
for ½π ≤ x ≤
11.
cos 2x =
2sin2 x - 1
1 – 2cos2 x
12.
sin (½π + x) =
The number of x-intercepts
of f(x) = x4 - 6x2 – 16 is
ln(e3 - e2) =
The solution of the equation
e2x = 9 is
Which of the equalities is/are
correct?
(more than 1 answer is
possible)
Which of the equalities is/are
correct
for all x  0, 1, -1 ?
Which of the equalities is /are
correct for p > 0
Write
as (where a and b
- sin x
- cos x
sin2 x - 2 cos2
x
cos x
0
2
4
1
ln(3)
ln(4.5)
2 + ln(e - 1)
2ln(e - 1)
ln 3
ln 4.5
(ln 3) 2/2
ln 81
7.
8.
9.
13.
14.
15.
16.
17.
18.
19.
20.
are
the smallest possible positive
integers)
If
, express R in c
Answer:
Answer:
and m
Exercises inverse functions and logarithmic functions:
2cos2 x - 1
sin x
1. Find the exact value of each expression:
a. log2 (64)
b.
c.
d.
2. Express the given quantity as a single logarithm: a. 2ln(4) - ln(2) b. ln (x) +a ln (y) – b ln(z)
3. Solve each equation for x: a.
b. ln(x) = -1 c.
d. ln(x) – ln(x-1) = 1
4. a. What is a one-to-one function?
b. How can you tell from the graph that the inverse function exists?
5. Determine whether f(x) is one-to-one if
a. f(x) = 7x -3
b. g(x) = |x|
6. Find the formula of the inverse function if a.
b.
c.
7. If a bacteria population starts with 100bacteria and doubles every three hours then the number of bacteria
after t hours is
a. Find the inverse of this function and explain its meaning.
b. When will the population reach 50000?
Answers:
A
B
D
1.1
1.2
1.3
1.4
1.5
1.7
1.8
1.9
1a. 5 - √5
1c. x2 + 1
2c. x > 3
1b. x + 1 for x ≥ -1
2a. x > -2
2d. -1 ≤ x < ½
-x – 1 for x < 1
2b. x ≥ -1
2e. x ≤ 1
1. 5
2. -9/2
3a. y = 6x - 5
1a. sin(¾π) = ½√2 , cos(¾π) = -½√2, tan(¾π) = -1
1b. sin(9π/2) = 1, cos(9π/2) = 0, tan(9π/2): undefined
1c. ½ , ½√3, ⅓√3
2f. x < 0 or x > ¼
3a. x = ±3/2
3b. x ≤ -7 or x ≥ -3
3b. y = 3x - 3
2. cos θ = 4/5,
tan θ = ¾
3. 5√3
4. x > (c-b)/a
5a. 5
5b. 1, -9
3c. 5x – 2y + 1= 0
4a. ⅓π, 5π/3
4b. ¼ π , ¾π,
5π/4, 3π/2
5c. –⅓
5d. none (D = 9-40< 0)
6. [0, 3] or 0 ≤ t ≤ 3 (t > 0!)
4c. 0 ≤ x ≤ π/6
and 5 π/6 ≤ x ≤ 2π
4d. ⅔π, 4π/3
a. D = { x | x  ±1}
b. D = (-∞, ∞)
a. D = [5, ∞) , R = [0, ∞)
b. D = (-∞, ∞) , R = (-∞, ∞)
a. Even (graph by reflecting about the y-axis) b. Neither c. Odd (graph by reflecting
about the origin)
a. root b. algebraic c. polynomial, degree 9 d. rational e. trigonometric e. logarithmic
a. y = f(x) + 3 b. y = f(x) - 3 c. y = f(x - 3) d. y = f(x + 3) e. y =- f(x) f. y = f(-x) g. y =
3f(x) h. y = ⅓f(x)
a. Start with y = 1/x and reflect it about the x-axis
b. Start with y = cos x and stretch horizontally by a factor 2
c. Start with y = 1/x and shift it to the right 3 units.
d. y = 1 + 2x – x2 = 2 - (x2 – 2x +1) = 2 – (x - 1)2 , so:
start with y = x2 , shift it 1 to the right, reflect about the x-axis and shift it 2 upward.
f+g (x)= x3 + 3x2 + x– 1, f-g (x)= x3 - 3x2 + x + 1,
and
a.
and
b.
2(3x+2)2 – (3x+2) ,
=3(2x2 – x) + 2 ,
= 9x + 8 (D= (-∞, ∞) for all)
(D = [1, ∞) ),
(D = [2, ∞) ), and
= 2(2x2 – x)2 – (2x2 – x)
= x2 - 1(D = [1, ∞) ),
= x4 (D = (-∞, ∞) )
=
(D = {x| x  0}),
c.
(D = {x| x  0}) and
= x-3+ x-1
(D = {x| x  0}) ,
x
= (x3 + x)3 + (x3 + x) (D = R)
(D = [1, ∞) )
1.11 f(x) = a , D = (-∞, ∞), R = (0, ∞)
(the graph is increasing for a > 1, y = 1 for a = 1 and decreasing for 0 < a < 1
1.12 a. Graph
(it`s an increasing graph) and shift 1 unit upward b.
1.10
x
(decreasing graph)
c. Graph
, reflect it about the x-axis and shift it 3 units upward
(decreasing graph, y-intercept = 2).
1.13 a = 2, C = 3
Answers to the test:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
18
d b c a d d b c c a d b b c a c b, c, d b, c
19.
20.
if
Answers Exercises logarithm and inverse function
1. a. 6
b. -2
c. 15
2. a. ln(8) = 3 ln(2)
b.
3. a. 4 ln(2)
b. 1/e
d. 8
c. 5 + log2(3) = 5+ ln(3)/ln(2)
d. ½ + ½
(x > 1 because of the domain of ln(x-1) !)
4. a. If x1  x2 implies f (x1)  f(x2) b. Every horizontal line intersects the graph at most once.
5. a. Yes
b. No
6. a.
b.
7. a. f -1 ( n) = 3log2(n/100) = 3ln(n/100)/ln(2)
(x ≥ 0)
c.
b. After about 26.9 hours (f -1(50000))