MA45W Pre-calculus ... A large pond is stocked with fish. The fish P

MA45W Pre-calculus
Name
Semester 1 Review
1. A large pond is stocked with fish. The fish
4. Find all real solutions of the equation.
population P is modeled by the formula
P = 4t + 14 t + 150 , where t is the number of days
since the fish were first introduced into the pond.
How many days will it take for the fish population
to reach 400? Please round the answer to the
nearest day.
x
x+4
−
=1
3x + 8 x + 3
t = __________ days
5. Find all real solutions of the equation.
2. A small–appliance manufacturer finds that the
profit P (in dollars) generated by producing x
microwave ovens per week is given by the formula
1
P=
x(200 − x) provided that 0 ≤ x ≤ 60.
10
x 6 − 9x 3 − 10 = 0
How many ovens must be manufactured in a given
week to generate a profit of $750?
6. Find all real solutions of the equation.
P = __________ ovens per week
x − 7 x + 12 = 0
3. Solve the equation by factoring.
3x 2 + 8x = 3
7. Jack invests $2,000 at a certain annual interest rate,
and he invests another $3,000 at an annual rate that
is one-half percent higher. If he receives a total of
$265 interest in one year, at what rate is the $2,000
invested?
__________ %
1
8. A box with a square base and no top is to be made from a square piece of cardboard by cutting 4-in. squares from
each corner and folding up the sides, as shown in the figure. The box is to hold 100 in 3 . How big a piece of
cardboard is needed?
__________ in. by __________ in.
9. Find the x- and y-intercepts and determine if the
12. Solve the equation algebraically.
equation is odd, even or neither.
3
6
6
−
=
x + 2 3x 3x + 6
y = x 4 − 16
(a) Find the x-intercepts.
(b) Find the y-intercepts.
(c) Is the function odd, even, or neither and why?
10. Do the graphs intersect in the given viewing
rectangle? If they do, how many points of
intersection are there?
y=
25 − x 2 , y =
1
(53 − 7x) ; [–6, 6] by [–1. 6]
8
11. Find all real solutions of the equation, correct to
two decimals.
x 3 − 3x 2 − x − 3 = 0
x = __________
2
15. Find the domain of the function.
13. Evaluate the piecewise defined function at the
indicated values.
h(x) =
ÔÏÔ 2
ÔÔ x + 4x if x ≤ −3
ÔÔ
f(x) = ÌÔ x
if − 3 < x ≤ 1
ÔÔ
ÔÔ
if x > 1
ÔÓ −9
8x − 7
(a) Evaluate f(−4) .
f(−4) = __________
16. Find
ÁÊ 7 ˜ˆ
(b) Evaluate f ÁÁÁÁ − ˜˜˜˜ .
Ë 2¯
ÊÁ 7 ˆ˜
f ÁÁÁ − ˜˜˜˜ = __________
ÁË 2 ¯
g(x + h) − g(x)
if g(x) = 4x 2 - 3x + 5
h
17. The graph of a function is given. Determine the
average rate of change of the function between the
indicated values of the variable.
(c) Evaluate f(−3) .
f(−3) = __________
(d) Evaluate f(0) .
f(0) = __________
(e) Evaluate f(35) .
f(35) = __________
14. Find the inverse of the function and its domain of
the function.
f(x) = x 2 + 7, 0 ≤ x ≤ 9
3
22. Describe the transformations applied
18. A function is given. Determine the average rate of
to y =
x in the appropriate
2
5 − x +4
order given y = −
3
change of the function between the values of the
variable.
h(t) = t 2 + 7t; t = −1, t = 1
19. A function is given. Determine the average rate of
change of the function between the values of the
variable.
23. A rectangle has a perimeter of 30 ft. Find a function
f(x) = x + x 4 ; x = −1, x = 3
that models its area A in terms of the length x of
one of its sides.
20. A function f is given, and the indicated
transformations are applied to its graph (in the
given order). Write the equation for the final
transformed graph.
f(x) =
24. Use the given graphs of f and g to evaluate the
3
x reflect in the y-axis, shrink vertically by
1
5
a factor of , and shift upward unit
2
7
expression.
21. Find the maximum or minimum value of the
function.
f (t) = 7t 2 + 14t + 101
(a) (g û f)(3) = ______________
(b) (g û g)(−2) = ______________
4
28. The graph of a polynomial function
−1
25. Draw f (x)
P(x) =
1 3 3
x − x + 3 is given.
18
2
26. Find the domain of the function.
f(x) =
x +
8−x
(a) From the graph, find the x-intercept(s).
x = __________
(b) From the graph, find the y-intercept(s).
−1
27. If g(x) = x + 8x with x ≥ −4 , find g (9) .
2
y = __________
(c) From the graph, find the coordinates of all local
maxima.
(d) From the graph, find the coordinates of all local
minima.
(e) From the graph, find the intervals on which the
function is:
increasing, decreasing, and constant
5
33. Find a polynomial of the specified degree that has
29. Graph the polynomial by finding an appropriate
window.
the given zeros.
y = 2x 3 + 3x 2 − 12x − 40
Degree 3; zeros –3, 3, 5
(a) From the graph, find the coordinates of all local
maxima.
(b) From the graph, find the coordinates of all local
minima.
34. Find all zeros of the polynomial.
P(x) = x 4 − 8x 3 + 24x 2 − 32x + 16
30. Use synthetic division to find the remainder when
dividing by (x - 3).
P(x) = x 3 + 2x 2 − 7x + 7
35. Find all zeros of the polynomial.
P(x) = x 4 − 2x 3 − 8x 2 + 18x − 9
31. Use synthetic division to find P(-8)
P(x) = x 3 + 8x 2 − 6, c = −8
P (−8) = __________
36. Find all zeros of the polynomial.
P(x) = 9x 4 − 85x 2 + 36
32. Two polynomials P and D are given. Use long
division to divide P (x ) by D (x ) , and express P in
the form P (x ) = D (x ) ⋅ Q (x ) + R (x ) .
P(x) = x 5 + x 4 − 8x 3 + x + 2, D(x) = x 2 + x − 7
37. Find all zeros of the polynomial.
P(x) = −x 3 + 3x + 2
6
42. Solve for all real solutions.
38. Find all the zeros of the polynomial.
P(x) = 5x 4 + 36x 3 + 32x 2 − 1
3x = x 5
x = __________
43. If $3,000 is invested in an account for which
interest is given, find the amount of the investment
at the end of 3 years for the following interest rates.
39. Find all zeros of the polynomial.
(a) 6% compounded quarterly. Please give the
answer to two decimal places.
P(x) = x − x − 6
3
A(3) = __________
1
(b) 6 % compounded monthly. Please give the
2
answer to two decimal places.
40. Find all zeros of the polynomial.
A(3) = __________
P(x) = x 5 − 29x 4 + 2x 3 − 58x 2 + x − 29
(c) 7% compounded daily. Please give the answer
to two decimal places.
A(3) = __________
(d) 8% compounded continuously. Please give the
answer to two decimal places.
41. Find the intercepts and asymptotes and graph.
r(x) =
A(3) = __________
3x(x + 2)
(x − 1)(x − 6)
44. How long will it take for an investment of $3,000
(a) Determine the x-intercept(s).
(b) Determine the y-intercept(s).
to double in value if the interest rate is 6.5% per
year, compounded continuously? Please round your
answer to two decimal places.
(c) Determine the vertical asymptote(s).
__________ years
(d) Determine the horizontal asymptote(s).
7
50. Use the Laws of Logarithms to expand the
45. Solve for x.
expression.
(a)log x 729 = 3
ˆ
Ê
log2 ÁÁÁ x 4 y ˜˜˜
¯
Ë
x = __________
(b)log x 16 = 2
x = __________
46. Find the domain of the function.
51. Use the Laws of Logarithms to expand the
expression. Assume x > 0, y > 0 and z > 0.
f(x) = log8 (x + 3)
ÊÁ 2
Á x
loga ÁÁÁÁ 4
Á yz
Ë
ˆ˜
˜˜
˜˜
˜˜
¯
47. Evaluate the expression.
log3 6 − log 3 8 + log 3 108
52. Use the Laws of Logarithms to combine the
expression.
ln(x + y) + ln(x − y) − 2ln z
48. Evaluate the expression.
log4 64 600
49. Evaluate.
log7 539
8
57. Solve the equation. Please round the answer to four
53. Use the Laws of Logarithms to combine the
expression.
decimal places, if necessary.
2 ÊÁË log 3 x + 2 log 3 y − 4log 3 z ˆ˜¯
e 4 x + 2e 2 x − 8 = 0
x = __________
58. Solve the logarithmic equation for x. Please round
the answer to four decimal places, if necessary.
log(9x + 5) = 3
54. Find the solution of the exponential equation,
x = __________
correct to four decimal places.
e −3x = 9
59. Solve the logarithmic equation for x. Please round
x = __________
the answer to four decimal places, if necessary.
2 − ln(7 − x) = 0
55. Find the solution of the exponential equation,
x = __________
correct to four decimal places.
3 5x − 2 = 4
60. Solve the logarithmic equation for x. Please round
x = __________
the answer to four decimal places, if necessary.
log3 x + log 3 (x + 1) = log 3 6
56. Find the solution of the exponential equation,
x = __________
correct to four decimal places.
Ê
ˆ
5 ÁÁ 1 + 106x ˜˜ = 12
Ë
¯
x = __________
9
65. Each time your heart beats, your blood pressure
61. The point P is on the unit circle. Find P(x,y) from
first increases and then decreases as the heart rests
between beats. The maximum and minimum blood
pressures are called the systolic and diastolic
pressures, respectively. Your blood pressure
reading is written as systolic/diastolic. A reading of
120/80 is considered normal.
the given information.
3
and P lies below the
4
x-axes. Find the y-coordinate.
The x-coordinate of P is −
A certain person's blood pressure is modeled by the
function
p(t) = 110 + 15sin(150π t)
where p(t) is the pressure in mmHg, at time t
measured in minutes.
62. Find the exact value of the trigonometric function
(a) Find the period of p.
at the given real number.
csc
period = __________ minute
3π
2
(b) Find the number of heartbeats per minute.
(c) Find the blood pressure reading.
__________ / __________
63. Find the exact value of the trigonometric function
at the given real number.
(d) How does this compare to normal blood
pressure?
sec(−4π )
66. Find an angle between 0° and 360° that is
64. From the information given, find the quadrant in
coterminal with the given angle.
which the terminal point determined by t lies.
770°
sint < 0 and cos t > 0
__________ °
10
67. Find an angle between 0° and 360° that is
70. The wheels of a car have radius 11 in. and are
coterminal with the given angle.
rotating at 1500 rpm. Find the speed of the car in
mi/h.
1,290°
Please round the answer to the nearest tenth. Use π
=3.1416.
__________ °
__________ mi / h
68. Find the length of the arc s in the figure.
71. Find the exact value for each trigonometric
function.
Please give the answer to one decimal place.
69. Find the area of the sector shown in the figure.
Please give the answer to two decimal places.
11
(a) sin
2π
3
(b) sec
4π
3
(c) cot
9π
2