Download Logarithmic and Exponential Functions

Review for Final Exam
Note: all students will take final exam in two parts, over two days.
Logarithmic and Exponential Functions
1.
Determine a formula for the exponential function given in the table below.
X
70
71
72
73
74
F(x)
32
8
2
0.5
0.125
1
𝑓(𝑥 ) = 32 ( )𝑥−70
4
2. Write and solve an exponential equation. Round answer to the next whole number.
The population of a town in Alaska was 9500 in 1985. Since then, the population has decreased
an average of 1.3% per year. In what year do you predict that the population will drop below
3000 people?
In the year 2073
3. Use properties of logarithms to write the expression as a sum or difference of logarithms or
multiples of logarithms:
a. log
𝑥
log 𝑥 − 2 log 𝑦
𝑦2
1
1
ln 5 + ln 𝑦
3
3
3
b. ln √5𝑦
4. Use properties of logarithms to write the expression as a sum or difference of logarithms or
multiples of logarithms. Then, list the transformations that change f(x) = log x into g(x).
a. log
1000
3
√𝑥
1
3 − 3 log 𝑥
1
translate up 3 units; reflect over x-axis; vertical shrink factor of 3
b. log
(𝑥−1)2
10
2 log(𝑥 − 2) − 1
translate down 1 unit and right 2 units; vertical stretch factor of 2
5. A bacteria population is increasing 12.2% each day. How long will it take the population to
quadruple? Write an equation and solve it. Round to the nearest whole day.
≈ 12 or 13 days
6. Solve the equation. Round to the thousandths place.
a.
2(3𝑥−2) = 23
x ≈ 2.175
b.
3(2)𝑥 = 51
x ≈ 4.087
Review for Final Exam
7. Georgia needs $5000 for a down payment on a car in 6 years. Her bank promises an interest rate
of 1.15% yearly. How much should she invest now to guarantee that she will have enough
money for the car in 6 years?
Principal is ≈ $ 4668.47
Trigonometry Basics
5𝜋
9
8. Convert 100° into radians. Give an exact, simplified answer:
𝑟𝑎𝑑𝑖𝑎𝑛𝑠
9. Find all six trigonometric functions of A in ∆ABC:
5
Csc A =
13
5
5 cm Cos A = 12
Sec A =
13
12
C
Cot A =
12
5
B
Sin A = 13
13
A
12 cm
5
Tan A =
12
B
10. Solve the right ∆ABC if a = 25 and b = 7.3.
16.28 °
11. Solve the right ∆ABC if B = 48° and a = 7
B
26.044
units
48 °
25 units
10.46
units
A
7 units
C
7.77
units
73.72 °
A
7.3 units
C
Review for Final Exam
12. The point (4,9) is on the terminal side of angle θ. Evaluate the six trigonometric functions for θ:
Sin θ =
Tan θ =
9
Csc θ =
√97
9
Cot θ =
4
√97
9
Cos θ =
4
√97
Sec θ =
√97
4
4
9
13. Find one positive angle and one negative angle that are co-terminal with an angle of
19𝜋
−13𝜋
8
8
3𝜋
8
2𝜋
radians:
14. Calculate the length of an arc intercepted by a central angle of
radians in a circle with a radius
3
4𝜋
of 2 centimeters.
𝑐𝑒𝑛𝑡𝑖𝑚𝑒𝑡𝑒𝑟𝑠
3
Sinusoidal Functions and Problem-solving
15. Determine whether f(x) is a sinusoid. If yes, determine the values of a, b, and h so that
f(x) ≈ a sin (b (x – h)) + 0. Round to two decimal places if necessary.
a. f(x) = sin x – 3 cos x
= 3.16 sin 1 ( x – 1.25)
b. f(x) = π sin3 x – 4π sin 2x
not sinusoidal because coefficients are 3 and 2
Review for Final Exam
c. f(x) = 3 cos 2x – 2 sin 2x
= -3.61 sin 2 ( x – 0.491)
16. In the equation sin x = 0.5, solve for x in the interval
𝜋
2
≤ 𝑥 ≤ 𝜋.
x=
5𝜋
6
17. The average daily air temperature (℉) for Fairbanks, Alaska, from 1975 to 2004, can be
2𝜋
modeled by the equation: 𝑇(𝑥) = 37.3 sin [ (𝑥 − 114)] + 26, where x is the time in days.
365
January 1 is represented by x = 1. On what days do you expect the average temperature to
be 32℉?
Day 123 and Day 287, which by the way is May 3rd and October 14th in the year 2014
Review for Final Exam
18. On a particular Labor Day, the high tide in southern California occurs at 7:12 AM. At that time,
you measure the water at the end of the Santa Monica Pier to be 11 feet deep. At 1:24 PM it is
low tide and you measure the water to be only 7 feet deep. Assume the depth of the water is
modeled by a sinusoidal function of time.
a. At what time on that Labor Day does the first low tide occur? 1 AM
b. What was the approximate depth of the water at 9:00 PM? 10.5 feet
c. What is the first time on that Labor Day that the water is 9 feet deep? 4:06 AM
d.
2
19. Specify the period and amplitude of the function f(x) = 4 sin 3 (𝑥):
Amplitude is 4; period is 3π
20. State the amplitude, period, phase shift, and vertical displacement of the function
2
𝑥−3
f(x) = 3 cos ( 4 ) + 1 as compared to the parent function f(x) = cos x:
Amplitude is
2
,
3
period is 8π, phase shift is 3, and vertical displacement is 1
21. Write an equation of the sine function with amplitude
vertical translation down 5 units:
1
f(x) = 2 sin 𝜋(𝑥 + 3) − 5
½ , period 2, phase shift left 3 units, and
Review for Final Exam
22. Solve ∆ABC, if a = 3.2 inches, b = 7.6 inches, and c = 6.4 inches:
C
7.6 inches
74.7 °
3.2 inches
A
80.8 °
24.6 °
B
6.4 inches
23. Solve ∆DEF if d = 41 meters, D = 22.9°, and F = 33°:
F
33 °
87.2
meters
41 meters
D
124.1 °
22.9 °
27.0
meters
E
24. Two triangles can be formed using the measurements C = 68°, a = 19 feet, and c = 18 feet. Solve
both triangles:
Review for Final Exam
B
B
10.15 °
33.85 °
180 - Angle A =
new angleA
18.0 feet
19.0 feet
18 feet
19 feet
101.85 °
78.15°
3.42 feet
C
10.8 feet
C
A
68 °
A
68 °
25. What is the measure of the largest angle in a triangle with sides 117 cm, 102 cm, and 160 cm?
160 cm
117 cm
93.62 °
102 cm
Trigonometric Identities
𝜋
26. Simplify the expression: cot ( 2 − 𝑥) csc (-x)
=
− sec x
Review for Final Exam
27. Simplify the expression csc θ cot2 θ + csc θ
28. Simplify the expression
=
𝑠𝑖𝑛2 𝛽+ 𝑐𝑜𝑡 2 𝛽+ 𝑐𝑜𝑠2 𝛽
𝑐𝑠𝑐 2 𝛽
csc3 θ
=
1
29. Determine all the solutions to the equation 4 cos2 x – 4 cos x + 1 = 0 in the interval [0, 2π):
X=
𝜋
3
𝑎𝑛𝑑
5𝜋
3 radians
30. Write the expression 5 – 6 sin2 θ – 5 cos θ in factored form as an algebraic expression of a single
trigonometric function:
2
6 cos x – 5 cos x – 1
31. Determine all the solutions to the equation tan ρ sin2 ρ = tan ρ in the interval [0, 2π):
X = 0π ,
𝜋
2
,𝜋
𝑎𝑛𝑑
3𝜋
radians
2