Download Outcomes for Exam 2

Mth 112 Outcomes for Exam 2
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To be successful on Exam 2 you should be able to …
Prerequisite Material
1. Solve a linear or quadratic equation algebraically. (QF is adequate)
2. Graph a line from its equation without the aid of a graphing calculator.
3. Find the equation of a line from (a) two points, (b) slope and a point, (c) graph
Circles
1. Find the arc length, circumference or sector area of a circle from partial information.
2. Switch among alternate rotational velocities (ω). e.g. rpm ↔ rad/sec
3. Find linear velocities of a rotating object. e.g. velocity of vehicle given tire rpm
4. Find rotational velocities in a combination of connected gears. (Indirect proportion)
5. Convert angles among various formats: ±θ in radians, ±θ in DMS, bearing, azimuth.
6. Use sin θ, cos θ, tan θ to find coordinates on the unit circle (r = 1).
7. Use sin θ, cos θ, tan θ to find coordinates on edge of circle when r ≠ 1.
Triangles & Trigonometry
1. Apply the Pythagorean Theorem to find missing dimensions.
2. Apply similar triangles to find missing dimensions.
3. Apply trigonometry to generate trig relationships from a partially labeled triangle.
4. Apply trigonometry to generate trig relationships from partial information. e.g. sin θ = 4/5, cos θ = ?
5. Apply sin θ, cos θ, tan θ to find missing dimensions.
6. Apply sin-1 y/r, cos-1 x/r, tan-1 y/x to find missing angles.
7. Apply the Law of Sines to find missing dimensions or angles.
8. Apply the Law of Cosines to find missing dimensions or angles.
9. Apply right triangle trigonometry to solve applications.
10. Apply non-right triangle trigonometry to solve applications.
Functions
1. Use geometric and trigonometric relationships to find functional relationships.
2. Rewrite an implicit function in explicit form. i.e. F(x,y) = 0 → y = f(x).
3. Understand function notation/vocabulary. e.g. domain, range, f(x), f(g(x)), f(2), f(a + b)
4. Use appropriate notation to describe an interval. e.g. 0 ≤ tan x < ∞ or range of tan x is [0, ∞)
5. Apply inverse notation correctly. tan-1 x ≠ cot x.
6. Distinguish between f-1(x) vs. [f(x)]-1
7. Solve (a) f(t) = g(t) by the intersection method, (b) Solve f(t) = 0 by the root method, (c) Solve f(t) = k by tables.
8. Apply trigonometric relationships to simplify expressions. e.g. tan[arctan (4 + x)] − 4 = x, sin x cot x = cos x
9. Apply inverse trig relationships to generate new trig relationships. e.g. sin-1 x = θ, cos θ = ?
10. Give the domain, range and ±sign of trigonometric functions and their inverses.
Some Sample Problems
1)
y = sin x ; tan x = ?
2)
sin θ = 2t/ 3 ; tan θ = ?
3)
Solve x2 − 3x = 10
4)
Solve for y: 3 sin y = x + 2
5)
Solve for y: arctan (y + 1) = x/5
6)
Solve for y: 10 − 8(3x − 5y)/4 = 2x − 3y + 7
7)
In what quadrant(s) is (sin θ)(cos θ) positive?
8)
At what θ-values are tan θ undefined?
9)
Where is cos θ = 1?
10)
Where does sin θ = cos θ ?
11)
Where is arcsin x undefined?
12)
What is the domain of arcos x?
13)
What is the range of invtan x?
14)
What is the range of sin x?
15)
tan x = a/b; cot x = ?
16)
arctan (P/Q) = t; sin t = ?
17)
Is sin(sin-1(x)) = x? Why, why not? Is sin-1(sin(x)) = x? Why, why not? Is sin-1(sin(x)) = sin(sin-1(x))?
18)
Find all solutions where x ≥ 0 give answer as x.xx.
19)
A marlin is hooked and running (swimming away) at 30 m/sec. How fast is the reel spinning (rpm) if the
spool diameter is 10 cm? Assume the size of the line does not affect the diameter.
20)
An engine is turning at 4500 rpm. The main pulley is 20 cm, the AC pulley is 8 cm and the alternator
pulley is 15 cm. Note: The main pulley turns at engine speed.
(a) How fast in deg/sec is the main pulley turning?
(b) How fast in rad/sec is the main pulley turning?
(c) How fast in rpm is the AC pulley turning?
(d How fast in m/sec is the belt moving?
(e) If a 4th pulley is included how large must it be so that it turns at 400 rad/sec?
21)
A railroad track heading due north makes a curving turn and heads N 13° E. The inner radius of the circular arc is 100 m. The
rails are 2 m apart. How much longer is the outer rail?
22)
Through how many radians will each of the following hands of a clock rotate from 12 noon to 6:30? Leave your answer in
terms of . (a) Hour hand _________ (b) Minute hand _________ (c) Second hand ________
23)
What is the angle between the hands of a clock at 5:22?
24)
Find A, a, b
b
25)
Find a, b, C
85'
26)
Find the height of the mountain
θ1 = 31.25°
110°
105'
A
θ2 = 32.5° x=1,542'
125'
a
125'
a
32°
27)
ex + 5sin (x + 1) = 8
C
Find h as a function of θ and x
28)
b
Find r as a function of R and θ.
29)
Find y as a function of θ and x.
r

5
R
h

30)

x
Find height of the flagpole
a=43°, b =34°, Find h(x).
31)
Tower A: Fire is sighted at N 53° E
Tower B: Fire is sighted at N 58° W
Find the coordinates of the fire. A = (0, 0)
Find x, θ, d
32)
6
228 98.4°
108.5°
a
33)
x
d
1455
h
b
y
x
x

Tarzan wants to retrieve a brilliant red orchid for Jane's hair. He leaves Jane and runs due north for 600 m then turns and runs
1200 m at N 65° W. There he swims a 100 m wide river that flows due SW. Once on dry land he runs due south for 2500 m.
What direction and how far is it for Tarzan to get back to Jane, the love of his life?
Answers (unchecked)
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17a)
17b)
17c)
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20a)
20b)
20c)
20d)
20e)
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33)
y
1 − y2
2t
tan θ =
3 − 4t2
x = -2, 5
x+2
y = arcsin
3
y = tan (x/5) − 1
8x − 3
y=
13
I, III
±90° ± n(360°) or ± π/2 ± n(2π)
± n (360°) or ± n(2π)
±45° ± n(180°) or ±π/4 ± n(π)
x > 90° , x < 90° or x > π/2, x < π/2
[0°, 180°] or [0, π]
(-90°, 90°) or (-π/2, π/2)
[-1, 1]
cot x = b/a
P
sin t =
2
P + Q2
Yes because Range of invsin (x) is in the Domain of sin (x)
No because invsin (sin (2π)) = 0 ≠ 2π !
Only when x is in the Range of invsin (x). i.e. -π/2 ≤ x ≤ π/2
x ≈ 1.9589
18,000/π rpm ≈ 5730 rpm
27,000 deg/sec
150π rad/sec
11,250 rpm
15π m/sec
7.5π cm
13π/90 m ≈ 45.38 cm
(a) 195° = 13π/12; (b) 6.5 rev = 13π; (c) 390 rev = 780π
29°
a ≈ 96.80°; A ≈ 159.27; b ≈ 51.20°
a ≈ 38.48°; b ≈ 31.52°; C ≈ 188.75'
h ≈ 19, 703'
h(x) = x tan (θ)
r(R, θ) = Rcos(θ)
y = x tan (θ) − 5
h(x) = x [tan 77° − tan 43°]
Location ≈ (9.58 mi, 7.22 mi)
x ≈ 3177.98; θ = 63.1°; d ≈ 2444.90
(x, y) ≈ (-1178, -1351) D ≈ 1792 m; θ ≈ 49°; → N 41° E
tan x =
[
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