Download Notes Exam 3 to Exam 4

M130 Course Notes Exam 3 to Exam 4
4.1 Exponentials
1. The exponential function with base a is f (x) = ax for a > 0, a 6= 1.
2. Graphs of exponential when 0 < a < 1 and a > 1.
(a) Domain
(b) Range
(c) Passes through (0, 1)
(d) Always increasing/decreasing
(e) Horizontal asymptote
(f) Transformations
3. Base e
4.2 Logarithms
1. Note f (x) = ax is injective. Hence f −1 exists.
2. Defn: Let a 6= 1 be a positive number. The logarithmic function with base a is denoted
by loga and is defined by
loga x = y ⇐⇒ ay = x
3. Evaluating logs
(a) log1 = 0
(b) loga a = 1
(c) loga ax = x
(d) aloga x = x
4. Graphing (mono-inc.), passes through (1, 0)
5. Natural log
E.g.,
1. Simplify log4 (1/2)
2. Solve for x: log4 (x) = 2
3. Convert to log form 811/2 = 9
4. Convert to exp form log3 81 = 4
4.3 Logarithm Laws
1. log xy = log x + log y
2. log(x/y) = log x − log y
3. log xy = y log x
4. logb x =
logB x
logB b
5. Expand/Contract
4.4 Exponential and Logarithmic Equations
E.g.,
1. 2x = 8
2. 23x = 34
3. x2 2x − x2x = 2 · 2x
4. e2x − ex − 6 = 0
5. ln(2 + x) = 1
6. log2 (x2 − x − 2) = 2
7. ln(x − 1) + ln(x + 2) = 1
5.1 Unit Circle
1. x2 + y 2 = 1
2. is P on S 1 ?
3. Locating a point on S 1 given x/y value and Quadrant ...
4. Defn: The terminal point P (x, y) is the value obtained by moving t units from (1, 0) on
S 1 counterclockwise if t > 0 and clockwise for t < 0.
(a) t = π
(b) t = 0
(c) t = π/2
5. Defn: The reference number t corresponding to the terminal point for t is the shortest
distance on S 1 between T P (t) and the x-axis
5.2 Trigonometric Functions
1. On S 1 , sin θ = y, csc θ = 1/y, etc.
2. Consider rt. 4
3. Evaluating trig. function
4. Where each is positive/negative
5. Domain
(a) sin, cos: R
(b) tan, sec: R\ {π/2 + nπ}n∈Z
(c) cot, csc: R\ {nπ}n∈Z
6. csc θ = 1/ sin θ, etc.
7. Even/Odd
8. Pythagorean Identities
(a) sin2 θ + cos2 θ = 1
(b) tan2 θ + 1 = sec2 θ
(c) 1 + cot2 θ = csc2 θ
E.g.,
1. sin, cos, tan, cot, tan, sec, csc @0, π, π/
2. (−3/5, 4/5) is terminal point evaluate trig functions.
3. Sign of tan t sec t in Q IV
4. Write first in terms of second when in Q III tan t, cos t
5. Write first in terms of second when in Q IV sin t, sec t
5.3 Trigonometric Graphs
1. Periodicity Identities:
(a) sin(t + 2πn) = sin t for n ∈ Z
(b) cos(t + 2πn) = cos t for n ∈ Z
2. Graphs of Sine, Cosine
3. Ψ(x) = A sin k(x − γ) and χ(x) = A cos k(x − γ) for positive k have:
(a) Amplitude |A|
(b) Period 2π/k
(c) Phase Shift γ
(d) One period appears in the interval [γ, γ + 2π/k]
E.g.,
1. Find the amplitude, period, phase shift, and graph one period.
(a) y = 2 sin(x − π/3)
(b) y = cos[1/2(x + π/4)]
(c) y = 2 sin(2/3x − π/6)
2. The graph of one period is given. Determine the period, phase shift, and amplitude; and
give an equation.
5.4 More Trigonometric Graphs
1. Periodicity Identities:
(a) tan(t + π) = tan t
(b) cot(t + π) = cot t
(c) csc(t + 2π) = csc t
(d) sec(t + 2π) = sec t
2. Graphs of sec x, csc x, tan x, cot x
(a) On (−π/2, π/2), y = tan t is %.
(b) On (0, π), y = cot t is &.
(c) y = csc t:
(0, π) ∼ ∪
(π, 2π) ∼ ∩.
(d) y = sec t:
(0, π/2) ∼ RHS(∪)
(π, 3π/2) ∼ ∩
(3π/2, 2π) ∼ LHS(∪).
3. The functions y = a tan(kx) and y = a cot(kx) have period π/k.
4. Then functions y = a csc(kx) and y = a sec(kx) have period 2π/k.
E.g.,
1. Find the period and graph the function.
(a) y = tan(x/2)
(b) y = tan(x − π/4)
(c) y = 2 csc(x − π/3)
(d) y = sec[2(x + π/2)]
(e) y = cot(x − π/2)
(f) y = 1/2 sec(2πx − π)
6.1 Angle Measure
1. Convert between radians/degrees.
2. Defn: An angle is in standard position if one side of the angle is the x-axis.
3. Defn: Two angles in standard position are coterminal if their (non x-axis) sides coincide
4. How to reduce angles. (Review)
5. Arc length s = rθ
6. Area of a sector A = 21 r2 θ
E.g.,
1. (Length of arc/area of sector) that subtends central angle of 45◦ in a circle of radius 10.
2. A sector of a circle of radius 24 has an area of 288. Find the central angle.
6.2 Right Triangle Trigonometry
1. Trig. functions in reference to a triangle.
2. Special triangles
E.g.,
1. From the top of a 200ft building you check out hot c/hicks. If you spy a hottie and the
angle of depression to the hottie is 23◦ . How far are you from the hottie? How far is the
hottie from the building?
6.3 Trigonometric Functions of Angles
1. Trig. functions in reference to an imbedded triangle in a circle of radius r.
E.g.,
1. Find the values of the trig functions of θ if cos θ = −7/12 with θ ∈ QIII.
2. Find the values of the trig functions of θ if cot θ = 1/4 with sin θ < 0.
7.1 Trigonometric Identities
Pythagorean
1. sin2 θ + cos2 θ = 1
2. tan2 θ + 1 = sec2 θ
3. cot2 θ + 1 = csc2 θ
Reciprocal
1. tan θ =
sin θ
cos θ
2. cot θ =
cos θ
sin θ
3. sec θ =
1
cos θ
4. csc θ =
1
sin θ
=
1
tan θ
Cofunction
1. sin( π2 − θ) = cos θ
2. cos( π2 − θ) = sin θ
3. tan( π2 − θ) = cot θ
4. sec( π2 − θ) = csc θ
5. csc( π2 − θ) = sec θ
6. cot( π2 − θ) = tan θ
Even/Odd
1. sin(−θ) = − sin θ
2. cos(−θ) = cos θ
3. tan(−θ) = − tan θ
E.g.,
1. Simplify cos t csc t
2. Simplify cos3 t + sin2 t cos t
3. Simplify
1+cot A
csc A
4. Verify cos x/ sec x + sin x/ csc x = 1
5. Verify
1 − sin x
= (sec x − tan x)2
1 + sin x
6. Verify sec t csc t(tan t + cot t) = sec2 t + csc2 t
7. Verify
tan x + tan y
= tan x tan y
cot x + cot y
7.2 Addition and Subtraction Formulae
1. sin(α ± β) = sin α cos β ± cos α sin β
2. cos(α ± β) = cos α cos β ∓ sin α sin β
3. tan(α ± β) =
tan α ± tan β
1 ∓ tan α tan β
E.g.,
1. Determine sin 15◦
2. Determine cos 17π/12
3. Verify cot(π/2 − u) = tan u
4. Verify cos(x − π) = − cos x
5. Verify 1 − tan x tan y =
cos(x + y)
cos x cos y
6. Verify cos(x + y) cos(x − y) = cos2 x − sin2 y
7. Verify cos(x + y) + cos(x − y) = 2 cos x cos y
7.3 Double-Angle and Half-Angle Formulae
Double-Angle
1. sin 2x = 2 sin x cos x
2. cos 2x = cos2 x − sin2 x = 1 − 2 sin2 x = 2 cos2 x − 1
3. tan 2x =
2 tan x
1 − tan2 x
Half-Angle
r
1 − cos x
2
r
1 + cos x
2. cos(x/2) = ±
2
1. sin(x/2) = ±
3. tan(x/2) =
1 − cos x
sin x
=
sin x
1 + cos x
E.g.,
1. Find sin 2x if x is in QII and tan x = −4/3
2. Determine tan 15◦
3. Determine cos 3π/8
4. Verify sin 8x = 2 sin 4x cos 4x
5. Verify tan2 (x/2 + π/4) =
1 + sin x
1 − sin x
7.4 Inverse Trigonometric Functions
1. a
7.5 Trigonometric Equations
1. a