Acellus Precalculus Semester 1

Find the amount of money accumulated after investing a principle (P) for years (t) at interest rate (r), compounded continuously. P = $15,500 (starting number) r = 9.5%, 0.095 (rate) t = 12 (time) Answer: A = Pe^rt 15500e^0.095*12 = $48464.91 (Correct)

$48464.91

Find the amount of money accumulated after investing a principle (P) for years (t) at interest rate (r), compounded continuously. P = $5 (starting number) r = 3%, 0.03 (rate) t = 10 (time) Answer: A = Pe^rt 5e^0.03*10 = $6.75 (Correct)

$6.75

Find the amount of money accumulated after investing a principle (P) for years (t) at interest rate (r), compounded continuously. P = $20,000 (starting number) r = 6%, 0.06 (rate) t = 25 (time) Answer: A = Pe^rt 20000e^0.06*25 = $89633.78 (Correct)

$89633.78

Find the amount of money accumulated after investing a principle (P) for years (t) at interest rate (r), compounded continuously. P = $10,000 (starting number) r = 11%, 0.11 (rate) t = 20 (time) Answer: A = Pe^rt 10000e^0.11*20 = $90250.13 (Correct)

$90250.13

Find the x-coordinate (and y-coordinate) of this function's relative minimum. f(x) = | 2x - 6 | + 4 (3, 4) (Correct)

(3, 4)

Find the vertex of this parabola on the coordinate plane. y = 1/2(x - 4)^2 + 4 (4, 4) (Correct)

(4, 4)

Find formulas for (f + g), (f - g), and (f *g). f(x) = √x and g(x) = sin x (f + g) = √x + sin x (Correct) (f - g) = √x - sin x (Correct) (f * g) = √x ?

(f + g) = √x + sin x (f - g) = √x - sin x (f * g) = √x ?

Perform the indicated operation and simplify. Leave answer in factored form. x^2 + 7x + 6 / x^2 + x - 6 / x^2 + 5x - 6 / x^2 + 5x + 6 (x + 1) (x + 2) / (x - 2) (x - 1)

(x + 1) (x + 2) / (x - 2) (x - 1)

Fill in the question mark cos 7π/3 = ±√x/x

+1/2

Fill in the question mark cos 23π/6 = ±√x/x

+√3/2

Find the zeros. F(x) = 9x^2 - 3x - 2 List your answers in ascending order. - 1 / 3, 2 / 3 (Correct)

- 1 / 3, 2 / 3

Point P is on the terminal side of angle θ Find cos θ Point P is located at (-3, 0).

-1

Simplify the following expression as much as possible (sin x) * (csc(-x)) Answer: -1 (Correct)

-1

Evaluate. log 0.5 12 (Round your answer to the nearest hundredth).

-3.58

Find the sum of the first n terms using the formula: a (1 - r^n) / 1 - r 1 + -2 + 4 + -8 + 16 + -32 .... (10 terms) Answer: a = 1 (starting number) r = -2 (rate) n = 10 (number of terms) 1 (1 - (-2)^10) / 1 - (-2) = -341 (Correct)

-341

If tan A = 4/3 and tan B = 12/5, calculate and simplify the following: tan (A + B) = ? ?/? Answer: tan(A + B) = tan A + tan B / 1 - tan A * tan B 4/3 + 12/5 / 1 - (4/3*12/5) = -56/33 (Correct)

-56 / 33

Find the exact value for θ if -π/2 < θ < π / 2 tan^-1 (0) = θ Answer: 0 (Correct)

0

Use synthetic division to find the remainder. 2x^4 - 9x^3 + 23x^2 - 31x + 15 / x - 1 Remainder is 0. (Correct)

0

Determine the following without using a calculator tan -15π/4

1

Simplify the following expression as much as possible sin^2 (-x) + cos^2 (-x) Answer: 1 (Correct)

1

Simplify (2 + i)^2(-1) / 1 + I 1 / 2 - 7 / 2 I (Correct)

1 / 2 - 7 / 2 i

Find the amount accumulated after investing a principal (P) for (t) years at an interest rate compounded (k) times per year. P = $25,300, r = 4.5% (0.045), t = 25, k = 12 A = P(1 + r / k)^kt, A = $? Answer: 25300(1 + 0.045 / 12)^12*25 A = $77,765.69 (Correct)

A = $77765.69

Compound interest questions:

A = P(1 + r / k)^kt

Continuous compounding questions:

A = Pe^rt

Write as a power function. The area, A, of an equilateral triangle varies directly as the square of the length, s, of its sides. A = ks^2 (Correct)

A = ks^2

Find the missing information. Arc length = 3 m Radius = 1 m Central angle = x (Give your answer in radians) Answer: S = r * θ s = 3 r = 1 θ = x 1 = 3 * θ x = 3 (Correct)

Central angle = 3

Angles questions:

Degrees: 360º 60 mins = 1 deg 60 sec = 1 min 42º 15' 12'' 42 + 15/60 + 12/3600 3169/75 = 42.2533º 57º 30' 57 + 30/60 57.5 Radians: C = 2πr 41.25º 41 + 25/100 25/100 = x/60 1/4 = x/60 x = 15 41º 15' π/6 = 1/6 = 30º π/4 = 1/4 = 45º π/3 = 1/3 = 60º π/2 = 1/2 = 90º π = 180º 2π = 360º

Find the first 5 terms a(0) = 3 a(n) = 2a(n-1) Answer: 2 * a a(0) = 3 a(1) = 6 a(2) = 12 a(3) = 24 a(4) = 48 (All correct)

a(0) = 3 a(1) = 6 a(2) = 12 a(3) = 24 a(4) = 48

Find the first 5 terms a(0) = 3 a(n) = a(n-1)+4 Answer: a + 4 a(0) = 3 a(1) = 7 a(2) = 11 a(3) = 15 a(4) = 19 (All correct)

a(0) = 3 a(1) = 7 a(2) = 11 a(3) = 15 a(4) = 19

Find the first 5 terms a(0) = 6 a(n) = a(n-1) + 0.5 Answer: n + 0.5 a(0) = 6 a(1) = 6.5 a(2) = 7 a(3) = 7.5 a(4) = 8 (All correct)

a(0) = 6 a(1) = 6.5 a(2) = 7 a(3) = 7.5 a(4) = 8

Find a general term definition for this sequence. a(0) = 5 a(n) = 2a(n-1) Answer: a(0) = 5 a(1) = (2 * 5) = (10) a(2) = (2^2 * 5) = (20) Equation: a(n) = 5*2^n (Correct)

a(n) = 5*2^n

The function y = 3 cos (x + 3) - 2 has: amplitude = ? period = ? horizontal shift = ? vertical shift = ?

amplitude = 3 period = 2π horizontal shift = -3 vertical shift = -2

The function: y = 7/3 sin (x - 5/2) - 1 amplitude = x period = xπ horizontal shift = x vertical shift = x Answer: amplitude = 7/3 period = 2π horizontal shift = 5/2 vertical shift = -1 (Correct)

amplitude = 7/3 period = 2π horizontal shift = 5/2 vertical shift = -1

The function y = 7/3 sin (x - 5/2) - 1 has: amplitude = ?/? period = ? horizontal shift = ? vertical shift = ?

amplitude = 7/3 period = 2π horizontal shift = ? vertical shift = ?

Find a formula for the exponential function if, f(0) = 12 and f(1) = 4 a. f(x) = 4(1/3)^x b. f(x) = 12^x c. f(x) = 12(1/3)^x (Correct) d. f(x) = 4^x

c. f(x) = 12(1/3)^x

Find a formula for the exponential function if, f(0) = 6 and f(-2) = 2/3 a. f(x) = 6(1/3)^x b. f(x) = 3(6)^-X c. f(x) = 6(3)^x (Correct) d. f(x) = -2(6)^X

c. f(x) = 6(3)^x

Write the expression as a single log. 1/3 log x Answer: log a^x = x log a a. log x^-3 b. log x^3 c. log x^1/3 (Correct) d. log x^-1

c. log x^1/3

Which of the following expressions is not equal to the others? a. cos 40º b. cos 60º cos (-20º) - sin 60º sin (-20º) c. sin 60º cos 20º + cos 60º sin 20º (Correct) d. cos 60º cos 20º + sin 60º sin 20º

c. sin 60º cos 20º + cos 60º sin 20º

Graph one period of: y = 0.5 cos 3x a. starts (0.05, 0.5), ends (18.85, 0.5) b. starts (0.05, -0.5), ends (1.05, -0.5) c. starts (0.05, 0.5), ends (2.09, 0.5) (Correct) d. starts (0, 0) ends (2.09, 0)

c. starts (0.05, 0.5), ends (2.09, 0.5)

Given the graph, find a formula for the function (the intersections are π/4 and 3π/4) a. y = 2cos (x/2) b. y = cos (x/2) c. y = 2cos (2x) (Correct) d. y = 2sin (2x)

c. y = 2cos (2x)

Find the exact answer cos (2 sin ^-1)(1/2)) = ? Answer: 1/2 (Correct)

cos (2 sin ^-1)(1/2)) = 1/2

Write as a single function of a single angle cos 20º cos 30º + sin 20º sin 30º = [ ? ] ( ?º ) Answer: cos (A - B) = cos A cos B + sin A sin B cos 10º (Correct)

cos 10º

Write as a single function of a single angle cos 20º cos 30º - sin 20º sin 30º = [ ? ] ( ?º ) Answer: cos (A + B) = cos A cos B - sin A sin B cos 50º (Correct)

cos 50º

Write as a single function of a single angle cos 94º cos 18º + sin 94º sin 18º = ? ?º Answer: 94 - 18 = cos 76º (Correct)

cos 76º

Simplify the following expression as much as possible cos x - cos^3 x Answer: cos x (cos^2 x) (Correct)

cos x (cos^2 x)

Assume θ is an acute angle. cot θ = 11/3 Answer: also don't know how to do this and it's cos θ = 11 / √130

cos θ = 11 / √130

Assume θ is an acute angle sin θ = 3 / 8 cos θ = √? / ? Answer: sin θ = o/h cos θ = a/h tan θ = o/a (a) opposite = 3 (b) adjacent = x (c) hypotenuse = 8 c^2 = b^2 + a^2 8^2 = b^2 + 3^2 64 = b^2 + 9 55 = b^2 b = √55 (a) opposite = 3 (b) adjacent = √55 (c) hypotenuse = 8 sin θ = 3 / 8, so, cos θ = √55 / 8 (Correct)

cos θ = √55 / 8

Assume θ is an acute angle. sin θ = 3/8 Answer: sin θ = o/h cos θ = a/h tan θ = o/a (a) opposite = 3 (b) adjacent = x (c) hypotenuse = 8 c^2 = b^2 + a^2 8^2 = b^2 + 3^2 64 = b^2 + 9 55 = b^2 b = √55 (a) opposite = 3 (b) adjacent = √55 (c) hypotenuse = 8 cos θ = √55/8 (correct)

cos θ = √55/8

Fill in the question mark cos π/3 (60º) = x / x

cos π/3 = 1 / 2

Find the exact answer cos^-1 0 = π / ? Answer: π / 2 (Correct) (I GOT A 97 ON THE TEST BTW IM SCREAMINGGG) (Also I changed the answer that I got wrong a few questions back so everything is correct)

cos^-1 0 = π / 2

Find the exact answer cos^-1 0 = π / ? Answer: π / 2 (Correct)

cos^-1 0 = π / 2

Find the exact answer cos^-1 1 / 2 = π / ? Answer: π / 3 (Correct)

cos^-1 1 / 2 = π / 3

Trig functions of acute angles questions:

cot x = cos x / sin x sec x = 1 / cos x csc x = 1 / sin x

Point P is on the terminal side of angle θ cot θ = ? Point P is located at (0, 5) Answer: cot θ = 0 (Correct)

cot θ = 0

Fill in the question mark csc π/3 (60º) = x/√x (Notice that we don't rationalize the denominator)

csc π/3 = 2/√3

Compute the exact value without using a calculator. f(x) = 6 * 3^x, find f(-2) a. f(-2) = 1 / 54 b. f(-2) = 1 / 324 c. f(-2) = 54 d. f(-2) = 2 / 3 (Correct)

d. f(-2) = 2 / 3

Use the factor theorem to determine which of these binomials is a factor of 4x^3 - 11x^2 - 104x - 105. a. x + 7 b. x + 5 c. x - 5 d. x + 3 (Correct)

d. x + 3

Given the function below, choose the interval on which it will be one-to-one and still have the same range. f(x) = x^2 - 4x + 4 a. x ≤ 0 b. x ≥ 1 c. x ≤ -4 d. x ≤ 2 (Correct) e. x ≥ -2

d. x ≤ 2

e^ln6

eln6 = 6

3 + 2e^-x = 6

x = -0.41

Solve the equation and round to the nearest hundredth. 7 - 3e^-x = 2

x = -0.51

2 (10^-x/3) = 20

x = -3

log x = -2

x = 0.01

log x = -3

x = 1 / 1000

Solve the equation. log x = 4

x = 10000

Find the missing information Arc length: 40 cm Radius: x Central Angle: 20º (Round to the nearest thousandth) Answer: S = r * θ s = 40 r = x θ = 20º 40 = x * 20º x = 2 (Correct)

x = 2

log^4 (x - 5) = -1

x = 21 / 4

Find the missing information Arc length: 3 m Radius: 1 m Central Angle: x (Give your answer in radians) Answer: S = r * θ s = 3 r = 1 θ = x 3 = 1 * x x = 3 (Correct)

x = 3

Solve the equation and round to the nearest hundredth. ln(4x - 12) + 8 = 6

x = 3.03

50e^0.035x = 200

x = 39.61

Solve the equation and round to the nearest hundredth. 3 ln (x - 3) + 4 = 5

x = 4.40

1.06x = 41

x = 63.73

Solve for x. 10^x = 3 Answer: log a/b = log a - log b x = ln 3 / ln 10 (Correct)

x = ln 3 / ln 10

Solve the equation and check for extraneous solutions. 3 / x + 2 + 6 / x^2 + 2x = 3 - x / x (If you get no solutions, just type none.) x = none

x = none

Find two functions defined implicitly by this equation: 3x^2 + y^2 = 25 y = -√25 - 3x^2 (Correct) y = +√25 - 3x^2 (Correct)

y = -√25 - 3x^2 y = +√25 - 3x^2

Find a formula for this function (0.8, 1.5), (1.25, 0.7), (1.7, 1.5) y = ? cos (2π / ? (x - ?)) + ? Answer: Period: (Distance between the two peaks, (1.7 - 0.8 = 0.9)) Vertical (or phase) shift: (Halfway in between the two y coordinate numbers, then average the two, (1.5 + 0.7 / 2 = 1.1)) Amplitude: (Distance from 1.1 to 1.5, (1.5 - 1.1 = 0.4)) Horizontal shift: (The starting peak I think?, (0.8)) This helped me a lot with plugging in: https://www.omnicalculator.com/math/phase-shift y = 0.4 (AMP) cos (2π / 0.9 (PER) (x - 0.8 (HORI)) + 1.1 (VERT) (Correct)

y = 0.4 cos (2π / 0.9 (x - 0.8)) + 1.1

Find a formula for this function (0.5, 0.9), (1.05, 0.1), (1.6, 0.9) y = ? cos (2π / ? (x - ?)) + ? Answer: Period: 1.6 - 0.5 = 1.1 Vertical shift: 0.9 + 0.1 / 2 = 0.5 Amplitude: 0.9 - 0.5 = 0.4 Horizontal shift: 0.5 y = 0.4 (AMP) cos (2π / 1.1 (PER) (x - 0.5 (HORI)) + 0.5 (VERT) (Correct)

y = 0.4 cos (2π / 1.1 (x - 0.5) + 0.5

Find a formula for this function (1.2, 3.4), (2.3, 0.7), (3.4, 3.4) y = ? cos (2π / ? (x - ?)) + ? Answer: Period: 3.4 - 1.2 = 2.2 Vertical shift: 3.4 + 0.7 / 2 = 2.05 Amplitude: 3.4 - 2.05 = 1.35 Horizontal shift: 1.2 y = 1.35 (AMP) cos (2π / 2.2 (PER) (x - 1.2 (HORI)) + 2.05 (VERT) (Correct)

y = 1.35 cos (2π / 2.2 (x - 1.2)) + 2.05

Find a formula for this function (2.3, 3.7), (3.1, 0.3), (3.9, 3.7) y = ? cos (2π / ? (x - ?)) + ? Answer: Period: 3.9 - 2.3 = 1.6 Vertical shift: 3.7 + 0.3 / 2 = 2 Amplitude: 3.7 - 2 = 1.7 Horizontal shift: 0.9 y = 1.7 (AMP) cos (2π / 1.6 (PER) (x - 2.3 (HORI)) + 2 (VERT) (Correct)

y = 1.7 cos (2π / 1.6 (x - 2.3)) + 2

Find the formula for this function (0.4, 5.6), (1.3, 2.2), (2.2, 5.6) y = ? cos (2π / ? - ?) + ? Answer: Period: 2.2 - 0.4 = 1.8 Vertical shift: 5.6 + 2.2 / 2 = 3.9 Amplitude: 5.6 - 3.9 = 1.7 Horizontal shift: 0.4 y = 1.7 (AMP) cos (2π / 1.8 (PER) (x - 0.4 (HORI)) + 3.9 (VERT) (Correct)

y = 1.7 cos (2π / 1.8 (x - 0.4) + 3.9

Find an equation for this graph starts at 0.5 and does a little doopity do

y = 1/2 sin (1/3 x)

Find a formula for this function (1, 5), (5, 1), (9, 5) y = ? cos (2π / ? (x - ?)) + ? Answer: Period: 9 - 1 = 8 Vertical shift: 5 + 1 / 2 = 3 Amplitude: 5 - 3 = 2 Horizontal shift: 1 y = 2 (AMP) cos (2π / 8 (PER) (x - 1 (HORI)) + 3 (VERT) (Correct)

y = 2 cos (2π / 8 (x - 1)) + 3

Find an equation for this graph peaks: (π/4, 1), (9π/4,1)

y = 2 cos (x - π/4) - 1

Find the formula for this function (0.7, 20.4), (3.8, 2.4), (6.9, 20.4) y = ? cos (2π / ? - ?) + ? Answer: Period: 6.9 - 0.7 = 6.2 Vertical shift: 20.4 + 2.4 / 2 = 11.4 Amplitude: 20.4 - 11.4 = 9 Horizontal shift: 0.7 y = 9 (AMP) cos (2π / 6.2 (PER) (x - 0.7 (HORI)) + 11.4 (VERT) (Correct)

y = 9 cos (2π / 6.2 (x - 0.7) + 11.4

Find a formula for this function. x = π / 6, x = 7 * π / 6, x = 13 * π / 6 y = csc (x - π / ? ) It's 6

y = csc (x - π / 6 )

Find the formula for this function x = π / 6, x = 7*π / 6, x = 13*π / 6 y = csc (x - π / ?) Answer: 6 (Correct)

y = csc (x - π / 6)

Find a formula for this function. Looks like blue zebra stripes y = tan (?x)

y = tan (2x)

Find a direct relationship between y and x. x = e^2t and y = e^t y = √x (Correct)

y = √x

Convert degrees to radians 90º = π / ? radians Answer: π / 2 radians (Correct)

π / 2 radians

Convert degrees to radians. 60º = π/? radians

π/3 radians

Special angles questions:

π/6 = 1/6 = 30º π/4 = 1/4 = 45º π/3 = 1/3 = 60º π/2 = 1/2 = 90º sin θ = x 0º = 0, 30º = 1/2, 45º = √2/2, 60º = √3/2, 90º = 1 cos θ = x 0º = 1, 30º = √3/2, 45º = √2/2, 60º = 1/2, 90º = 0 tan θ = x 0º = 0, 30º = √3/3, 45º = 1, 60º = √3, 90º = undef. csc θ = x 0º = undef., 30º = 2, 45º = √2, 60º = 2√3/3, 90º = 1 sec θ = x 0º = 1, 30º = 2√3/3, 45º = √2, 60º = 2, 90º = undef. cot θ = x 0º = undef., 30º = √3, 45º = 1, 60º = √3/3, 90º = 0

Given z1 and z2, find the distance between them. z1 = 3 + 7i and z2 = -5 - 2i √?

√?

Find the first 5 terms a(0) = 2 a(n) = 3a(n-1) Answer: a * 3 a(0) = 2 a(1) = 6 a(2) = 18 a(3) = 54 a(4) = 162 (All correct)

a(0) = 2 a(1) = 6 a(2) = 18 a(3) = 54 a(4) = 162

Find the amount of money accumulated after investing a principle (P) for years (t) at interest rate (r), compounded continuously. P = $3,200 (starting number) r = 8%, 0.08 (rate) t = 4 (time) Answer: A = Pe^rt 3200e^0.08*4 = $4406.81 (Correct)

$4406.81

Find the amount of money accumulated after investing a principle (P) for years (t) at interest rate (r), compounded continuously. P = $500 (starting number) r = 5%, 0.05 (rate) t = 20 (time) Answer: A = Pe^rt 500e^0.05*20 = $1359.14 (Correct)

$1359.14

Find the amount of money accumulated after investing a principle (P) for years (t) at interest rate (r), compounded continuously. P = $100 (starting number) r = 4%, 0.04 (rate) t = 15 (time) Answer: A = Pe^rt 100e^0.04*15 = $182.21 (Correct)

$182.21

Find the amount of money accumulated after investing a principle (P) for years (t) at interest rate (r), compounded continuously. P = $12,000 (starting number) r = 7.5%, 0.075 (rate) t = 7 (time) Answer: A = Pe^rt 12000e^0.075*7 = $20285.51 (Correct)

$20285.51

Find the amount of money accumulated after investing a principle (P) for years (t) at interest rate (r), compounded continuously. P = $300,000 (starting number) r = 7%, 0.07 (rate) t = 30 (time) Answer: A = Pe^rt 300000e^0.07*30 = $2449850.97 (Correct)

$2449850.97

Fill in the question mark tan π/6 (30º) = x/√x (Notice that we don't rationalize the denominator)

tan π/6 = 1/√3

Convert radians to degrees 3π / 5 = ?º Answer: 108º (Correct)

108º

Convert degrees to radians. 71.72º = ? radians (Give your answer as a decimal rounded to the nearest thousandth.)

1.252 radians

Find the sum of the first n terms using the formula: a (1 - r^n) / 1 - r 1 + 2 + 4 + 8 + ... (10 terms) Answer: a = 1 (starting number) r = 2 (rate) n = 10 (number of terms) 1 (1 - 2^10) / 1 - 2 = 1023 (Correct)

1023

A baker puts 2.5 grams of yeast into a batch of bread dough. After 2 days, there are 3.1 grams of yeast. Assuming exponential growth, how many days until there are 8 grams of yeast? Answer: Finding rate: P = 2.5 (starting number) r = x (rate) t = 2 (time) f(t) = 3.1 (total) 3.1 = 2.5e^r*2 r = 0.10755568 Days until 8 grams of yeast: P = 2.5 (starting number) r = (rate) t = x (time) f(t) = 8 (total) 8 = 2.5e^*x x = 10.8, 11 days

11 days

Find the sum of the first n terms using the formula: a (1 - r^n) / 1 - r 1, 3/2, 9/3, 27/8, 81/6 .... (10 terms) Answer: a = 1 (starting number) r = 3/2 (rate) n = 10 (number of terms) 1 (1 - (3/2)^10) / 1 - 3/2 = 113.33 (Correct)

113.33

Convert radians to degrees 2 = ?º Give your answer as a decimal rounded to the nearest thousandth.

114.592º

Convert to decimal form. (Don't round) 118º 44' 15''

118.7375º

Convert this decimal angle to degrees, minutes, and seconds. 118.32º = ?º ?' ?''

118º 19' 12''

An invasive species has an initial population of 20 organisms. In five years, its population has grown to 130. Assuming exponential growth, how many years will it take for its population to reach 5,000? Answer: Finding rate: P = 20 (starting number) r = x (rate) t = 5 (time) f(t) = 130 (total) 130 = 20e^r*5 r = 0.37 Finding years to reach 5,000: P = 20 (starting number) r = 0.37 (rate) t = x (time) f(t) = 5000 (total) 5000 = 20e^0.37*x x = 14.92, 15 years (correct)

15 years

If the half life of a radioactive substance is 5.3 years and there are 8.4 grams present initially, when will there be 1 gram remaining? Answer: (I don't know how to do this one but it's 16 years)

16 years

Find the exact answer tan^-1 (-1) = π / ? Answer: - π / 4 (Correct)

tan^-1 (-1) = - π / 4

The population of New York state can be modeled by: P(t) = 19.71 / 1 + 61.22e-0.03513t P is the population in millions and t is the number of years since 1800, What was the population in 2020? (Round to nearest thousandth) Answer: Okay so I think we have to plug in 220 for t because there were 220 years since 1800 P(220) = 19.71 / 1 + 61.22e-0.03513(220) So the population in 2020 would be 19.193 million. (Correct)

19.193 million

The population of New York state can be modeled by: P(t) = 19.71 / 1 + 61.22e-0.03513t P is the population in millions and t is the number of years since 1800, What is New York's maximum sustainable population? (Round to nearest thousandth) Answer: I think it's 19.71 because it's in the numerator and the video said that the 100 in the numerator was the maximum capacity. (Correct)

19.71 million

Find the sum of the first n terms using the formula: a (1 - r^n) / 1 - r 100 + 50 + 25 + .... (8 terms) Answer: a = 100 (starting number) r = 1/2 (rate) n = 8 (number of terms) 100 (1 - (1 / 2)^8) / 1 - 1/2 = 199.22 (Correct)

199.22

Determine the following without using a calculator sec π/3 (60º)

2

log x2 / y3

2 log x - 3 log y

If the population of a city is 475,000 and is increasing 3.75% each year, when will it reach 1 million? (Use the function f(t) = Pe^rt and round your answer to the nearest year.) Answer: Population in 2020: P = 475000 (starting number) r = 0.0375 (rate) t = x (time) f(t) = 1000000 (total) 1000000 = 475000e^0.0375*x x = 19.8, 20 years

20 years

Convert this decimal angle to degrees and minutes. 21.2º = ?º ?'

21º 12'

Simplify the following expression as much as possible 1 + tan^2 x / csc^2 x Answer: tan^2 x (Correct)

tan^2 x

If the population of Centerville in 1910 was 4,200 and 5,000 in 1920, assuming exponential growth, what would the population be in 2020? Answer: Finding rate: P = 4200 (starting number) r = x (rate) t = 10 (time) f(t) = 5000 (total) 5000 = 4200e^r*10 r = 0.01743533 Population in 2020: P = 4200 (starting number) r = 0.01743533 (rate) t = 110 (time) f(t) = x (total) x = 4200e^0.01743533*0.017 x = 28587.5 people

28587 people

Evaluate the expression without using a calculator. ln e^3

3

Perform the indicated operation and simplify. Leave in factored form. 3x^2 / 2x - 1 - 9 / 2x - 1 3(x^2 - 3) / 2x - 1

3(x^2 - 3) / 2x - 1

Convert radians to degrees. π/6 = ?º Answer: π/6 = 180/6 = 30º (Correct)

30º

Find the sum of the first n terms using the formula: a (1 - r^n) / 1 - r 85 + 85(0.9) + 85(.9)^2 + .... (5 terms) Answer: a = 85 (starting number) r = 0.9 (rate) n = 5 (number of terms) 85 (1 - 0.9^5) / 1 - 0.9 = 348.08 (Correct)

348.08

In 1900, Kansas City had a population of 163,000. In 2020, it had a population of 1,686,000. Assuming exponential growth, how many years after 1900 did it take for the population to double? Answer: Finding rate: P = 163,000 (starting number) r = x (rate) t = 120 (time) f(t) = 1,686,000 (total) 1686000 = 163000e^r*120 r = 0.019 Finding years it took for the population to double: (I didn't understand how to do this bu 36 years is the correct answer) P = 163,000 (starting number) r = 0.019 (rate) t = x (time) f(t) = 326000, (163,000*2) (total) 326000 = 163000e^0.02*x 36 years

36 years

If there are initially 4.8 grams of a radioactive substance initially and 0.4 grams remain after 13 days, what is the half life? Answer: Finding rate: P = 4.8 (starting number) r = x (rate) t = 13 (time) f(t) = 0.4 (total) 0.4 = 4.8e^r*13 r = -0.19114666 Finding half life: P = 4.8 (starting number) r = -0.19114666 (rate) t = x (time) f(t) = 2.4 (total) 2.4 = 4.8e^-0.19114666*x t = 3.8, 4 days

4 days

Convert this decimal angle to degrees and minutes. 49.7º = ?º ?'

49º 42'

7^log 7^3

7^log 7^3 = 3

Find the sum of the first n terms using the formula: a (1 - r^n) / 1 - r 1, 5/3, 25/27, 125/27, 625/81 .... (7 terms) Answer: a = 1 (starting number) r = 5/3 (rate) n = 7 (number of terms) 1 (1 - (5/3)^7) / 1 - 5/3 = 52.08 (Correct)

52.08

5^log⁵8

5log⁵8 = 8

Convert degrees to radians. 150º = ?π / ? radians

5π / 6 radians

The population of New York state can be modeled by: P(t) = 19.71 / 1 + 61.22e-0.03513t P is the population in millions and t is the number of years since 1800, What was the population in 1900? (Round to nearest thousandth) Answer: Okay so I think we have to plug in 100 for t because there were 100 years since 1800 P(100) = 19.71 / 1 + 61.22e-0.03513(100) So the population in 1900 would be 6.977. (Correct)

6.977 million

If there are initially 3.5 grams of a radioactive substance and 2 grams remain after 7 days, what is the half life? Answer: (I don't know how to do this one either but it's 9)

9 days

If the population of Alaska in 2000 was 622,000 and 733,000 in 2020, assuming exponential growth, what would the population be in 2050? Answer: Finding rate: P = 622000 (starting number) r = x (rate) t = 20 (time) f(t) = 733000 (total) 733000 = 622000e^r*10 r = 0.00821028 Population in 2050: P = 622000 (starting number) r = 0.00821028 (rate) t = 50 (time) f(t) = x (total) x = 622000e^0.00821028*50 x = 937722 people

937722 people

Convert this decimal angle to degrees, minutes, and seconds. 99.37º = ?º ?' ?''

99º 22' 12''

For the fifth degree polynomial graphed, how many non-real zeros does it have? ?

?

Find the amount accumulated after investing a principal (P) for (t) years and an interest rate compounded annually. P = $12,000, r = 7.5% (0.075), t = 7 A = P(1 + r / k)^kt, A = $? Answer: 12000(1.075)^7 A = $19908.59 (Correct)

A = $19908.59

Find the amount accumulated after investing a principal (P) for (t) years at an interest rate compounded (k) times per year. P = $1,500, r = 7% (0.07), t = 5, k = 4 A = P(1 + r / k)^kt, A = $? Answer: 1500(1 + 0.07 / 4)^4*5 A = $2122.17 (Correct)

A = $2122.17

Find the amount accumulated after investing a principal (P) for (t) years and an interest rate compounded annually. P = $3,200, r = 8% (0.08), t = 4, k = 1 A = P(1 + r / k)^kt, A = $? Answer: 3200(1.08)^4 A = $4353.56 (Correct)

A = $4353.56

Find the amount accumulated after investing a principal (P) for (t) years at an interest rate compounded (k) times per year. P = $3,350, r = 6.2% (0.062), t = 8, k = 365 A = P(1 + r / k)^kt, A = $? Answer: 3350(1 + 0.062 / 365)^365*8 A = $5500.94 (Correct)

A = $5500.94

Find the amount accumulated after investing a principal (P) for (t) years at an interest rate compounded (k) times per year. P = $3,500, r = 5% (0.05), t = 10, k = 4 A = P(1 + r / k)^kt, A = $? Answer: 3500(1 + 0.05 / 4)^4*10 A = $5752.67 (Correct)

A = $5752.67

Evaluate. log 0.2 29 Answer: log a/b = log a - log b log 29 / log 0.2 = -2.09 (Correct)

Answer: -2.09

Evaluate. log 12 259 Answer: log a/b = log a - log b log 259 / log 12 = 2.24 (Correct)

Answer: 2.24

Evaluate. log2 7 Answer: log a/b = log a - log b log 7 / log 2 = 2.81 (Correct)

Answer: 2.81

Find the missing information. Arc length = x Radius = 5 ft Central angle = 18º (Round to the nearest thousandth) Answer: S = r * θ s = x r = 5 θ = 18º 18º * π/180 = π/10 radians x = 5 * π/10 x = 1.5707, 1.571 rounded (Correct)

Arc length = 1.571

Find the missing information. Arc length = x Radius = 33 m Central angle = π rad (Round to the nearest thousandth) Answer: S = r * θ s = x r = 33 θ = π x = 33 * π x = 103.6725, 103.673 rounded (Correct)

Arc length = 103.673

Find the missing information. Arc length = x Radius = 2 in Central angle = 25 rad Answer: S = r * θ s = x r = 2 θ = 25 rad x = 2 * 25 x = 50 (Correct)

Arc length = 50

Find the missing information. Arc length = x Radius = 7 in Central angle = 60º (Round to the nearest thousandth) Answer: S = r * θ s = x r = 7 θ = 60º (Convert degrees to radians.) 60º * π/180 = π/3 radians x = 33 * π/3 x = 7.330 rounded (Correct)

Arc length = 7.330

Find the missing information. Arc length = x Radius = 1 cm Central angle = 70 rad Answer: S = r * θ s = x r = 7 θ = 70 rad x = 1 * 70 x = 70 (Correct)

Arc length = 70

Logarithmic questions:

Natural Logarithm: ln(2) = loge(2) = 0.6931 ln(3) = loge(3) = 1.0986 ln(4) = loge(4) = 1.3862 ln(5) = loge(5) = 1.609 ln(6) = loge(6) = 1.7917 ln(10) = loge(10) = 2.3025 (e ≈ 2.718)

Is the following an exponential function? y = 4^2

No

Is the following an exponential function? y = x^8

No

Model the following expression with a function. The revenue when each item sells for $3.75. R(x) = 3.75x (Correct) (when x represents the number of items sold and R(x) represent the total revenue)

R(x) = 3.75x

Divide f(x) and d(x). Your answer should be in the following format: f(x) / d(x) = Q(x) + R(x) / d(x) f(x) / d(x) = 2x^3 - 5x^2 + 10x - 16 / 2x - 1 R(x) = ?

R(x) = ?

Divide f(x) and d(x). Your answer should be in the following format: f(x) / d(x) = Q(x) + R(x) / d(x) f(x) = x^3 - 1, d(x) = x + 1 R(x) = ?

R(x) = ?

Find the missing information. Arc length = 40 cm Radius = x Central angle = 20º (Round to the nearest thousandth) Answer: S = r * θ s = 40 r = x θ = 20º (Convert degrees to radians.) 20º * π/180 = π/9 radians 40 = x * π/9 or 40 / π/9 x = 114.5915, 114.592 rounded (Correct)

Radius = 114.592

Find the missing information. Arc length = 2.5 cm Radius = x Central angle = π/3 (Round to the nearest thousandth) Answer: S = r * θ s = 2.5 r = x θ = π/3 2.5 = x * π/3 or 2.5 / π/3 x = 2.3873, 2.387 rounded (Correct)

Radius = 2.387

Arc length questions:

S = r * Θ

Sum and difference formulas - sine and cosine questions

SUM: cos (A + B) = cos A cos B - sin A sin B DIFF: cos (A - B) = cos A cos B + sin A sin B SUM: sin (A + B) = sin a cos B + cos A sin B DIFF: sin (A - B) = sin A cos B - cos A sin B

Sum and difference formulas - tangent questions

SUM: tan(A + B) = tan A + tan B / 1 - tan A tan B DIFF: cos(A + B)= cos A cos B + sin A + sin B

For the given logistic function, f(x) = 5 / 1 + 4e^-2x The y-intercept = (0, ?) The horizontal asymptote is y = ? Answer: Okay so in the video, he plugged in 0 for x so I'm gonna do that and see what happens. f(0) = 5 / 1 + 4e^-2(0) 2(0) = 0 e0 = 1 4(1) = 4 1 + 4 = 5 5 / 5 = 1 Y-intercept is 1, so (0, 1). And I think the horizontal asymptote is 5 because of the numerator thing from the video. The y-intercept = (0, 1) (Correct) The horizontal asymptote is y = 5 (for x ≥ 0) (Correct)

The y-intercept = (0, 1) The horizontal asymptote is y = 5 (for x ≥ 0)

For the given logistic function, f(x) = 16 / 1 + 7e-3x The y-intercept = (0, ?) For x ≥ 0, the horizontal asymptote is y = ? Answer: Same thing again, I'm gonna plug in 0 for x. f(x) = 16 / 1 + 7e-3x 16/8 = 2 The y-intercept = (0, 2) (Correct) For x ≥ 0, the horizontal asymptote is y = 16 (Correct)

The y-intercept = (0, 2) For x ≥ 0, the horizontal asymptote is y = 16

For the given logistic function, f(x) = 6 / 1 + 2e^-2x The y-intercept = (0, ?) The horizontal asymptote is y = ? Answer: You know the drill, 0 for x. f(x) = 6 / 1 + 2e^0 6 / 3 = 2 The y-intercept = (0, 2) (Correct) For x ≥ 0, the horizontal asymptote is y = 6 (Correct)

The y-intercept = (0, 2) The horizontal asymptote is y = 6 (for x ≥ 0)

For the given logistic function, f(x) = 18 / 1 + 5(0.2)x The y-intercept = (0, ?) For x ≥ 0, the horizontal asymptote is y = ? Answer: You know the drill, 0 for x. f(x) = 18 / 1 + 5(0.2)x 18/6 = 3 The y-intercept = (0, 3) (Correct) For x ≥ 0, the horizontal asymptote is y = 18 (Correct)

The y-intercept = (0, 3) For x ≥ 0, the horizontal asymptote is y = 18

For the given logistic function, f(x) = 9 / 1 + 2e^x The y-intercept = (0, ?) The horizontal asymptote is y = ? Answer: You know the drill, 0 for x. f(x) = 9 / 1 + 2e^0 9/3 = 3 The y-intercept = (0, 3) (Correct) For x ≥ 0, the horizontal asymptote is y = 9 (Correct)

The y-intercept = (0, 3) The horizontal asymptote is y = 9

For the given logistic function, f(x) = 12 / 1 + 2(0.9)^x The y-intercept = (0, ?) For x ≥ 0, the horizontal asymptote is y = ? Answer: You know the drill, 0 for x. f(x) = 12 / 1 + 2(0.9)^x 12/3 = 4 The y-intercept = (0, 4) (Correct) For x ≥ 0, the horizontal asymptote is y = 12 (Correct)

The y-intercept = (0, 4) The horizontal asymptote is y = 12

For the given logistic function, f(x) = 16 / 1 + 3e^-2x The y-intercept = (0, ?) The horizontal asymptote is y = ? Answer: You know the drill, 0 for x. f(x) = 16 / 1 + 3e^0 16/4 = 4 The y-intercept = (0, 4) (Correct) The horizontal asymptote is y = 16 (Correct)

The y-intercept = (0, 4) The horizontal asymptote is y = 16

For the given logistic function, f(x) = 16 / 1 + 3(0.17)^x The y-intercept = (0, ?) The horizontal asymptote is y = ? Answer: You know the drill, 0 for x. f(x) = 16 / 1 + 3(0.17)^0 16/4 = 4 The y-intercept = (0, 4) (Correct) The horizontal asymptote is y = 16 (Correct)

The y-intercept = (0, 4) (Correct) The horizontal asymptote is y = 16 (Correct)

For the given logistic function, f(x) = 42 / 1 + 6(8.4)-x The y-intercept = (0, ?) For x ≥ 0, the horizontal asymptote is y = ? Answer: Okay same thing, I'm gonna plug in 0 for x. f(x) = 42 / 1 + 6(8.4)-x 42/7 = 6 The y-intercept = (0, 6) (Correct) For x ≥ 0, the horizontal asymptote is y = 42 (Correct)

The y-intercept = (0, 6) For x ≥ 0, the horizontal asymptote is y = 42

For the given logistic function, f(x) = 21 / 1 + 2(0.94)x The y-intercept = (0, ?) For x ≥ 0, the horizontal asymptote is y = ? Answer: Okay so kinda the same thing in the last problem, except there is no e, so I'm gonna plug in 0 for x. f(0) = 21 / 1 + 2(0.94)0 0.940 = 0 1 +2 = 3 21 / 3 = 7 The y-intercept = (0, 7) (Correct) For x ≥ 0, the horizontal asymptote is y = 21 (Correct)

The y-intercept = (0, 7) For x ≥ 0, the horizontal asymptote is y = 21

For the given logistic function, f(x) = 36 / 1 + 3(5.7)^-x The y-intercept = (0, ?) For x ≥ 0, the horizontal asymptote is y = ? Answer: You know the drill, 0 for x. f(x) = 36 / 1 + 3(5.7)^0 36/4 = 9 The y-intercept = (0, 9) (Correct) For x ≥ 0, the horizontal asymptote is y = 36 (Correct)

The y-intercept = (0, 9) For x ≥ 0, the horizontal asymptote is y = 36

Mid-term exam (50 questions)

Units 1, 2 and 3

Find the horizontal and vertical asymptotes for the function. f(x) = (x^2 + 2) / (x^2 - 1) Vertical asymptotes: x = -1, x = 1 (Correct) Horizontal asymptote: y = 1 (Correct)

Vertical asymptotes: x = -1, x = 1 Horizontal asymptote: y = 1

Is the following an exponential function? y = 3^x

Yes

Sum of a geometric sequence questions:

a (1 - r^n) / 1 - r

Sketch the graph of: y = 5 sin 2x the one that starts at zero and has two arches

a is correct

Find the first 5 terms a(0) = 1 a(n) = -2a(n-1) Answer: (I didn't know how to do this one so I used a calculator but these are right) a(0) = 1 a(1) = -2 a(2) = 4 a(3) = -8 a(4) = 16 (All correct)

a(0) = 1 a(1) = -2 a(2) = 4 a(3) = -8 a(4) = 16

Find the first 5 terms a(0) = 100 a(n) = a(n-1)-7 Answer: a - 7 a(0) = 100 a(1) = 93 a(2) = 86 a(3) = 79 a(4) = 72 (All correct)

a(0) = 100 a(1) = 93 a(2) = 86 a(3) = 79 a(4) = 72

Find the first 5 terms a(0) = 2 a(n) = a(n-1) + 3 Answer: n + 3 (the starting value was 2 so just add 3 to the it and then just keep adding 3 from there) a(0) = 2 a(1) = 5 a(2) = 8 a(3) = 11 a(4) = 14 (All correct)

a(0) = 2 a(1) = 5 a(2) = 8 a(3) = 11 a(4) = 14

Describe a sequence of transformations that will turn y = x^2 into graph of 3y = (x + 7)^2. a. compress vertically by a factor of 1/3, slide 7 left (Correct) b. compress horizontally by a factor of 1/3, slide 7 left c. compress horizontally by a factor of 1/3, slide 7 right d. compress vertically by a factor of 1/3, slide 7 right

a. compress vertically by a factor of 1/3, slide 7 left

Find a formula for the exponential function if f(0) = 5 f(3) = 40 a. f(x) = 5(2)^x (Correct) b. f(x) = 3(1/5)^x c. f(x) = 30(5)^x d. f(x) = 5(1/3)^x

a. f(x) = 5(2)^x

What type of relative extrema does this function have? f(x) = | 2x - 6 | + 4 a. minimum (Correct) b. maximum

a. minimum

Complete the factoring if needed, then solve the polynomial inequality. (2x - 7)(x^2 - 4x + 4) > 0 a. x > 7 / 2 (Correct) b. x < 2 or x > 7 /2 c. 2 < x < 7 / 2 d. x < 2 or 2 < x < 7 / 2

a. x > 7 / 2

What is this function? Looks like red zebra stripes a. y = cot (3x) (Correct) b. y = cot (2x) c. y = cot (x/3) d. y = cot (6x)

a. y = cot (3x)

Find the formula for this function intersects at (0,0), and (2π, 0) a. y = tan x/2 (Correct) b. y = cot x c. y = tan x d. y = sec x

a. y = tan x/2

Find the formula for this function. ok this graph looks starts at 0,0 like a baseball if that helps, I guess on this one a. y = tan x/2 (Correct) b. y = cot x c. y = tan x d. y = sec x

a. y = tan x/2

Use a sum or difference identity to find the exact value tan 5π / 12 a. √3 + 3 / 3 - √3 (Correct) b. √6 - √2 / 4 c. √3 - 3 / 3 + √3 d. √2 + √6 / 4

a. √3 + 3 / 3 - √3

Graph one period of: y = -8 cos 5x idk how to describe the graphs but its b

b is correct

Which sequence below would best model the following description? A newly planted tree releases 250 pounds of oxygen over the period of its first year. The amount of oxygen released by the tree grows at a rate of 12% per year, but the tree cannot generate more than 6000 pounds of oxygen per year. a) f(n) = 1.12f(n - 1)(1 + f(n - 1) / 6000) b) f(n) = 1.12f(n - 1)(1 - f(n-1)/6000) (correct) c) f(n) = 1.12f (n - 1)(6000 / f(n-1)) d) f(n) = 1.12f (n - 1)(f(n - 1) - 6000)

b) f(n) = 1.12f(n - 1)(1 - f(n-1)/6000)

Find the intercepts and asymptotes. f(x) = (2x + 1)(x - 2) / (x - 1)( x + 1) (Intercepts, Horizontal Asymptotes, Vertical Asymptotes) a. (0, 2)(1/2, 0)(2, 0), x = 2, y = -1, y = 1 b. (0, 2)(-1/2, 0)(2, 0), y = 2, x = -1, x = 1 (Correct) c. (0, 2)(0, -1/2)(2, 0), y = -1, y = 1, x = 2 d. (0, 2)(0, 1/2)(2, 0), x = -1, x = 1, y = 2

b. (0, 2)(-1/2, 0)(2, 0), y = 2, x = -1, x = 1

Solve the following rational inequality. x^2 - 1 / x^2 + 1 ≤ 0 a. x ≤ -1 or x ≥ 1 b. -1 ≤ x ≤ 1 (Correct) c. x ≤ 1 d. x ≥ 1

b. -1 ≤ x ≤ 1

Use a sum or difference identity to find the exact value cos 7π / 12 a. √6 - √2 / 4 b. -√6 + √2 / 4 (Correct) c. -√2 - √6 / 4 d. √6 + √2 / 4

b. -√6 + √2 / 4

Use synthetic division to find all the real zeros of the polynomial. f(x) = x^3 - 6x^2 + 7x + 4 a. 1, 4 b. 1 + √2, 1 - √2, 4 (Correct) c. -1, 1, 2 d. -1, 4

b. 1 + √2, 1 - √2, 4

Write as a sum or difference of logs or multiples of logs. log 4√x/y Answer: log a/b = log a - log b a. 1/4 log y - 1/4 log x b. 1/4 log x - 1/4 log y (Correct) c. 4 log x - 4 log y d. 4 log y - 4 log x

b. 1/4 log x - 1/4 log y

Describe how the graph of y = x^3 can be translated to the graph of y = (x - 2)^3. a. 2 up b. 2 right (Correct) c. 2 left d. 2 down

b. 2 right

Classify as = 1, < 1, or > 1. For n > 1. | (1 + i)^n | a. < 1 b. > 1 (Correct) c. = 1

b. > 1

Find intervals where the following function is increasing, decreasing, or constant. f(x) = (x + 2)^2 - 3 a. increasing from -∞ to -2 and decreasing from -2 to +∞ b. decreasing from -∞ to -2 and increasing from -2 to +∞ (Correct) c. decreasing from -∞ to 2 and increasing from 2 to +∞

b. decreasing from -∞ to -2 and increasing from -2 to +∞

Compute the exact value without using a calculator. f(x) = 3 * 5^x, find f(0) a. f(0) = 15 b. f(0) = 3 (Correct) c. f(0) = 1 d. f(0) = 0

b. f(0) = 3

Graph one period of: y = -8 cos 5x a. starts (0.05, 8), ends (1.26, 8) b. starts (0.05, -8), ends (1.26, -8) (Correct) c. starts (0.02, -8), ends (0.63, -8) d. starts (0.2, -8), ends (5, -8)

b. starts (0.05, -8), ends (1.26, -8)

Which of the ten basic functions in our toolkit have all real numbers for their range? a. √x, x, |x| b. x, x^3, ln(x) (Correct) c. sin(x), cos(x), x^2 d. x^2, ln(x), |x|

b. x, x^3, ln(x)

What is this function? It looks like a frownie face a. y = csc x/2 b. y = sec x/2 (correct) c. y = sec x/4 d. y = csc x/4

b. y = sec x/2

Find the domain. f(x) = √x^2 - 9 a. x ≥ 3 b. |x| ≥ 3 (Correct) c. |x| > 3 d. x > 3

b. |x| ≥ 3

Find the amplitude, period, horizontal shift, and vertical shift y = 5 cos (3x - π/6) + 0.5 a. π/2 b. π/18 (Correct) c. -π/2 d. π/6 (horizontal shifts only cuz they'll all the same)

b. π/18

Find the amplitude, period, horizontal shift, and vertical shift y = -2 sin (x - π/4) + 1 whichever one says amplitude = 2 period = 2π horizontal shift = π/4 vertical shift = 1

c is correct

Graph one period of: y = 0.5 cos 3x its the one only one that starts at y = 0.5

c is correct

Find all the zeros of the following function: f(x) = x^3 + 4x - 5 a. 1, -1 ± 19i / 2 b. ±1, ±5 c. 1, -1 ± √19i / 2 (Correct) d. -10, 0, 1

c. 1, -1 ± √19i / 2

Find the amplitude, period, horizontal shift, and vertical shift y = -2 sin (x - π/4) + 1 a. -2, 2π, π/4, 1 b. 2, π, π/4, 1 c. 2, 2π, π/4, 1 (Correct) d. -2, 2π, -π/4, 1

c. 2, 2π, π/4, 1

Express the following using interval notation. -2 ≤ x ≤ 5 a. (-2, 5) b. [2, 5] c. [-2, 5] (Correct) d. [-2, 5)

c. [-2, 5]

Which is a graph of the following? f(x) = [ x^2 if x ≤ 2 ] [ x + 2 if x > 2 ] a. open circle, going up b. closed circle over an open circle c. closed circle, going up (Correct) d. open circle over a closed circle

c. closed circle, going up

Compute the exact value without using a calculator. f(x) = 48 * (1/2)^x, find f(4) a. f(4) = 768 b. f(4) = 16 c. f(4) = 3 (Correct) d. f(4) = 96

c. f(4) = 3

Call the first term f(0) and give the general term definition for this sequence: 5, 10, 20, 40, 80,... a. f(n) = 2n + 5 b. f(n) = n + 5 c. f(n) = 5 * 2^n (Correct) d. f(n) = 2n

c. f(n) = 5 * 2^n

A certain forest can support a population of 800 deer. There are currently 200 deer in the forest and their population is growing at a rate of 2% per year. Complete the sequence below that models this population growth. f(0) = ? f(n) = f(n -1) + ? f (n-1) (1 - f (n - 1) / ? ) Answer: Okay I think I have to make an equation oh wait no i think i have to plug in the numbers So the max is 800 deer There are currently 200 deer and the rate growing by 2% per year f(0) = 200 (Correct) f(n) = f(n -1) + 0.02 f (n-1) (1 - f (n - 1) / 800 ) (Correct) Ok so I don't have a clue how to do this next part so I'm gonna go in and do 200*0.02=x then adding x to the previous number and subtracting one. (Why is this how my brain works) f(1) = 203 f(2) = 206 f(3) = 209 (All correct)

f(0) = 200 f(n) = f(n -1) + 0.02 f (n-1) (1 - f (n - 1) / 800) next part: f(1) = 203 f(2) = 206 f(3) = 209

A river has a population of 700 salmon. Due to overfishing, this population is currently declining at a rate of 5% per year. Use this sequence to find the population over the next five years. Round to the nearest whole number. Answer: Okay I think I made an equation that fits it f(n) = -0.05f(n - 1)(1 - f(n-1)/700) No I don't think that 700 is in the right stop because that's where the max is supposed to be and there is no max so I'm gonna put 0 in because that's the least there can be. Ok never mind it says undefined. Im gonna do it my way then, My tourwn *gru accent* 700 * 0.05 = 35, 700-35 = 665 665 * 0.05 = 33.25, 665-33.25 = 631.75 632 * 0.05 = 31.6, 632 - 31.6 = 600 600 * 0.05 = 30, 600 - 30 = 570 570 * 0.05 = 28.5, 570 - 28.5 = 542 (I'm so sorry this is so confusing but this is the only way I can do it, don't learn from me but if you're stuck, use these!) f(1) = 665 f(2) = 632 f(3) = 600 f(4) = 570 f(5) = 542 (All correct)

f(1) = 665 f(2) = 632 f(3) = 600 f(4) = 570 f(5) = 542

In a Petri dish, there were initially 4 bacteria. Five hours later, there were 972 of them. Find the exponential function that satisfies the given conditions. f(t) = _e^_*t Answer: P = 4 (starting number) r = x (rate) t = 5 (time) f(t) = 972 (total) 972 = 4e^r*5 x = 1.099

f(t) = 4e^1.099*5

Find the exponential function that satisfies the given conditions. Initial value = 52 Increasing at a rate of 2.3% per day f(t) = _e^_*t Answer: P = 52 (starting number) r = 2.3, 0.023 (rate) f(t) = 52e^0.023*t

f(t) = 52e^0.023*t

Exponential modeling questions:

f(t) = Pe^rt

Write an equation for this linear function. f(-4) = 0 and f(0) = 2 f(x) = 1/2x + 2 (Correct)

f(x) = 1/2x + 2

Write an equation for the parabola that has the given vertex and passes through the given point. Vertex: (-1, -3) Point: (1, 5) f(x) = 2(x + 1)^2 - 3 (Correct)

f(x) = 2(x + 1)^2 - 3

Find the inverse function: f(x) = 2 (3x - 7) f^-1(x) = (x/2) + 7 / 3 (Correct)

f^-1(x) = (x/2) + 7 / 3

Find f(x) and g(x) so that the function given can be described as y = f(g(x)). y = e^sin x g(x) = sin x (Correct) f(x) = e^x * (Correct)

g(x) = sin x f(x) = e^x

ln 1 / √e 7

ln 1 / √e 7 = -7 / 2

ln 1

ln 1 = 0

Write as a sum, difference or multiple of logarithms. ln 2/y = ln ? - ln ? Answer: log a/b = log a - log b ln 2/y = ln 2 - ln y (Correct)

ln 2/y = ln 2 - ln y

ln(8x)

ln 8 + ln x

ln^e-4

ln e-4 = -4

Fill in the question mark. log 3 x = ln ? / ln ?

ln x / ln 3

ln(1/e)

ln(1/e) = -1

Write as a sum, difference or multiple of logarithms. log (9y) = log ? + log ? Answer: log a/b = log a - log b log (9y) = log 9 + log y (Correct)

log (9y) = log 9 + log y

2 log x + 3 log y

log (x^2 / y^3)

Properties of Log Functions questions:

log a^x = x log a log a/b = log a - log b

4 log x - 5 log z

log x4 / y5

Write as a sum, difference or multiple of logarithms. log x^3 y^2 = log ? + ? log ? Answer: log a/b = log a - log b log x^3 y^2 = 3 log x + 2 log y (Correct)

log x^3 y^2 = 3 log x + 2 log y

Fill in the question mark. log2 (a + b) = ln ? / ln ? Answer: log a/b = log a - log b log2 (a + b) = ln a + b / ln 2 (Correct)

log2 (a + b) = ln a + b / ln 2

log^6 1

log^6 1 = 0

Simplify the following expression as much as possible sin x (tan x + cot x) Answer: sec x (Correct)

sec x

Assume θ is an acute angle. tan θ = 9/5 Answer: sin θ = o/h cos θ = a/h tan θ = o/a (a) opposite = 9 (b) adjacent = 5 (c) hypotenuse = x c^2 = b^2 + a^2 c^2 = 5^2 + 9^2 c^2 = 25 + 81 c^2 = 106 c = √106 (a) opposite = 9 (b) adjacent = 5 (c) hypotenuse = √106 cos θ = 5 /√106, so, sec θ = √106 / 5 (correct)

sec θ = √106 / 5

Fill in the question mark sec π/4 (45º) = √x

sec π/4 = √2

Write as a single function of a single angle sin π/3 cos π/7 - cos π/3 sin π/7 = ? Answer: sin (A - B) = sin A cos B - cos A sin B

sin 4π / 21

Write as a single function of a single angle sin 20º sin 30º + cos 20º cos 30º + = [ ? ] ( ?º ) Answer: sin (A - B) = sin A sin B + cos A cos B sin 50º (Correct)

sin 50º

Fill in the question mark sin 75º = √2 + √? / ?

sin 75º = √2 + √6 / 4

Find the function tan x * cos x = ? Answer: sin x (correct)

sin x

Simplify the following expression as much as possible sec^2 x csc x / sec^2 x + csc^2 x Answer: sin x (Correct)

sin x

Sums and differences II questions

sin θ = cos (π/2 - θ) cos θ = sin (π/2 - θ) tan θ = cot (π/2 - θ) cot θ = tan (π/2 - θ) sec θ = csc (π/2 - θ) csc θ = sec (π/2 - θ)

Assume θ is an acute angle sec θ = 17 / 5 sin θ = ?√? / ? Answer: sin θ = 2√66 / 17 (Correct)

sin θ = 2√66 / 17

Assume θ is an acute angle. sec θ = 17/5 Answer: idk how to do this but it's sin θ = 2√66 / 17 (correct)

sin θ = 2√66 / 17

The point (3, 4) is on the terminal side of angle θ, find sin θ = ? / ? Answer: sin θ = 4 / 5 (Correct)

sin θ = 4 / 5

Assume θ is an acute angle. sec θ = 23/9

sin θ = 8√7 / 23

Fill in the question mark sin π/4 = ? /√? (Notice that we didn't rationalize the denominator) Answer: sin π/4 = 1 /√2 (Correct)

sin π/4 = 1 /√2

Fill in the question mark sin π/4 (45º) = x/√x (Notice that we don't rationalize the denominator)

sin π/4 = 1/√2

Find the exact answer sin^-1 √3 / 2 = π / ? Answer: π / 3 (Correct)

sin^-1 √3 / 2 = π / 3

Write as a single function of a single angle tan -34º - tan 110º / 1 + tan -34º * tan 110º = ? Answer: -34 - 110 = tan (-144º) (Correct)

tan (-144º)

Write as a single function of a single angle tan 55º - tan 17º / 1 + tan 55º * tan 17º = Answer: 55 - 17 = tan (38º) (Correct)

tan (38º)

Write as a single function of a single angle tan 145º - tan 85º / 1 + tan 145º * tan 85º = ? Answer: 145 - 85 = tan (60º) (Correct)

tan (60º)

Write as a single function of a single angle tan 19º + tan 47º / 1 - tan 19º * tan 47º = ? ?º Answer: 19 + 47 = tan 66º (Correct)

tan 66º

Write as a single function of a single angle tan 55º + tan 17º / 1 - tan 55º * tan 17º = ? Answer: 55 + 17 = tan (72º) (Correct)

tan 72º

Simplify the following as much as possible. tan x / csc^2 x + tan x / sec^2 x Answer: tan x (Correct)

tan x

Simplify the following expression as much as possible 1 + tan x / 1 + cot x Answer: tan x (Correct)

tan x

Fill in the question mark tan π / 6 = ? /√? (Notice that we didn't rationalize the denominator) Answer: tan π / 6 = 1 /√3 (Correct)

tan π / 6 = 1 /√3