Honors Precalc (Caron) Chapter 12 and 5 Notes

What does logb(bˣ) equal for all real numbers?

logb(bˣ) = x

Simplify the product of (log₂10)(log₁₀2)

log₁₀10/log₁₀2 * log₁₀2/log₁₀10 cancels out so = 1

Express log₂5 in terms of base 10 logs

log₁₀5/log₁₀2

Solve the following: log₁₀50 + log₁₀2

log₁₀50 + log₁₀2 log₁₀(50 * 2) log₁₀(100) = 2

Solve the following: log₂⁵√16

log₂⁵√16 log₂(16¹/⁵) 1/5 log₂16 1/5 * 4 = 4/5

Compare two logs using the definition: Which is larger, log₃10 or log₇40?

log₃10 is greater than 2 log₇40 is less than 2 therefore, log₃10 is greater

What is y in logarithmic functions?

the exponent raised to the yield x

Compare the graphs 1.5ˣ, 2ˣ, and 4ˣ

4ˣ is the steepest and 1.5ˣ is the widest

How could you find the real root in the equation x⁵ - 3x² + 4x + 2 = 0?

1) graph; line crosses x axis between 0 and -1 2) table; (-1, -6) and (0, 2) so switches to positive and negative between there. Use tenths to narrow it down (-.1 . . . -.9) and further; tableset on graphing calculator helps

State the multiplicity of the following roots: (x - 4)² (x - 5)³ = 0

4, multiplicity of 2 5, multiplicity of 3

State the domain, range, intercepts, asymptotes, and whether it is growth or decay for the graph y = 2ˣ

domain: (-∞,∞) range: (0,∞) y-int: (0,1) x-int: none asymptote: y = 0 b: growth

Place the equation x² -4x + 5 in aₙ(x - r₁) (x - r₂) form

find roots using quadratic equation; 4 ±√-4/2 simplifies to 2±i (x - (2 + i))(x - (2 - i)) (x - 2 - i)(x - 2 + i)

How can you graph logarithmic functions?

flip the ordered pairs of an exponential function

How is this inverse written?

f⁻¹(x) = log(b)x

When does exponential growth occur and when does exponential decay occur?

growth: b > 1 decay:0 < b < 1 (fractions less than 1)

What is compound interest?

interest compounded annually

You invest $100- into a Certificate of Deposit (CD) earning 3.8% interest compounded quarterly. How much do you have in teh CD after 3 years?

A = 1000(1 + (.038/4))⁴*³ = $1120.15

Quotient Property

the log of a quotient is the log of the numerator minus the log of the denominator logb(P) - logb(Q) = logb(P/Q)

What can r be referred to as?

the zero of the function f

When does the graph decrease?

when 0 < b < 1

Determine the rational roots or show that they exist for 1/4x⁴ - 3/4x³ + 17/4x² + 4x + 5 = 0

**must be integers for rational roots theorem so multiply by 4 x⁴ - 3x³ + 17x² + 16x + 20 = 0 p/q = ±1, 2, 4, 5, 10, 20 lower bound -1 (3rd row 1, -4, 21, -5, 25) upper bound 4 (3rd row 1, 1, 21, 100, 420) **no rational roots

What are other names for zeroes?

- solutions - roots - x-intercepts

Solve the following: . 3x(10ˣ) + 10ˣ = 0 . 4x²(2ˣ) - 9(2ˣ) = 0 . 3(3ˣ) -5x(3ˣ) + 2x²(3ˣ) . x + 4)10ˣ/(x - 3) = 2x(10ˣ)

. 10ˣ(3x + 1); x = -1/3 . (2ˣ)(2x + 3)(2x - 3); 3/2, -3/2 . 3ˣ(2x - 3)(x - 1) = 1, 3/2 . (x + 4)10ˣ/(x - 3) = 2x(10ˣ) x = 4, -1/2

Convert the following exponential functions to logarithmic: . 8 = 2³ . (1/9) = 3⁻² . 1 = e⁰ . a = bᶜ

. 3 = log(2)8 . -2 = log(3)1/9 . 0 = log(e)1 . c = log(a)b

Solve the following: . log₉²⁷ . log₄¹/³² . log₅⁵√⁵

. 3/2 . -5/2 . 3/2

Solve 2³ʸ⁺¹ = √2

. 3y + 1= 1/2 3y = -1/2 y = -1/6

Use the properties of exponents to simplify the following: . (3^√²)^√² . (3^√²)²

. 3² = 9 . switch exponents; (3²)^√² = 9^√²

Describe the domain, range, y-intercept, x-intercept, and asymptote of the following: . y = eˣ⁻¹ . y = -eˣ⁻¹ . y = -eˣ⁻¹ + 1

. D: (-∞,∞) R: (0,∞), y-int: (1/e), x-int: none, asymptote: y = 0 graph is y = eˣ shifted one unit to the right . D: (-∞,∞), R: (-∞,0). y-int: (-1/e), x-int: none, asymptote: y = 0 graph is reflected over the y axis . D: (-∞,∞), R: (-∞, 1), y-int: (-1/e + 1), asymptote y = 0 x-int: 0 = -eˣ - 1; -1 = -eˣ⁻¹, so e must = 1 so that the exponent is 0

Calculate the y-intercepts for the following: . aˣ . -aˣ

. a⁰ = 1 . -a⁰ = -1

Show that the equation x³ - 2x² + x - 1 = 0 has at least one rational root

. factoring doesn't work . possible roots ±1 don't work (show using synthetic division) has 2 imaginary roots so at least one real the real can be rational or irrational, so it has at least one irrational root and potentially 3

What are properties of logs with the same bases and bases of 1?

. log(b)b = 1 . log(b)1 = 0

State the periodic interest rates (n) for the following terms: . annually . semi annually . quarterly . monthly . daily

. n = 1 . n = 2 . n = 4 . n = 12 . n = 365

State the asymptote, intercept, and translated form of the following: . logarithmic function . exponential function

. x = bʸ ⁻ ᵏ + h vertical asymptote x-int . y = aˣ⁻ʰ + k horizontal asymptote y-int

Describe how y, b, and x shift when converting from exponential to logarithmic

. x goes from exponent to isolated on opposite side of equation . y goes from isolated to the number . b stays the base

How can you find the x-intercept for the exponential function y = 2⁻ˣ - 2?

0 = 2⁻ˣ - 2 2 ⁻ˣ = 2 for y to be 0, 2 -x must be 2(-1)

Express the following equation as a single logarithm with a coefficient of 1: ln (x² - 9) + 2 ln 1/(x+3) + 4 ln x

= ln (x+3)(x-3)x⁴/(x+3)(x+3) = ln (x⁵ - 3x⁴)/(x + 3)

Solve for x: 4⁵ˣ ⁺ ² = 70. Evaluate the final answer with a calculator

5x + 2 log 4 = log 70 5x + 2 = log 70/log 4 x = ((log 70/log 4)-2)/5 = 0.213

If $850 is invested at an annual interest rate of 6% compounded continuously, what is the amount in the account after 10 years?

A = 850e.⁰⁶ * ¹⁰ = $1548.80

A bank offers a nominal interest rate of 6% per annum for certain accounts. Compute the effective rate if interest is compounded monthly.

A = P(1 + (.06/12))¹²*¹ = P(1.0617) effective interest rate is 6.17% no matter how much is invested

periodic compound interest formula

A = P(1 + (r/n))ⁿᵗ A = amount P = principal investment r = annual rate n = number of times it is compounded

compound interest formula (annually)

A = P(1 + r)ᵗ A = amount P = initial investment r = rate ᵗ = time (years)

compound interest formula (continuously)

A = Peʳᵗ A = amount P = initial investment e = e variable r = rate t = time in years

State the domain, range, y-intercept, whether it is growth or decay, and the asymptote of the exponential function y = eˣ

Domain: (-∞,∞) Range: (0,∞) y-int: 1 growth because e is larger than 1 asymptote: x-axis

Fundamental Theorem of Algebra

Every polynomial equation of the form aₙxⁿ + aₙ_₁xⁿ⁻¹ + . . . a₁x + a₀ = 0 where n ≥ 1, aₙ ≠ 0 has at least one root within the complex number system (it may be a real number)

Solve the equation 2x⁴ - 3x³ + 12x² + 22x - 60 = 0, given that one root is 1 + 3i.

If one root is 1 + 3i, then its conjugate is 1 - 3i Expand: (x - 1 + 3i)(x - 1 - 3i) = x² - 2x + 10 Divide by the original function using long division = 2x² + x - 6 (2x - 3)(x + 2) Roots: -2, 3/2, 1 + 3i, 1 - 3i

Linear Factors Theorem

Let f(x) = aₙxⁿ + aₙ_₁xⁿ⁻¹ + . . . a₁x + a₀ = 0 where n ≥ 1, aₙ ≠ 0. Then f(x) can be expressed as a product of n linear factors: f(x) = aₙ(x - r₁) (x - r₂). . . (x - rₙ)

Conjugate Root Theorem (roots)

Let f(x) be a polynomial in which all coefficients are rational. Suppose that a + b√c is a root of the equation f(x) = 0 where a, b, and c are rational and √c is irrational. Then a - b√c is also a root of the equation.

Descartes's Rule of Signs (positive)

Let f(x) be a polynomial, all of whose coefficients are real numbers, and consider the equation f(x) = 0. a) The number of positive roots either is equal to the number of variations in sign of f(x) or is less than that of an even integer (you can find the number of positive roots by how many sign changes there are or with a difference of 2 from the original degree until 0; ex. 4, 2, 0)

Descartes's Rule of Signs (negative)

Let f(x) be a polynomial, all of whose coefficients are real numbers, and consider the equation f(x) = 0. b) The number of negative roots is either equal to the number of variations in sign of f(-x), or is less than that by an even integer. (you can find the negative roots by how many sign changes there are when x is -x with a difference of 2 from the original degree until you get to 1)

The Location Theorem

Let f(x) be a polynomial, all of whose coefficients are real numbers. If a and b are real numbers such that f(a) and f(b) have opposite signs, then the equation f(x) = 0 has at least one real root between a and b.

Conjugate Root Theorem (imaginary numbers)

Let f(x) be a polynomial, all of whose coefficients are real numbers. Suppose that a + bi is a root of the equation f(x) = 0, where a and b are real and b ≠ 0. Then a - bi is also a root of the equation.

Factor Theorem

Let f(x) be a polynomial. If f(r) = 0 then x - r is a factor of f(x). Conversely, if x - r is a factor of f(x), then f(r) = 0

Use the property log(b)Pⁿ = n log(b)P to solve the problem y = 2ˣ - 3

Method 1: log 3 = x log 2 log 3/log 2 = 1.58 Method 2: log₂3 = log₂2ˣ log₂3 = x

Find a quadratic equation given 1 irrational root with rational coefficients and a leading coefficient of 1 such that one of the roots is r₁ = 4 + 5√3.

Method 1: x² -bx + c = 0 (4 + 5√3)(4 - 5√3) = -59 4 + 5√3 + 4 - 5√3 = 8; -8 x² - 8x - 59 Method 2: (x - 4 + 5√3)(x - 4 - 5√3) and solve

Given that log(b)2 = A and log(b)6 = B, express the following in terms of A and B: log(√b)2

Method 1: √bˣ = 2 b¹/² * ˣ = 2 bˣ/² = 2 log(b)2 = x/2 A = x/2, so = 2A Method 2: (√bˣ = 2)² = bˣ = 4 log(b)4 = x log(b)2² 2 log(b)2, so 2A

Evaluate the logarithmic expression log₄32

hint: 4¹/² = √4 since 2⁵ = 32 = 5/2

How do complex roots occur when present and when is this the case?

in conjugate pairs; for all polynomial equations with real coefficients

How can you find the inverse of the exponential function f(x) = bˣ?

interchange x and y in the equation f⁻¹ = x = bʸ

If b denotes an arbitrary positive real number, what do the properties of rational exponents apply to?

irrational exponents

What does a multiplicity look like when graphed?

it bounces on the x axis (resembles a quadratic equation)

What does the horizontal line test say about the function f(x) = bˣ?

it is one-to-one and has an inverse

exponential function with base b (b > 0, b ≠ 1)

let b denote an arbitrary positive constant other than 1. the exponential function with base b is defined by the equation y = bˣ ex. 3(2)ˣ, (1/2)ˣ NOT x²

Why does ln 1 - 0?

ln 1 stands for log(e)1, which - 0 (the exponential form of ln 1 = 0 is e⁰ = 1)

Why does ln e = 1?

ln e stands for log(e)e, which is 1

Why does ln(e²) = 2?

ln(e²) stands for loge(e²), which is 2

Solve the following: 3 log₃7 = 7

log 3 * log₃7 = log 7 log₃7 = log 3/log 7

Given that log(b)2 = A and log(b)6 = B, express the following in terms of A and B: log(b)b/36

log(b)b - log(b)36 log(b)b - log(b)6² = 1 - 2B

What is the inverse of exponential functions?

logarithmic functions

Express the following equation as a single logarithm with a coefficient of 1: 1/2 logb(x) - logb(1 + x²)

logb(x¹/²) - logb(1 + x²) = logb(x¹/²/(1 + x²) OR logb(√x/(1 + x²)

Solve log₂16/16

log₂16 can be simplified to 4; 4/16 = 1/4

Solve the following: log₈56 - log₈7

log₈56 - log₈7 log₈(56/7) log₈8 = 1

Describe the logarithmic change of base property

logₐx = log(b)x/log(b)a

Describe the shape of most exponential graphs

most exponential graphs will have the same arcing shape

Solve the problem (3²⁺√⁵)(3²⁻√⁵)

multiplied together so add; 2 + √5 + 2 - √5√ = 4 3⁴ = 81

Power Property

n logb(P) = logb(Pⁿ) ex. -logb(2) = logb(1/2)

Will the sign change with an even exponent?

no

Should you use tables to find points when plotting exponential functions?

no, unless she wants specific ordered pairs; just find the asymptote and the y-intercept and do your best to sketch it out quickly

Are the complex numbers of r(k) necessarily all distinct?

no; some of all of them may be real numbers

Find the rational roots (if any) of the equation 2x³ - x² - 9x - 4 = 0.

p = ±4, 1, 2 q = ±2, 1 p/q = ±4, 2, 1, 1/2 -1/2 is a root (use synthetic division) leftover 2x² - 2x - 8 use quadratic equation; 1±√17/2

Describe rational and irrational roots in the polynomial equation aₙxⁿ + aₙ_₁xⁿ⁻¹ + . . . a₁x + a₀ = 0 where all the coefficients are integers. If p/q is a rational number where p and q have no common factors other than ±1 and p/q is a root of the equation, what are p and q factors of?

p is a factor of a₀ q is a factor of aₙ

How do you find the y-intercept?

plug in 0 for x; ex. y = 4ˣ y = 4⁰ y = 1

What are P and Q assumed to be for the above properties?

positive

What is the root of the equation provided that f(r) = 0?

r

What is the difference between x as a rational and x as an irrational number?

rational: ends irrational: continues on indefinitely (do not end)

If r₁ and r₂ are the roots of the equation x² + bx + c, how do you find c and b?

r₁r₂ = c r₁ + r₂ = -b

What is a property of exponents?

the ability to switch exponents ex. (3⁴)⁵ = (3⁵)⁴

What must you know to graph an exponential function?

the asymptote (usually the x axis unless otherwise specified) and the y-intercept

What does the asymptote start on?

the ending point for the range; it should always be a part of the range

What are the only functions with this property?

the exponential function f(x) = eˣ and the constant multiples of this function (y = ceˣ where c is a constant)

What happens the greater the base, b, is?

the faster the graph rises from left to right

Write an exponential function that represents the following situation: Mrs. Caron wants to improve class attendance. She offers you 2 cents on the first day, 4 cents on the second day, eight cents on the third day, and so on.

the function f(X) = 2ˣ; day__amount 1____2¹ = 2 2____2² = 4 3____2³ = 8 4____2⁴ = 16

What is log₂?

the name of the function; cannot be cancelled from the numerator and denominator ex. log₂8/log₂4 = 3/2 NOT 8/4

What is ln x?

the natural log of x; means log(e)x

How can you find the number of rational roots in a polynomial?

the number of times it crosses the x axis

multiplicity

the number of times that a root is repeated

Product Property

the of the product is the sum of the logs of the factors logb(P) + logb(Q) = logb(PQ)

What is e defined as in calculus?

the target value or limit of the function f(x) = (1 + (1/x))ˣ

What is the instantaneous rate of change on the graph f(x) = e?

the y-coordinate of the point

What does it mean if there is no sign variation?

there are no positive real roots

What is the difference between the graphs (1/2)ˣ and 2⁻ˣ?

they are the same

Use the remainder theorem to evaluate whether 3 is a zero of 2x³ - 5x² + x - 6

use synthetic division = 2x² - 5x + 1, R 6 x - 3 is not a factor

How can you test if a value is a root?

use synthetic division ex. x - 1 and 2x² - 3x + 4; = 2x - 1, R 3 1 is not a zero of the equation

Take the graph y = x³ - 5x + 2. It shows that they are no muliplicities and that it has at least three roots, one of which is 2 or is close to 2. Use synthetic division to confirm that x = 2 is a root of equation (3), then solve.

use synthetic division; 2 is a root remaining equation is x² + 2x - 1

What is an example of x as an irrational number as represented through rational approximations?

values of 2ˣ for rational numbers x approaching √2 = 1.41421356. . . x__1.4___1.41__1.414___1.4142___1.41421 2ˣ_2.6..._2.65..._2.664..._2.6651..._2.66514...

How do we describe the equations y = log(b)x and x = bʸ?

we say y = log(b)x is in the logarithmic form and x = bʸ is in the exponential form

Remainder Theorem

when a polynomial f(x) is divided by x - r, the remainder is f(r) ex. remainder is 3 and f(1) = 2(1)² - 3(1) + 4 = 3

When can exponential functions have x-intercepts?

when the x-axis isn't an asymptote

Find the x-intercepts of the following: . y = 4ˣ⁺²

x - 2 = 1 3

If b denotes an arbitrary positive real number where bˣ = bʸ and b≠1, what is x equal to?

x = y ex. b^√² = bʸ y = √2

Find a polynomial f(x) with leading coefficients such that the equation f(X) = 0 has only those roots specified in the following: (root, multiplicity) (3, 2)(-2, 1)(0, 2) What is the degree of their polynomial?

x²(x - 3)(x - 3) (x + 2) highest degree is 5 x⁵ - 4x⁴ - 3x³ + 18x²

What are examples of polynomials and what do they have?

x³ - 17x² + 6x - 1 = 0 √2x⁴⁷ - πx²⁵ + √3 = 0 they have at least one root

Use the fact that 2ˣ is always positive to solve an equation

x³2ˣ - 3(2ˣ) = 0 (2ˣ)(x³ - 3) = 0 x = ³√3

What is the difference between the graphs y = 2ˣ and y = (1/2)ˣ?

y = 2ˣ stretches upward on the right side and close to the asymptote on the left y = (1/2)ˣ is reflected across the y axis; stretches upward on the left side and close to the asymptote on the right same y-intercept

What are the tangents for y = eˣ, y = 3ˣ, and y = 2ˣ ?

y = 2ˣ: tangent of 0.7 y = 3: tangent of 1.1 y = eˣ: tangent of 1

What can the equation y = bˣ also be represented as?

y = abᵏˣ, where a, b, and k are constants and b ≠ 1

Describe the domain, range, y-intercept, x-intercept, and asymptote for exponential functions of the form y = bˣ.

y = bˣ domain: (-∞,∞); always all real numbers range: (0,∞) y-int: (0,1) x-int: none asymptote: x axis

Describe the graphs of the following: . y = ln x . y = ln (x - 1)

y = ln x is the inverse of y = eˣ y = ln (x-1) is the same graph shifted one unit to the right with an asymptote at x = 1

What is x = bʸ equivalent to?

y = log(b)x

What is the inverse of y = eˣ?

y = log(e)x; rewrite as ln x

Do the properties of exponents continue to hold if we use an approximation of an irrational number as a rational number?

yes

In the Lower Bound Theorem for Real Roots, how should you determine whether the signs alternate in the third row?

zeroes are counted as either + or -

Graph the function y = -3⁻ˣ + 1

⁻ˣ so decay asymptote y = 1 reflextion over x axis y-int (0,0)

What can 'x' also be?

an irrational number

Solve the equation 10ˣ/² = 16: 1) use a graphing calculator to show it has one root, then estimate that root 2) solve the given equation algebraically by rewriting it in log form. Give the exact form and a calculated approximate as an answer, then check to see that it is consistent with the graphed value

1) = 2.4 2) log₁₀(x/2) = log 16 (x/2) log 10 = log 16 x/2 = log16/log10 x = 2 log(16)/log(10)

A principal of $1600 is placed in a savings account at 5% per annum compounded continuously. When will the balance reach $2400?

2400 = 850e.⁰⁶ᵗ 2400/850 = e.⁰⁶ᵗ ln(2.824) = .06t t = 17.3 years

Upper Bound Theorem for Real Roots

Consider the polynomial equation f(x) = aₙxⁿ + aₙ_₁xⁿ⁻¹ + . . . a₁x + a₀ = 0 where all of the coefficients are real numbers and aₙ is positive. 1) if we use synthetic division to divide f(x) by x - B, where B > 0, and we obtain a third row containing no negative numbers, then B is an upper bound for the real roots of f(x) = 0. (if you try a positive number and all numbers are positives, do not go higher)

Lower Bound Theorem for Real Roots

Consider the polynomial equation f(x) = aₙxⁿ + aₙ_₁xⁿ⁻¹ + . . . a₁x + a₀ = 0 where all of the coefficients are real numbers and aₙ is positive. 2) IF we use synthetic division to divide f(X) by x - b, where b < 0, and we obtain a third row in which the numbers are alternatively positive and negative, then b is a lower bound for the real roots f(x) = 0 (if numbers alternate, try higher numbers)

Describe the domain, range, y-intercept, x-intercept, and asymptote of logarithmic functions

D: (0,∞) R: (-∞,∞) y-int: none x-int: (1,0) asymptote: y axis

Get information regarding the roots in the equation x³ + 8x + 5 = 0 with at least one irrational root

Positive: x³ + 8x + 5 = 0 no sign change so no positive roots Negative: -x³ - 8x + 5 changes once so 1 negative root Total: 0 positive, 1 negative, 2 imaginary

Use Descartes's rule to obtain information regarding the roots of x³ - x² + 3x + 2 = 0.

Positive: x³ - x² + 3x + 2 = 0. 2 or 0 positive roots (2 sign changes) Negative: -x³ - x² + -3x + 2 = 0. 1 sign change so 1 negative root Total: 2 positive and 1 negative or 0 positive, 1 negative, 2 imaginary

Use Descartes's rule to obtain information regarding the root x⁴ + 3x² - 7x - 5 = 0 with at least one irrational root

Positive: x⁴ + 3x² - 7x - 5 = 0 1 sign change so 1 positive root Negative: x⁴ + 3x² + 7x - 5 1 sign change so 1 negative root Total: 1 positive, 1 negative, 2 imaginary

asymptote

a line that the graph gets very close to but never cross or touches

What is an exception to the Lower Bound Theorem for real roots?

a number may fail the lower bound test but still be a lower bound

root

a number r substituted for x that makes the statement true

What should you be able to solve?

a polynomial equation of any degree

If b denotes an arbitrary positive real number, when what is the quantity bˣ for each real number x?

a unique real number

Use log properties to expand expressions by writing as sums and differences in such a way that there is no log of products, quotients, or powers: a) log₁₀√3x b) log₁₀³√2x/(3x²+1) c) ln x²√(2x-1)/(2x+1)³/²

a) 1/2 log₁₀3 + 1/2 log₁₀x b) 1/3 log₁₀2 + 1/3 log₁₀x - 1/3 log₁₀(3x²+1) OR 1/3(log₁₀2 + log₁₀x - log₁₀(3x²+1)) c) 2 ln x + 1/2 ln (2x-1) - 3/2 ln (2x+1)

Describe the graphs of the following: a) y = log₁₀x b) y = -log₁₀x

a) 10ʸ = x vertical asymptote x = 0, x-int (1,0) **increases b) D: (0,∞), R:(-∞,∞), y = -10ˣ reflected over the x axis **decreases

Suppose that $1000 is placed in a savings account at 10% per annum. How much is in the account at the end of 1 year if the interest is: a) compounded once each year (n = 1) AND b) compounded quarterly (n = 4) c) How much interest is earned under the compounded quarterly? d) Based on your answer to c), what is the effective rate?

a) A = 1000(1 + (.10/1))¹*¹ = $1100 b) A = 1000(1 + (.10/4))⁴*¹ = $1103.81 c) $1103.81 - $1000 = $103.81 d) 103.81 is 10.381% of $1000 so the effective rate is 10.381%

You invest $800 into a savings account earning a 2.5% annual interest rate. a) How much money will you have in 5 years? b) How long will it take to double your money?

a) A = 800(1 + .025)⁵ = $905.13 b) 1600 = 800(1.025)ᵗ 2 = 1.025ᵗ t = log₁.₀₂₅2 t = 28.07 years

In a biology experiment, N(t) is the number of bacteria in a colony after t hours, and t = 0 corresponds to when the experiment begins. Suppose the period t = 4 to t = 8 hours is modelled by the exponential function y = eˣ. a) find the average rate of change of the population over time from 5 ≤ t ≤ 7 b) find the instantaneous rate of change at t = 7 hours

a) e⁷-e⁵/(7-5) = 474 bacteria/hour b) e⁷ = 1,097 bacteria/hour

Given that log(b)2 = A and log(b)6 = B, express the following in terms of A and B: a) log(b)8 b) log(b)√6 c) log(b)12 d) log(b)3

a) log(b)2³ = 3 log(b)2 = 3A b) log(b)6¹/² = 1/2B c) log(b)2 + log(b)6 = logb(12) so = A + B d) log(b)6 - log(b)2 = B-A

Solve the following using the change of base property: a) log₂3 b) log₁₀e

a) log₁₀3/log₁₀2 = ln 3/ln 2 b) ln e/ln 10 = 1/ln 10

Check to see if the following are zeroes: a) -3 of the function f defined by f(x) = x⁴ + x² - 6 b) √2 of the equation x⁴ + x² - 6

a) plug it in; (-3)⁴ + (-3)² - 6 81 + 9 - 6 ≠ 0, so it is not a zero b) (√2)⁴ + (√2)² - 6 0 = 0 so it is a zero

State the domain, range, intercepts, and asymptote for the following: a) y = -2ˣ b) y = 2⁻ˣ c) y = (1/2)ˣ d) y = 2⁻ˣ - 2

a) reflection over x axis d: all reals, r: (-∞, 0), no x-ints, y-int (0, -1), asymptote y = 0 b) d: all reals, r (0, ∞), no x-int, y-int (0,1), asymptote y = 0 c) same as b) d) d: all reals, r: (-2, ∞), y-int (0, -1), need to calculate x-int, asymptote y = -2

What is e?

an irrational unit which is approximately 2.7182818284

nominal rate

annual rate the given rate of 10% per annum compounded once a year

effective rate

annual yield (interest earned/original amount)*100

What should you remember regarding the y-intercepts?

any number to the zero power is 1

If b denotes an arbitrary positive real number, what can we do when x is irrational?

approximate bˣ as closely as we wish by evaluating bʳ, where r is a rational number sufficiently close to the number x

Describe the function f(x) = (1 + (1/x))ˣ in terms of the exponential function f(x) = eˣ

as x grows larger and larger the value of f(x) approaches the value of e

Solve the following: bˡᵒᵍᵇ⁽ᴾ⁾ = ?

bˡᵒᵍᵇ⁽ᴾ⁾ = P

What are other properties of exponents?

common bases, multiplication, addition, etc.

What do exponential graphs grow by?

common factors over equal intervals

What is the compound interest formula?

compounded continuously

What are irrational roots in?

conjugate pairs: there are always 2

Find the domain of the function f(x) = log₂(12 - 4x)

domain can be defined by an asymptote; change to exponential to easily see it 2ʸ = 12 - 4x 2ʸ - 12 = -4x divide by -4 to find x -1/4 * 2ʸ + 3 = x D: (-∞,3); reflection so decreasing

Describe the domain and range for exponential graphs

domain: all real numbers range: related to asymptotes, either all positives or all negatives before the asymptote

How many roots does every polynomial equation of the degree n ≥ 1 have where a root of multiplicity k is counted k times?

exactly n roots

What types of equations does the Fundamental Theorem of Algebra not apply to?

exponential equations where the variable is the exponent and inverse functions (no negative exponents allowed)

Describe the rate of change for exponential graphs

exponential graphs do not have constant rates of change, but they do have constant ratios

What is the instantaneous rate of change of the function f(x) = eˣ at x = a when a and b are real numbers?

eᵃ

Find the instantaneous rate of change of f(x) = eˣ at x = 1

f(1) = e¹ = e (at x = 2 the rate of change is e²)

Find a quadratic function that has zeroes of 3 and 5 and a graph that passes through the point (2, -9)

f(x) = aₙ(x - r₁)(x - r₂) y = aₙ(x - 3)(x - 5) substitute in the order pair for x and y and solve -9 = a(2 - 3)(2 - 5) a = -3 f(x) = -3(x - 3)(x - 5) f(x) = -3x² + 24x - 45

Express the following second-degree polynomial in the form aₙ(x - r₁) (x - r₂): 3x² - 5x - 2

factored as (x - 2)(3x + 1) 3(x + 1/3)(x - 2) OR 3(x - (-1/3))(x - 2)

Solve the equation 4ˣ = 8

find a common base first 2²ˣ = 2³ 2x = 3 x = 3/2