AP Calculus BC: formula quiz

Flashcards covering unit circle, trig identities, limits, asymptotes, logarithms, and absolute value based on Page 1 notes.

Unit circle definition and properties

The circle centered at the origin (0,0) with a radius of 1, represented by the equation x^2 + y^2 = 1. For any angle \theta measured counterclockwise from the positive x-axis, its coordinates on the unit circle are (cos \theta, sin \theta). Essential for understanding trigonometric functions and memorizing common angle values.

sin^2 x + cos^2 x

1 (This is the fundamental Pythagorean identity of trigonometry, essential for relating sine and cosine functions.)

1

sin^2 x + cos^2 x (This is the fundamental Pythagorean identity of trigonometry, essential for relating sine and cosine functions.)

tan^2 x + 1

sec^2 x (This is a Pythagorean identity relating tangent and secant functions.)

sec^2 x

tan^2 x + 1 (This is a Pythagorean identity relating tangent and secant functions.)

1 + cot^2 x

csc^2 x (This is a Pythagorean identity relating cotangent and cosecant functions.)

csc^2 x

1 + cot^2 x (This is a Pythagorean identity relating cotangent and cosecant functions.)

tan x

sin x / cos x (This quotient identity defines the tangent function in terms of sine and cosine.)

sin x / cos x

tan x (This quotient identity shows that tangent is the ratio of sine to cosine.)

cot x

cos x / sin x (This quotient identity defines the cotangent function in terms of cosine and sine.)

cos x / sin x

cot x (This quotient identity shows that cotangent is the ratio of cosine to sine.)

sin x

1/csc x (This reciprocal identity states that sine is the reciprocal of cosecant.)

1/csc x

sin x (This reciprocal identity states that cosecant is the reciprocal of sine.)

cos x

1/sec x (This reciprocal identity states that cosine is the reciprocal of secant.)

1/sec x

cos x (This reciprocal identity states that secant is the reciprocal of cosine.)

tan x

1/cot x (This reciprocal identity states that tangent is the reciprocal of cotangent.)

1/cot x

tan x (This reciprocal identity states that cotangent is the reciprocal of tangent.)

csc x

1/sin x (This reciprocal identity defines cosecant as the reciprocal of sine.)

1/sin x

csc x (This reciprocal identity defines sine as the reciprocal of cosecant.)

sec x

1/cos x (This reciprocal identity defines secant as the reciprocal of cosine.)

1/cos x

sec x (This reciprocal identity defines cosine as the reciprocal of secant.)

cot x

1/tan x (This reciprocal identity states that cotangent is the reciprocal of tangent.)

1/tan x

cot x (This reciprocal identity states that tangent is the reciprocal of cotangent.)

sin 2x

2 sin x cos x (This is the double angle identity for sine, expressing sin(2x) in terms of sin x and cos x.)

2 sin x cos x

sin 2x (This expresses 2 sin x cos x as the sine of a double angle.)

cos 2x

cos^2 x - sin^2 x (This is one form of the double angle identity for cosine.)

cos^2 x - sin^2 x

cos 2x (This expresses cos^2 x - sin^2 x as the cosine of a double angle.)

cos 2x

2 cos^2 x - 1 (This is another form of the double angle identity for cosine, useful when only cosine is known.)

2 cos^2 x - 1

cos 2x (This expresses 2 cos^2 x - 1 as the cosine of a double angle.)

cos 2x

1 - 2 sin^2 x (This is a third form of the double angle identity for cosine, useful when only sine is known.)

1 - 2 sin^2 x

cos 2x (This expresses 1 - 2 sin^2 x as the cosine of a double angle.)

tan 2x

2 tan x / (1 - tan^2 x) (This is the double angle identity for tangent, expressing tan(2x) in terms of tan x.)

2 tan x / (1 - tan^2 x)$$

tan 2x (This expresses 2 tan x / (1 - tan^2 x) as the tangent of a double angle.)

Even/Odd Identity: sin(-x)$$

- sin x (This identity shows that sine is an odd function, meaning f(-x) = -f(x).)

- sin x (Value for sine as an odd function)

sin(-x) (This shows that for an odd function like sine, changing the sign of the input changes the sign of the output.)

Even/Odd Identity: cos(-x)$$

cos x (This identity shows that cosine is an even function, meaning f(-x) = f(x).)

cos x (Value for cosine as an even function)

cos(-x) (This shows that for an even function like cosine, changing the sign of the input does not change the output.)

Even/Odd Identity: tan(-x)$$

- tan x (This identity shows that tangent is an odd function, derived from sine/cosine properties.)

- tan x (Value for tangent as an odd function)

tan(-x) (This shows that for an odd function like tangent, changing the sign of the input changes the sign of the output.)

sin x sin y

1/2 [cos(x - y) - cos(x + y)] (This product-to-sum identity converts a product of two sine functions into a sum/difference of cosine functions.)

1/2 [cos(x - y) - cos(x + y)]

sin x sin y (This identity converts a sum/difference of cosine functions into a product of two sine functions.)

cos x cos y

1/2 [cos(x - y) + cos(x + y)] (This product-to-sum identity converts a product of two cosine functions into a sum of cosine functions.)

1/2 [cos(x - y) + cos(x + y)]

cos x cos y (This identity converts a sum of cosine functions into a product of two cosine functions.)

sin x cos y

1/2 [sin(x + y) + sin(x - y)] (This product-to-sum identity converts a product of sine and cosine functions into a sum of sine functions.)

1/2 [sin(x + y) + sin(x - y)]

sin x cos y (This identity converts a sum of sine functions into a product of sine and cosine functions.)

sin(x + y)$$

sin x cos y + cos x sin y (This is the sine sum identity, expanding sin(x+y).)

sin x cos y + cos x sin y

sin(x + y) (This identity condenses the expansion of sin x cos y + cos x sin y.)

sin(x - y)$$

sin x cos y - cos x sin y (This is the sine difference identity, expanding sin(x-y).)

sin x cos y - cos x sin y

sin(x - y) (This identity condenses the expansion of sin x cos y - cos x sin y.)

cos(x + y)$$

cos x cos y - sin x sin y (This is the cosine sum identity, expanding cos(x+y).)

cos x cos y - sin x sin y

cos(x + y) (This identity condenses the expansion of cos x cos y - sin x sin y.)

cos(x - y)$$

cos x cos y + sin x sin y (This is the cosine difference identity, expanding cos(x-y).)

cos x cos y + sin x sin y

cos(x - y) (This identity condenses the expansion of cos x cos y + sin x sin y.)

tan(x + y)$$

(tan x + tan y)/(1 - tan x tan y) (This is the tangent sum identity, expanding tan(x+y).)

(tan x + tan y)/(1 - tan x tan y)$$

tan(x + y) (This identity condenses the expansion of (tan x + tan y)/(1 - tan x tan y).)

tan(x - y)$$

(tan x - tan y)/(1 + tan x tan y) (This is the tangent difference identity, expanding tan(x-y).)

(tan x - tan y)/(1 + tan x tan y)$$

tan(x - y) (This identity condenses the expansion of (tan x - tan y)/(1 + tan x tan y).)

sin^2 x

(1 - cos 2x)/2 (This power-reducing identity expresses sin^2 x in terms of cos 2x, useful for integration.)

(1 - cos 2x)/2

sin^2 x (This expresses (1 - cos 2x)/2 as sine squared, useful for reducing powers of trigonometric functions.)

cos^2 x

(1 + cos 2x)/2 (This power-reducing identity expresses cos^2 x in terms of cos 2x, useful for integration.)

(1 + cos 2x)/2

cos^2 x (This expresses (1 + cos 2x)/2 as cosine squared, useful for reducing powers of trigonometric functions.)

tan^2 x

(1 - cos 2x)/(1 + cos 2x) (This power-reducing identity expresses tan^2 x in terms of cos 2x, useful for integration.)

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

tan^2 x (This expresses (1 - cos 2x)/(1 + cos 2x) as tangent squared, useful for reducing powers of trigonometric functions.)

sin x + sin y

2 sin[(x + y)/2] cos[(x - y)/2] (This sum-to-product identity converts a sum of two sine functions into a product.)

2 sin[(x + y)/2] cos[(x - y)/2]$$

sin x + sin y (This identity converts 2 sin[(x + y)/2] cos[(x - y)/2] from a product of sines and cosines to a sum of sines.)

sin x - sin y

2 cos[(x + y)/2] sin[(x - y)/2] (This sum-to-product identity converts a difference of two sine functions into a product.)

2 cos[(x + y)/2] sin[(x - y)/2]$$

sin x - sin y (This identity converts 2 cos[(x + y)/2] sin[(x - y)/2] from a product of sines and cosines to a difference of sines.)

cos x + cos y

2 cos[(x + y)/2] cos[(x - y)/2] (This sum-to-product identity converts a sum of two cosine functions into a product.)

2 cos[(x + y)/2] cos[(x - y)/2]$$

cos x + cos y (This identity converts 2 cos[(x + y)/2] cos[(x - y)/2] from a product of cosines to a sum of cosines.)

cos x - cos y

-2 sin[(x + y)/2] sin[(x - y)/2] (This sum-to-product identity converts a difference of two cosine functions into a product.)

-2 sin[(x + y)/2] sin[(x - y)/2]$$

cos x - cos y (This identity converts -2 sin[(x + y)/2] sin[(x - y)/2] from a product of sines to a difference of cosines.)

Distance Formula between two points (x1, y1) and (x2, y2)$$

\sqrt{(x2 - x1)^2 + (y2 - y1)^2} (This formula calculates the length of the straight line segment connecting the two given points in a Cartesian coordinate system.)

Formula for the distance between two points (x1, y1) and (x2, y2): \sqrt{(x2 - x1)^2 + (y2 - y1)^2}$$

The distance between the points (x1, y1) and (x2, y2) on a Cartesian plane.

Midpoint Formula for a segment connecting (x1, y1) and (x2, y2)$$

((x1 + x2)/2, (y1 + y2)/2) (This formula gives the coordinates of the point that is exactly halfway between the two given endpoints of a line segment.)

Formula for the midpoint of a segment connecting (x1, y1) and (x2, y2): ((x1 + x2)/2, (y1 + y2)/2)$$

The precise center point of the line segment connecting (x1, y1) and (x2, y2)

Logarithm Product Rule: ln(ab)$$

ln a + ln b (The logarithm of a product of two numbers is the sum of the logarithms of the individual numbers.)

Sum of logarithms as a single logarithm: ln a + ln b

ln(ab) (This rule allows a sum of logarithms to be rewritten as the logarithm of a product.)

Logarithm Power Rule: ln(a^k)$$

k ln a (The logarithm of a number raised to an exponent is the exponent multiplied by the logarithm of the number.)

Exponent times logarithm as a single logarithm: k ln a

ln(a^k) (This rule allows an exponent times a logarithm to be rewritten as the logarithm of a number raised to that exponent.)

Logarithm Change of Base Formula: log_a b

ln b / ln a (Allows conversion of a logarithm from one base (a) to another, typically to the natural logarithm or common logarithm for calculation.)

Ratio of natural logarithms for change of base: ln b / ln a

log_a b (This formula allows changing the base of a logarithm by dividing the natural logarithm of the argument by the natural logarithm of the original base.)

Fundamental Concept: Definition of a Limit

The limit L of a function f(x) as x approaches a value a (denoted as lim_{x \to a} f(x) = L) means that as x gets arbitrarily close to a (but not equal to a), the values of f(x) get arbitrarily close to L. This describes the behavior of a function near a point.

Understanding a Two-Sided Limit

A two-sided limit lim_{x \to a} f(x) = L exists if and only if the function values f(x) approach the same value L as x approaches a from both the left-hand side (x < a) and the right-hand side (x > a).

Distinguishing One-Sided Limits

These limits describe the behavior of a function as x approaches a value a from only one direction.

• Left-hand limit: lim_{x \to a^-} f(x) means x approaches a from values less than a.
• Right-hand limit: lim_{x \to a^+} f(x) means x approaches a from values greater than a.

For a two-sided limit to exist, the left-hand and right-hand limits must be equal.

Sum Law for Limits: lim_{x \to a} (f(x) + g(x))

lim{x \to a} f(x) + lim{x \to a} g(x) (The limit of a sum of functions is the sum of their individual limits.)

Sum of individual limits for a sum of functions: lim{x \to a} f(x) + lim{x \to a} g(x)$$

lim_{x \to a} (f(x) + g(x)) (This law states that the sum of the individual limits of two functions is equal to the limit of their sum.)

Difference Law for Limits: lim_{x \to a} (f(x) - g(x))

lim{x \to a} f(x) - lim{x \to a} g(x) (The limit of a difference of functions is the difference of their individual limits.)

Difference of individual limits for a difference of functions: lim{x \to a} f(x) - lim{x \to a} g(x)$$

lim_{x \to a} (f(x) - g(x)) (This law states that the difference of the individual limits of two functions is equal to the limit of their difference.)

Product Law for Limits: lim_{x \to a} (f(x) g(x))

lim{x \to a} f(x) \cdot lim{x \to a} g(x) (The limit of a product of functions is the product of their individual limits.)

Product of individual limits for a product of functions: lim{x \to a} f(x) \cdot lim{x \to a} g(x)$$

lim_{x \to a} (f(x) g(x)) (This law states that the product of the individual limits of two functions is equal to the limit of their product.)

Quotient Law for Limits: lim_{x \to a} (f(x) / g(x))

lim{x \to a} f(x) / lim{x \to a} g(x) (The limit of a quotient of functions is the quotient of their individual limits, provided the limit of the denominator is not zero.)

Quotient of individual limits for a quotient of functions: lim{x \to a} f(x) / lim{x \to a} g(x) (where denominator limit is non-zero)

lim_{x \to a} (f(x) / g(x)) (This law states that the quotient of the individual limits of two functions is equal to the limit of their quotient, given the denominator's limit is non-zero.)

c \cdot lim_{x \to a} f(x)