AP Calculus AB Cumulative review

Range of tangent

(-∞,∞)

Area formula for a circle

(π)r^2 where r is the radius

The Range of the Cosine Function is what?

-1 ≤ cos(t) ≤ 1

The Range of the Sine Function is what?

-1 ≤ sin(t) ≤ 1

Lucas draws two vectors. The first vector L is ⟨6,−8⟩. He draws his second vector so that it is orthogonal to the first vector, has a magnitude of 20, and is in the 3rd quadrant. Write his second vector in linear combination form (use i and j to represent the standard unit vectors).

-16i -12j

0 degrees in radian

0 radian

What is the definition of a radian?

1 radian is the angle at which the arc length is equal to the radius of any given circle

When will the derivative of a function fail to exist?

1. Any points of discontinuity (Function must be continuous) 2. Function is not differentiable 3. Has a "corner" (like |x| at x=0) or a "cusp" (Like x^2/3), meaning the change in slope is instantaneous, not gradual. 4. Has an undefined slope at a certain point (Like the cube root of x)

Write an equation for the specified line: 1. through (1, -6) with slope 3 2. through (-1, 2) with slope -1/2 3. the vertical line through (0, -3)

1. y +6 = 3 (x-1) y = 3(x-1) - 6 y = 3x - 3 - 6 y= 3x - 9 2. y - 2 = -1/2 ( x +1) y = -1/2 (x + 1) + 2 y = -x/2 - 1/2 + 2 y = -x/2 + 3/2 3. x = 0

Suppose that a cannon fires a cannonball at a velocity of 20 mph at an angle of θ = 30 degrees over horizontal. Find the vector of the cannonball's flight in component form

20 = magnitude 30 degrees = angle 20<cos30, sin30> = 20<√3/2, 1/2> = <10√3, 10>

What are the identities for Tan(2u)

2tan(u) / (1 - tan^2(u))

How many radians are in a circle

2π radians, or approximately 6.28 radians.

120 degrees in radians

2π/3

Circumference formula

2πr where r is the radius

In the function f(x) = x+2, as the value of x approaches 1, what does the value of f(x) approach?

3. as x => 1, f(x) approaches 3.

135 degrees in radians

3π/4

Suppose Gabriel has 4 guppies. From previous experiments, Gabriel knows that the population will double every day. Create a function to represent the total number of guppies over t days

4(2)^t

150 degrees in radians

5π/6

Suppose a ball is thrown at an initial velocity of 70 feet per second, at an angle of 40 degrees with the horizontal. Find the vector of the ball's flight in component form

70*<cos(40), sin(40)>

Suppose the magnitude of E is 13, where <E(x), E(y)> are both integers, and E(x) < E(y) < 0, what is the vector in component form?

<-12,-5>

What is an identity for Sin(2u)?

= 2sin(x)cos(x)

What are the identities for Cos(2u)

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

What is a removable discontinuity?

A "hole" in a graph; a value of a graph which cannot be drawn because it is not defined

What is the equation to calculate half-life?

A = N(1/2)^(t/(t 1/2)) N(t) is the final amount N is the initial amount t is the time periods elapsed t 1/2 is the half life of the object

What is the equation for interest?

A = P(1 + rn)^(nt) A is final amount P is initial principal balance R is interest rate N is the number of times interest is applied over the period T is the number of time periods elapsed

Unit circle definition

A circle represented as x^2 + y^2 = 1, or a circle with radius 1 centered on the origin

What are piecewise functions?

A function defined by applying different equations to different parts of the domain

What is an inverse function?

A function that is derived by swapping the x and y values in a function. Denoted as f^-1(x).

What is a one-to-one function?

A function where each output is assigned to ONLY one input.

What is a limit?

A limit describes how the outputs of a function behave as we approach an input In other words, the y-value that a function approaches as x approaches a certain value Ex: In the equation sin(x)/x, as x=>0, f(x)=> 1

What is a neutral function?

A neutral function is where f(-x) is equal to neither f(x), nor -f(x). It becomes a new function entirely

What is a cosine

Adjacent side/hypotenuse

What is cotangent

Adjacent side/opposite side

What is the equation to find when an investment will multiply in value?

Ae^rt = nA A is the initial amount nA is the initial amount multiplied by n e is euler's number (2.71828...) r is the interest rate in decimal form t is the time elapsed

What is the definition of Whole Numbers?

All Positive integers including 0. Ex: 0, 1, 2, 3,...613,...

What is the definition of natural numbers

All Positive integers with the exception of 0. Colloquially referred to as the cardinal or counting numbers. Ex: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11...

What is the definition of Negative Numbers?

All numbers that are less than 0 Ex: -735,... -5, -4, -3, -2, -1

What is the definition of a Real Number?

All rational and Irrational numbers; any non-imaginary number

Domain of tangent

All real numbers except for x = π/2 + nπ, where n is an integer. On these intervals, the function is undefined

What is the definition of an even function?

An even function is a function where f(-x) = f(x). Always symmetric about the y-axis.

What is the definition of an odd function?

An odd function is a function where f(-x) = -f(x). Always symmetric about the origin

What is the definition of Integers

Any number that can be expressed without fractional components. Ex: 1, 0, and 8120 are all integers, but 2.1 is not an integer. It has a fractional component, thus it is not an integer.

What is the definition of Rational Numbers?

Any number that can be represented as a ratio between two integers a/b where b is NOT equal to 0. They may also be represented as infinitely repeating decimals, so long as it is possible to put it in rational fractional form Ex: 8/10, 4, 1, 934, 5, 0, 328734029/123, 9/4, 1/3, (0.33333)

What are complex numbers?

Any number that can be written as A + Bi, where A and B are any real number

What is the definition of Irrational Numbers?

Any number that cannot be represented as a ratio of two integers and can only be represented as non-repeating decimals. Ex: √2, π, e, √101

How do you find instantaneous velocity?

By drawing a tangent to the curve, and then finding the slope/instantaneous speed of that tangent.

How can you guesstimate the end behavior of a graph?

Check the overall degree of the function first. If the highest degree of the numerator minus the denominator is even, then the end behavior is usually the same for both sides. If it's odd, than usually it's different for both sides. Then check the sign of both of the variables with the highest degree. If it is +/+ or -/-, then the end behavior will be positive for the positive x axis. If it is -/+ or +/-, then the end behavior will be negative on the positive axis.

How to solve inverse trig functions

Ex: Cos^-1(3/5) = x x is in the first quadrant From this, we know that cos(x) is 3/5, meaning the "adjacent" value is 3 and the hypotenuse value is 5. Using pythagoreans theorem, we find 3^2 + b^2 = 5^2, so b is 4. You should also know this is a 3-4-5 unit triangle. From this, you have the info to solve for all other trig values cos(x) = 3/5 sin(x) = 4/5 tan(x) = 4/3 sec(x) = 5/3 csc(x) = 5/4 cot(x) = 3/4

If for point f(x) we have point (a, f(a)) where a is a constant, then we know (f(a), a) exists on f^-1(x). The derivative at (f(a), a) is the reciprocal of (a, f(a)), though not necessarily perpendicular

Ex: f(1) = 7, f'(1) = 3/5, find (f^-1)'(7) You know the slope at f'(1) is 3/5, therefore at the inverse function, the slope is the reciprocal. Therefore, (f^-1)'(7) = 5/3

True or false: 1/f(x) = f^-1(x)

False. f^-1(x) is to denote an inverse function, not a 1/x function.

How do you mathematically represent that a cusp exists?

Find the limit of the derivative as x => c+ and x => c-, and just show they approach different infinities from each side

How do you mathematically represent a vertical tangent?

Find the limit of the derivative as x => c+ and x => c-, and just show they approach the same infinity

How do you mathematically represent that a corner exists?

Find the limit of the derivative as x => c+ and x => c-, and just show they are not equal.

Cheat for remembering radian values of cos for degrees

For Cos write every 15 degrees an integer (4, 3, 2, 1, 0), take the square root of it, then divide by two

Cheat for remembering radian values of sin for degrees

For Sin write every 15 degrees an integer (0, 1, 2, 3, 4), take the square root of that number, then divide by two

When is function continuous at point c?

Functions are continuous if and only if at point c 1. f(c) exists 2. Lim f(x) exists as x => c 3. Lim f(x) as x => c is equal to f(c)

What is secant

Hypotenuse/adjacent side

What is cosecant

Hypotenuse/opposite side

What is the Law of Sines?

If ABC is a triangle with sides a, b, and c, then a/SinA = b/SinB = c/SinC or SinA/a = SinB/b = SinC/c Used with either ASA or SSA triangles

What is the Law of Cosines?

If ABC is a triangle with sides a, b, and c, then a^2 = b^2 + c^2 - 2(b)(c)cosA b^2 = a^2 + c^2 - 2(a)(c)cosB c^2 = a^2 + b^2 - 2(a)(b)cosC Used with either SAS or SSS triangles

What is Heron's Formula?

If ABC is a triangle with sides a, b, and c, then area of a triangle = square root of s(s-a)(s-b)(s-c) where s = (a + b + c)/2

What is the Law of Tangents?

If ABC is a triangle with sides a, b, and c, then (a + b)/(a - b) = tan[(A + B)/2]/tan[(A - B)/2] Used with SAS triangles

What is one sneaky way to find horizontal asymptotes?

If both polynomials are the same degree, divide the coefficients of the highest degree terms. Ex: (6x^2 - 3x + 4)/(2x^2 - 8), both polynomials are 2nd degree, so the asymptote is at y = 6/2 = 3 If the polynomial in the numerator is a lower degree than the denominator, the x-axis (y = 0) is the horizontal asymptote If the polynomial in the numerator is a higher degree than the denominator, there is no horizontal asymptote. There is a slant asymptote

How do you know if two composite functions are inverses?

If you solve the equation and both f[g(x)] and g[f(x)] are both equal to x, then the functions are inverses

How do you find the angle between Two Vectors?

If θ is the angle between two nonzero vectors u and v, then cos(θ) = (u*v) / |u|*|v| or u*v = cos(θ) * (|u|*|v|)

How does phase shift affect the sinusoidal graph

In (x-c), as c increases, graph shifts to the right, as c decreases, graph shifts to the left

How is a one-to-one function different from a normal function?

In a normal function, multiple inputs can yield the same output, such as in x^2

What conditions must be met for a two-sided limit to exist?

In order for a two-sided limit to exist, both one-sided limits must exist and equal each other. lim f(x) as x=> c- must be equal to lim f(x) as x=> c+

Find the limit of f(x) = sin(1/x) as x => ∞

Isolate 1/x temporarily. You know that as a x => ∞, 1/x approaches 0. Therefore, you can "imagine" sin(0) instead of sin(1/x). What does the sin of 0 approach? Zero. Therefore, we know that as x => ∞, sin(1/x) should approach 0

What does d/dx f(x) mean

It just means "Find the derivative of f(x)"

What does dy/dx mean?

It just means find the derivative.

What is IVT?

It stands for the Intermediate Value Theorem (IVT). The theorem is that, if a function f(x) is continuous on the closed interval [a, b], then f(x) takes on every value between f(a) and f(b)

How can you brute force write a log base in a calculator?

Just write

Find the limit of f(x) = (1/x+3) at x => -3+

Keep in mind that -3 is a vertical asymptote, so the answer is not defined, it must be ∞ or -∞, but random guessing will get us nowhere. What we can do is just plug in a few values. 1/-2+3 is 1, 1/-2.5+3 is 2, 1/-2.9+3 is 10, so we can see the values are increasing exponentially, so more likely than not, the limit approaches ∞ Therefore, f(x) => ∞ as x => -3+

The following limits are given to you: 1. lim f(x) = 2, x => 1 2. lim f(x) = -∞, x => -2+ 3. lim f(x)= -1, x => ∞ 4. lim f(x) = 0, x => -∞ 5. lim f(x) = ∞, x => -2- 6. lim f(x) = ∞, x => 5+ 7. lim f(x) = ∞, x => 5- Sketch a y = f(x) that satisfies these equations

Keep in mind that 3 and 4 are horizontal asymptotes. Draw these out. Then, notice that 2, 5, 6, and 7 are vertical asymptotes. Draw these. then, use 1 and try to connect the dots

How is a one-sided limit denoted?

Left hand (L => R): lim f(x), x => c- Right hand (L <= R): lim f(x), x => c+ Keep in mind the + and - signs are denoted to look almost like exponents

How do you find Secant

Let cos(t)=x. Sec(t)=1/x, where x≠0

How do you find cotangent?

Let sin(t)=y, cos(t)=x, and x≠0. Cot(t)=x/y, where y≠0

How do you find cosecant

Let sin(t)=y. Csc(t)=1/y, where y≠0

What are the Dot Product identities?

Let u, v, and w be vectors in the plane or in space and let c be scalar u*v = v*u 0*v = 0 u*(v + w) = u*v + u*w v*v = |v|^2 c(u*v) = cu*v = u*cv

How can we apply Logarithms?

Logarithms can be used to solve problems regarding exponents. Ex: 3^x = 2 x= log(2)/log(3) or x= ln(2)/ln(3)

If the limit is an infinite, do you put infinity or DNE?

Mathematically, both are acceptable. The AP exam may put either in the multiple choice section.

what are the free fall constants?

Metric units: g=9.8 m/s^2, s=1/2*(9.8)t^2 = 4.9t^2 English units= 32 ft/sec^2, s=1/2*(32)t^2 = 16t^2

What is the equation to model population growth?

N(t) = N*e^(rt) N(t) is the final population N is the initial population e is euler's number (2.71828...) T is the time periods elapsed R is the percentage growth

Is g[f(x)] necessarily equal to f[g(x)]?

No. Certain functions yield different results for g[f(x)] as opposed to f[g(x)]

What is the definition of Imaginary numbers?

Numbers that can be written as a real number multiplied by the imaginary unit "i". "i" is defined as √-1. Since a root of a negative number is not known, we use "i" as a placeholder Ex: √-9 = 3i

How many decimal places should you answer with?

On the AP exam, always answer with 3 rounded decimal places or more, but no less!

When can you round a decimal in a problem?

Only at the end. It must also contain at least 3 decimal points.

Which discontinuities does extended functions work for?

Only removable/hole discontinuities.

What is tangent

Opposite side/adjacent side

What is a sine

Opposite side/hypotenuse

What is the period of a sinusoidal function

Period is the distance between two points that repeat the sinusoidal function

How do you use point slope form?

Point slope form is useful if the slope m and a point (x1, y1) through which it passes are known. Form: y - y1 = m(x - x1) y and x do not change, y1 and x1 do change. m represents the known slope

Where are the trigonometric functions positive

Remember: All Students Take Calculus >All = all trig functions >Students = Sin functions (Sin, Csc) >Take = Tan functions (Tan, Cot) >Calculus = Cos functions (Cos, Sec)

What are the 6 trigonometric terms

Sine Cosine Tangent Cosecant Secant Cotangent

Which trigonometric functions are reciprocal?

Sine and Cosecant Cosine and Secant Tangent and Cotangent

What is the sandwich theorem?

Suppose that g(x) ≤ f(x) ≤ h(x), and lim g(x) = lim (h(x) = L as x => c, then lim f(x) = L as x => c Assume L and c are both real numbers

Converting degrees to radians

Take the degree and multiply it by π/180. Ex: 540 degrees is set to 540π/180, or 3π. 540 degrees is equal to 3π radians

Converting radians to degrees

Take the radian measure and multiply it by 180/π. Ex: 2π/5 rads is transformed to 360π/5π, or 72. Answer is 72 degrees

Solve log3(729) (Log base 3 of 729)

The answer is 6. 3 to the power of 6 is 729.

What is the amplitude of a sinusoidal function

The distance of the extrema to the midline

What is the phase shift of a sinusoidal function

The distance that the function "shifts" by horizontally

What is the vertical shift

The distance that the function "shifts" the midline vertically

What is the greatest integer function?

The greatest integer function is a function that gives the largest integer which is less than or equal to x. Denoted as f(x) = int(x)

What is the midline of a sinusoidal function

The imaginary line that the sinusoidal function passes through. Think half of the distance between extremas.

Definition of reference angle

The reference angle is the acute angle θ' formed by the terminal side of θ and the horizontal axis. θ' = 2pi - θ (degrees) θ' = 360 - θ (degrees) Ex: at θ=300, the reference angle θ' is 360-300. θ' is 60 degrees

Find the limit of f(x) = ((2+x)^3 - 8)/x as x approaches 0

There are two ways of solving this problem. You can either plug in values and see the result, but this is very time consuming, and doesnt work very well if you dont have a calculator. What you do is you'll expand and simplify the numerator, and you will end up with ((x^3 + 6x^2 + 12x)/x, which simplifies to (x(x^2 + 6x + 12))/x Now we have the means to remove the denominator, where we can finally get x^2 + 6x + 12 Using substitution, we find that the limit of f(x) as x approaches 0 is 12.

What is a derivative graph?

Think of it as a line telling you the tangent line slope. For example, speed.

True or false: The inverse of a function multiplied by its original function always yields x

True. f*f^-1(x) = x f^-1*f(x) = x

What are the types of discontinuities

Type 1 is a hole/removeable discontinuity. In this discontinuity, the limit of f(x) as x => c exists, but f(c) is not equal to the limit. Type 2 is a jump discontinuity. In this discontinuity, f(c) exists, but the limit of f(x) as x => c does not exist, since the limit as x => c- is not equal to x => c+ Type 3 is an infinite discontinuity, or vertical asymptote. In this discontinuity, either f(x) as x => c+ approaches +/- infinity, or f(x) as x => c- approaches +/- infinity

How can you find the limit on a calculator?

Use the table function and put in values increasingly close to the limit

f(2) = 3 and f'(2) = 5. Find the tangent line that passes through the point where x = 2

We are given the coordinates (2,3), and we know that the tangent line at this point has a slope of 5. As a result, we can plug it all in and get y-3 = 5(x-2)

In the function f(x) = ((x+8)(x-2))/((x+3)(x-2)), as the value of x approaches 2, what does the value of f(x) approach?

We get a removable discontinuity. However, if we remove (x-2) from both the numerator and the denominator of the equation, we get f(x) to equal 2 at x=2 for the new function. However, if the original equation is unchanged, we simply say that as x approaches 2, f(x) approaches, but never reaches, 2.

What are the key things to look out for when graphing a function

Write down the following A (vertical shift in absolute value)= ... B (horizontal shift, not period)= ... C (phase shift)= ... D (vertical translation)= ... Period= 2pi/B Invert/reflect= true/false

What is the formal definition of a Horizontal asymptote?

Y = b is a HA if either lim f(x) = b as x => ∞, or lim f(x) =b as x => -∞

Examples of piecewise function

Y = |x| { -x if x<0 Y = { { 2x if x≥0

Can certain functions cross the horizontal asymptote?

Yes, but not frequently. (Cos(x)/x) + 1 passes through the horizontal asymptote, but at infinite values it approaches the horizontal asymptote.

Find the limit of f(x) = (2x^4 - x^2 + 7)/(x^2 - 2x + 1) as x => ∞

You can't use substitution here, since infinity is an idea, and you can substitute variables for ideas. However, we can imagine the following. At infinite values, we can ignore the second and third values at both the numerator and denominator. Just worry about the terms with the highest degrees. We can guess that it looks something like what (2x^4)/(x^2) would look like, which is equal to 2x^2. In 2x^2, as x approaches infinity, f(x) approaches infinity as well. Therefore, limit of f(x) = ∞ as x => ∞

How do you find a perpendicular line?

You change the sign and flip the fraction. If slope is m, the perpendicular line will be -1/m

What is a composite function

a function obtained from two given functions, where the range of one function is contained in the domain of the second function, by assigning to an element in the domain of the first function that element in the range of the second function whose inverse image is the image of the element

What is a logarithm?

a quantity representing the power to which a fixed number (the base) must be raised to produce a given number.

Even trigonometric functions

cos(-t)=cos(t) sec(-t)=sec(t)

Sec(x) = 4. Solve for x when 0 ≤ x ≤ 2pi

cos(x) = 1/4 using a calculator, you know that x = cos^-1(1/4) = 1.318. To find the second angle, you subtract 1.318 from 2pi. The result is approximately 4.965. Don't forget to ONLY round at the end

What is the chain rule?

d/dx (f(g(x))) = f'(g(x))*g'(x)

What are the derivatives of the trig functions?

d/dx sin(x) = cos(x) d/dx cos(x) = -sin(x) d/dx tan(x) = sec^2(x) d/dx cot(x) = -csc^2(x) d/dx sec(x) = sec(x)tan(x) d/dx csc(x) = -csc(x)cot(x)

Which function acts as its own derivative?

e^x is its own derivative. If you find the derivative, you just end up with e^x again.

Example of a composite function

f(x)= 3x - 4 g(x) = x^2 f[g(x)] =3[g(x)] - 4 =3(x^2) - 4 =3x^2 - 4

How do you find acceleration?

final velocity-initial velocity/time

Find the formula for the extended function that is continuous at x = -2. f(x) = (x^2-4)/x+2

if f(x) = (x^2-4)/x+2, just factor the top. You end up

How do you denote a limit?

lim f(x) = L, x => c

{ 3x^2, x≤0 f(x) ={ { 5 + x, x>0 Find the limit of 0, 0+, and 0-

lim f(x) as x=> 0+ is 5 lim f(x) as x=> 0- is 0 lim f(x) as x=> 0 DNE

Arc Length Formula

s=r(theta), where r is the radian and theta is the measure of the angle. Ex: 240 degrees is the same as 4π/3 radians. at a radius of 4 inches, multiply 4(4π/3), and you get 16π/3 inches, or roughly 16.76 as the answer

Odd trigonometric functions

sin(-t)=-sin(t) csc(-t)=-csc(t) tan(-t)=-tan(t) cot(-t)=-cot(t)

What are the Sum and Difference Identities?

sin(u + v) = sin(u)cos(v) + sin(v)cos(u) sin(u - v) = sin(u)cos(v) - sin(v)cos(u) cos(u + v) = cos(u)cos(v) - sin(u)sin(v) cos(u - v) = cos(u)cos(v) + sin(u)sin(v) tan(u + v) = [tan(u) + tan(v)] / [1 - tan(u)tan(v)] tan(u - v) = [tan(u) - tan(v)] / [1 + tan(u)tan(v)]

What are the derivatives of Trig functions?

sin(x) = cos (x); cos (x) = -sin(x); tan(x) = sec^2(x); cot(x) = -csc^2(x); sec(x) = sec(x)tan(x); csc(x) = -csc(x)cot(x)

Pythagorean trigonometric identities

sin^2(θ) + cos^2(θ) = 1 sec^2(θ) - tan^2(θ) = 1 csc^2(θ) - cot^2(θ) = 1 1 + tan^2(θ) = sec^2(θ) 1 + cot^2(θ) = csc^2(θ) 1 - cos^2(θ) = sin^2(θ) 1 - sin^2(θ) = cos^2(θ) cos^2(θ) - 1 = sin^2(θ) sec^2(θ) - 1 = tan^2(θ) csc^2(θ) - 1 = cot^2(θ)

Definition of trigonometric functions of any angle

sinθ= y/r cosθ= x/r tanθ= y/x where x≠0 cotθ= x/y where y≠0 secθ= r/x where x≠0 cscθ= r/y where y≠0 r is radius of circle

How do you find speed?

speed= distance/time, or the absolute value of velocity

How do you find the radius of any circle with one pair of coordinates

sqrt(x^2 + y^2) = radius

Cheat for remembering values of tan

take the values of sin and cos numerators for each 15 degrees divide sin by cos

Quotient Identities

tanθ = sinθ/cosθ cotθ = cosθ/sinθ

Linear speed formula

v=s/t, where s is arc length and t is time

Angular speed

w=theta/t, where theta is the measure of the angle and t is time

Vertical asymptotes of tangent

x = π/2 + nπ, where n is any integer

Generic formula for cosine function

y = a*cos(b(x-c))+ d a is amplitude b is 2π/period c is phase shift d is vertical shift

Generic formula for sine function

y = a*sin(b(x-c))+ d a is amplitude b is 2π/period c is phase shift d is vertical shift

Generic formula for tangent function

y = cos(b(x-c))+ d tangent has no amplitude b is π/period c is phase shift d is vertical shift

How do you find the radian measure formula for angle θ

θ=s/r, where s is the length of the arc and r is the radius

180 degrees in radians

π

Period of tangent

π (pi)

15 degrees in radian

π/12 radian

90 degrees in radians

π/2

60 degrees in radians

π/3

45 degrees in radians

π/4

30 degrees in radians

π/6