Final

sin² θ + cos² θ =

sin² θ + cos² θ =

1

1 + tan² θ =

sec² θ

1 + cot² θ =

csc² θ

sin(-θ) =

-sin θ

cos(-θ) =

cos θ

tan(-θ) =

-tan θ

sin(A + B) =

sinAcosB + sinBcosA

sin(A - B) =

sinAcosB - sinBcosA

cos(A + B) =

cosAcosB - sinAsinB

cos(A - B) =

cosAcosB + sinAsinB

sin 2θ =

2sinθcosθ

cos 2θ =

cos² θ - sin² θ = 2cos² θ - 1 = 1 - 2sin² θ

tan θ =

sin θ / cos θ = 1 / cot θ

cot θ =

cos θ / sin θ = 1 / tan θ

sec θ =

1 / cos θ

csc θ =

1 / sin θ

cos² θ =

(1 / 2)(1 + cos 2θ)

sin² θ =

(1 / 2)(1 - cos 2θ)

d/dx (x^n) =

nx^(n - 1)

d/dx (fg) =

fg' + gf'

d/dx (f / g) =

(gf' - fg') / g^2

d/dx f(g(x)) =

f'(g(x))g'(x)

d/dx (sin x) =

cos x

d/dx (cos x) =

-sin x

d/dx (tan x) =

sec² x

d/dx (cot x) =

-csc² x

d/dx (sec x) =

secxtanx

d/dx (csc x) =

-cscxcotx

d/dx (e^x) =

e^x

d/dx (a^x) =

a^x * ln a

d/dx (ln x) =

1 / x

d/dx (Arcsin x) =

1 / √(1 - x²)

d/dx (Arctan x) =

1 / (1 + x²)

d/dx (Arcsec x) =

1 / (|x| * √(x² - 1))

d/dx (Arccos x) =

-1 / √(1 - x²)

d/dx (Arccot x) =

1 / (1 + x²)

d/dx (Arccsc x) =

-1 / (|x| * √(x² - 1))

d/dx [c] =

0

d/dx [cf(x)] =

cf'(x)

∫a dx =

ax + C

∫x^n dx =

(x^n+1) / (n + 1) + C, n ≠ 1

∫1 / x dx =

ln |x| + C

∫e^x dx =

e^x + C

∫a^x dx =

a^x / ln a + C

∫ln x dx =

xln x - x + C

∫sin x dx =

-cos x + C

∫cos x dx =

sin x + C

∫tan x dx =

ln |sec x| + C = -ln |cos x| + C

∫cot x dx =

ln |sin x| + C

∫sec x dx =

ln |sec x + tan x| + C

∫csc x dx =

-ln |csc x + cot x| + C

∫sec² x dx =

tan x + C

∫secxtanxdx =

sec x + C

∫csc² x dx =

-cot x + C

∫cscxcotx dx =

-csc x + C

∫tan² x dx =

tan x - x + C

∫1 / (a² + x²) dx =

(1 / a)(Arctan (x / a)) + C

∫1 / (√(a² - x²)) dx =

Arcsin (x / a) + C

∫1 / (x * √(x² - a²)) =

(1 / a)(Arcsec (|x| / a)) + C = (1 / a)(Arccos |a / x|) + C

A function y = f(x) is continuous at x = a if...

i) f(a) exists ii) lim (x→a) f(x) exists iii) lim (x→a) f(x) = f(a). Otherwise, f is discontinuous at x = a.

The limit lim(x→a) exists if and only if...

lim(x→a) f(x) = L → lim(x→a⁺) f(x) = L → lim(x→a⁻) f(x) = L

A function y = f(x) is even if...

f(-x) = f(x) for every X in the function's domain. Every even function is symmetric about the y-axis.

A function y = f(x) is odd if...

f(-x) = -f(x) for every X in the function's domain. Every odd function is symmetric about the origin.

A function f(x) is periodic with period p(p > 0) if...

f(x + p) = f(x) for every value of X.

Intermediate-Value Theorem:

A function y = f(x) that is continuous on a closed interval [a,b] takes on every value between f(a) and f(b).

The period of the function y = Asin(Bx + C) or y = Acos(Bx + C) is...

2π / |B|

The amplitude of the function y = Asin(Bx + C) or y = Acos(Bx + C) is...

|A|

The period of the function y = tan x is...

π

If f is continuous on [a,b] and f(a) and f(b) differ in sign, then...

the equation f(x) = 0 has at least one solution in the open interval (a,b).

lim(x→±∞) f(x) / g(x) = 0 if...

the degree of f(x) < the degree of g(x).

lim(x→±∞) f(x) / g(x) = ∞ if...

the degree of f(x) > the degree of g(x).

lim(x→±∞) f(x) / g(x) = [c] if...

the degree of f(x) = the degree of g(x).

A line y = b is a horizontal asymptote of the graph y = f(x) if either...

lim(x→∞) f(x) = b or lim(lim(x→-∞) f(x) = b. (Compare degrees of functions in faction)

A line x = a is a vertical asymptote of the graph y = f(x) if either...

lim(x→a⁺) f(x) = ±∞ or lim(x→a⁻) f(x) = ±∞. (Values that make the denominator 0 but not the numerator)

If (x₀,y₀) and (x₁,y₁) are points on the graph of y = f(x), then the average rate of change of y with respect to x over the interval [x₀,x₁] is...

((f(x₁) - f(x₀)) / (x₁ - x₀)) = ((y₁ - y₀) / (x₁ - x₀)) = ∆y / ∆x.

If (x₀,y₀) is a point on the graph of y = f(x), then the instantaneous rate of change of y with respect to x at x₀ is...

f'(x₀).

Definition of a Derivative:

f'(x) = lim(h→0) (f(x + h) - f(x)) / h or f'(a) = lim(x→a) (f(x) - f(a)) / (x - a). (The latter definition of the derivative is the instantaneous rate of change of f(x) with respect to x at x = a.)

lim(n→∞) (1 + (1 / n))^n =

e

lim(n→0) (1 + n)^(1 / n) =

e

Mean Value Theorem:

If f is continuous on [a,b] and differentiable on (a,b), then there is at least on number c in (a,b) such that (f(b) - f(a)) / (b - a) = f'(c).

Extreme-Value Theorem:

If f is continuous on a closed interval [a,b], then f(x) has both a maximum and minimum on [a,b].

To find the maximum and minimum values of a function y = f(x), locate...

i) the points where f'(x) is zero OR where f'(x) fails to exist ii) the end points, if any, on the domain of f(x). iii) Plug those values into f(x) to see which gives you the maximum and which gives you the minimum values (the x-values is where that value occurs).

Let f be differentiable for a < x < b and continuous for a ≤ x ≤ b, i) If f'(x) > 0 for every x in (a,b), then f is (increasing/decreasing) on [a,b]. ii) if f'(x) < 0 for every x in (a,b), then f is (increasing/decreasing) on [a,b].

i) increasing ii) decreasing

Suppose that f''(x) exists on the interval (a,b), i) If f''(x) > 0 in (a,b), then f is concave (upward/downward) in (a,b). ii) If f''(x) < 0 in (a,b), then f is concave (upward/downward) in (a,b).

i) upward ii) downward

To locate the points of inflection of y = f(x), find...

the points where f''(x) = 0 OR where f''(x) fails to exist. Then, test these points to make sure that f''(x) < 0 on one side and f''(x) > 0 on the other.

If a function is differentiable at the point x = a, it (is/is not) continuous at that point. The converse is (true/false). Continuity (does/does not) imply differentiability.

i) is ii) false iii) does not

Euler's Method:

Starting with the given point (x₁,y₁), the point (x₁ + ∆x, y₁ + f'(x₁,y₁)∆x) approximates a nearby point. (This approximation may then be used as the starting point to calculate a third point and so on.)

Logarithm functions grow (slower/faster) than any power function (xⁿ).

slower

Among power functions, those with higher powers grow (faster/slower) than those with lower powers.

faster

All power functions grow (faster/slower) than any exponential function (a^x, a > 1).

slower

Among exponential functions, those with larger bases grow (faster/slower) than those with smaller bases.

faster

We say, that as x→∞: f(x) grows (faster than/slower than/at the same rate as) g(x) if lim(x→∞) f(x) / g(x) = ∞ or if lim(x→∞) g(x) / f(x) = 0

faster than

We say, that as x→∞: f(x) grows (faster than/slower than/at the same rate as) g(x) if lim(x→∞) f(x) / g(x) = L ≠ 0.

at the same rate as

L'Hôpital's Rule:

If lim(x→a) f(x) / g(x) is of the form 0 / 0 or ∞ / ∞, and if lim(x→a) f'(x) / g'(x) exists, then lim(x→a) f(x) / g(x) = lim(x→a) f'(x) / g'(x).

Two functions f and g are inverses of each other if...

f(g(x)) = x for every x in the domain of g and g(f(x)) = x for every x in the domain of f.

To test if a graph has an inverse, use the...

horizontal line test

If f is strictly increasing or decreasing in an interval, then f (has/does not have) an inverse.

has

i) If f is differentiable at every point on an interval I, and f'(x) ≠ 0 on I, then g = f⁻¹(x) (is/is not) differentiable at every point of the interior of the interval f(I). ii) If the point (a,b) is on f(x), then the point (b,a) (is/is not) on g = f⁻¹(x).

i) is ii) is (Furthermore, g'(b) = 1 / f'(a).)

Properties of y = e^x:

i) y = e^x is the inverse function of y = ln x.
ii) The domain is -∞ < x

Properties of y = ln x:

i) The domain is x > 0. ii) The range is -∞ < y <∞. iii) y = ln x is continuous and increasing everywhere on its domain. iv) ln (a * b) = ln a + ln b. v) ln (a / b) = ln a - ln b. vi) ln a^r = r * ln a. vii) y = ln x < 0 if 0 < x < 1. viii) lim (x→∞) ln x = ∞ and lim(x→0⁺) ln x = -∞. ix) log_a x = ln x / ln a x) d/dx ln f(x) = f'(x) / f(x) and d/dx ln x = 1 / x