Calc Exam 1 Material

Cos

Cos

Adjacent / Hypotenuse

Sin

Opposite / Hypotenuse

Tan

Opposite / Hypotenuse

Sec

1 / cos

hypotenuse / adjacent

Csc

1 / sin

hypotenuse / opposite

Cot

1 / tan

adjacent / opposite

cos / sin

Unit Circle

• circle with the radius of 1 centered around the origin

• positive direction is counter clock wise

• negative direction is clock wise

Unit Circle coordinates

cos = x

sin = y

Unit Circle first quadrant

• When flipping values from sin to sec or cos to csc you keep the negative and just flip the number

note

Pythagorean identities

sin^2 + cos^2 = 1

1 + tan^2 = sec^2

cot^2 + 1 = csc^2

How do you get the Pythagorean identities?

You take the first on of sin^2 + cos^2 = 1 and you divided it by sin and then by cos to get the other two

Exponential Function

form y = b^x

horizonal asymptote at y=0

Logarithmic Function

logbX

vertical asymptote at x=0

Natural Log

log with base e

ln

Common Log

log with base 10

log10X

logb(XY)

logb(X) + logb(Y)

logb(X / Y)

logb(X) - logb(Y)

logb(X^r)

rlogb(X)

logbB = 1

ln(e) = 1

log10 = 1

note

Velocity

change in x / change in t

change in position over change in time

Instantaneous velocity

the velocity at one particular point

Average Rate of Change

f(x2) - f(x1) / x2 - x1

Secant line

a line which passes through two points on the curve

can find its slope by using the average rate of change equation

To determined the rate of change at one point instead of the rate of change between two points we need the instantaneous rate of change

note

The limit is the instantaneous rate of change

note

limit

suppose f(x) is defined when x is near the number a (this means that f is defined on some open interval contain a, except possible at a itself)

lim f(x) = L

x→ a

the limit of f(x) as x approaches a equals L

lim f(x) = L

x → a-

the left hand limit of f(x) as x approaches a is equal to L if we can made the values of f(x) arbitrarily close to L by restricting x to sufficiently close to a with x < a

lim f(x) = L

x → a+

the right handed limit of f(x) as x approaches a is equal to L if we can make the value of f(x) arbitrarily close to L by restricting x to be sufficiently close to a with x > a

if lim f(x) = L

x → a

then lim f(x) = lim f(x) = L

x → a- x → a+

if lim f(x) does not = lim f(x) = L

x → a- x → a+

lim f(x) = DNE

x → a

The outputs for the limit is the y output for the function

note

lim f(x) = infinity

x → a

let f(x) be defined on both sides of a expect possibly at a itself

means that the values of f(x) can be made arbitrarily large ( as large as we please by taking x sufficiently close to a, but not equal to a

lim f(x) = - infinity

x → a

means that the values of f(x) can be made arbitrarily large and negative by taking x sufficiently close to a but not equal to a

There is a vertical asymptote if one of these statements is true

lim f(x) = infinity

x → a

lim f(x) = infinity

x → a-

lim f(x) = infinity

x → a+

lim f(x) = - infinity

x → a

lim f(x) = - infinity

x → a-

lim f(x) = - infinity

x → a+

if y = f(x) / g(x) and g(a) = 0 while f(a) does not = 0 then f(x) has a vertical asymptote at x = a

you plug it what the lim is approaching towards to the f(x) function if the numerator is equal to positive and the denominator equal to positive than the lim is approaching positive infinity. If the numerator is approaching positive and the denominator is zero you are approaching positive infinity. If the numerator is negative and the denominator is negative, you are approaching negative infinity. If the numerator is negative and the denominator is zero you are approaching neative infinity

This is only if there is one function in f(x)

lim [f(x) + g(x)]

x → a

lim f(x) + lim g(x)

x → a x → a

lim [f(x) - g(x)]

x → a

lim f(x) - lim g(x)

x → a x → a

lim c[f(x)]

x → a

lim f(x) * c

x → a

lim [f(x) * g(x)]

x → a

lim f(x) * lim g(x)

x → a x → a

lim [f(x) / g(x)]

x → a

lim f(x) / lim g(x) when g(x) is not equal to 0

x → a x → a

lim f(x)^n

x → a

[lim f(x) ]^n

x → a

lim square root (f(x))

x → a

square root [lim f(x)]

x → a

Direct Substitution Property

Let f(x) be a polynomial or a rational function and a is in the domain of f then

lim f(x) = f(a)

x → a