Calc Final

To find the range of function

Change the variables

Change all X variables in original function to Y

Set the function equal to X

Solve for Y

G(x) = √(3-x) + √(2+x)

1. Set 3-x ≥ 0 / Solve for X / x ≤ 3
2. Set 2+x ≥ 0 / Solve for X / x ≥ -2
3. x ≤ 3 & x ≥ -2
4. -2 ≤ x ≤ 3
5. Domain: \[-2,3\]

Finding domain, avoid ______ in denom

Zero 0

Finding domain, avoid ______ in square root

Negatives

Finding domain, avoid ______ in log

0 and negatives

Even function

f(-x) = f(x)

Odd function

f(-x) = - f(x)

Y intercept of exponential functions

y = ab

Vertical shifts upward & downward

y = f(x) + c. <— up

y = f(x) - c. <— down

Horizontal shifts

y = f(x-c). <— right

y = f(x+c) <— left

Reflect across x axis

Y = - f(x)

Across the y axis

Y = f(-x)

Vertical stretch & shrink formula

Y = cf(x)

Vertical stretch

C> 1

Y = cf(x)

Vertical shrink

0 < c < 1

Y = cf(x)

Horizontal stretch

F(cx)

0 < c < 1

Horizontal shrink

Y = f(cx)

C>1

Find domain: log (x-2) / √(x^2 - x)

1. X-2 > 0 / x >2
2. X^2 - x ≥ 0 —> x(x-1) ≥ 0 / x ≥ 0,1 bad values


1. Domain = x >2

Domain of composite function { f o g)

f o g = f(g(x))

The domain is the domain of G

When finding limit and there’s a square root in function

Use the conjugate

Absolute value limit from the negative side

X → 2- |x-2| —> -(x-2)

Absolute value limit from the positive side

X → 2+ |x-2| —> (x-2)

when lim x → a = 0/0

Square root conjugation

Break down absolute values

Common Denominators

Lim x → 0 {sin x / x} =

1

Lim x → 0 { x / sin x } =

1/ ___ in other words, reciprocal

Squeeze Thrm

f(x) ≤ g(x) ≤ h(x)

f(x) = L & h(x) =L

g(x) = L

The Lim x—> 0 sin (1/x)

DNE

Sin 1/x is always between

\-1 < sin 1/x < 1

show that Lim x—> 0 {x^2 Sin(1/x) } = 0

1. Establish that Sin 1/x → -1 ≤ Sin 1/x ≤ 1
2. Multiply everything -1 ≤ Sin 1/x ≤ 1 by x^2
3. Lim x → 0 { -x^2 ≤ x^2 Sin 1/x ≤ x^2 }
4. Lim x→0 x^2 = 0
5. Lim x→0 -x^2 = 0
6. Answer= 0

Indeterminate forms

0/0

If lim x→a f(x) ≠ 0 and

lim x→a g(x) = 0, then f(x) / g(x)

could be -∞ or ∞ or DNE

Hole of the function

1. set denom = 0
2. the term the cancels out in the numer. and denom. is where the hole is
3. set the cancel term equal to zero
4. solve for x
5. x is where the hole is

\
cancelled out → (x-2)

x - 2 = 0

x =2 is the hole

lim x→ 0 cos x - 1 / x

0

Horizon Asymp

look at higher power term in numerator and denom.

degree of num > degree of denom

no HA so answer could be ∞ or -∞

degree of num < degree of denom

y = 0

x^3 + a^3

(a-b) (a^2 + ab + b^2)

degree of num = degree of denom

coeffic of num / coeffee of denom

with HA, you must look at

the term with the HIGHEST degree

when R>0 at lim x→∞ 1/xᴿ

lim x→∞ 1/xᴿ = 0

when lim x →∞ f(x) / g(x) = ∞/∞

limits as x → ∞ or -∞

Factor out the highest power of X of numerator.

Factor out the highest power of X of denom.

You do this by dividing every term by highest term

Then use limit laws

limits at infinity mean

HA

lim x→a C/0-

\-∞

lim x→a C/0

∞

When question asks to show that the function has roots/x intercept,

IVT

f(a) & f(b) → you need one negative and one positive answer

dy/dx | |

linearization formula

L(x) = f(a) + (f’a) (x-a)

With linearization problems, when it says approximate or estimate followed by a number

L(x) equals that number

You can find the X by L(x) = f(x)

When a in not given in the linearization problems

choose the whole number closest to the estimate

differentials

dy = f’(x)dx

surface area of cube

6a^2

volume of cube

a^3

one to one function never

takes the same y value twice

lim x→ sin x / x

1

lim x→0 (cos x - 1) / x

0

lim x → 0 (sin 7x / 7x )

1

d/dx ln x

1/x

d/dx logₐ \[f(X)\]

1/ f (x) ln a f’(x)

d/dx e ᶠ⁽ˣ⁾

f’(x) eᶠ⁽ˣ⁾

d/dx eᵏˣ

keᵏˣ

2ˣ

x ln 2

inverse formula

1/ f’ \[f ⁻¹(x)\]

with exponential func, x =

eˡⁿ⁽ˣ⁾

d/dx arcsin

1/ sqrt (1-x^2)

d/dx arccos

\- 1/sqrt(1-x^2)

d/dx arctan

1/1+x^2

d/dx arccot

\- 1/1+x^2

d/dx arcsec

1/ |x| sqrt(x^2 - 1)

d/dx arccsc

\- 1/ |x| sqrt(x^2 - 1)

lim x → ∞ x^2sin(1/x^2)

write as fraction:

\
sin (1/x^2) / 1/x^2

\
le hopital

MVT

f(x) is continuous on \[a,b\]

f(x) is differentiable on (a,b)

f(a) = f(b)

f(b) - f(a) / b - a =

f’(c)

Rolle’s Thrm steps

1. check if continuous
2. if differnetiable
3. if f(a) = f(b)

\
Then f’(c) = 0

\

Find global max/min of f(x) on closed interval

1. find derivative of f(x)
2. set equal to zero and find critical values
3. plug in critical values into f(x)
4. largest number is global max & smallest number is global min

Riemann summ width of rectangle

\[a,b\]

b - a / # of rectangles

\

antideriv of x^-1

ln x