calc midterm 2

Question 1

Product Rule

Answer 1

f’g + fg’

Question 2

Quotient Rule

Answer 2

f’g - fg’ / g^2

Question 3

Chain Rule

Answer 3

f’(g) * g’

Question 4

With Chain Rule, it goes

Answer 4

derivative of outer, keep inner, derivative of outer, keep inner, derivative of inner

Question 5

Whenever you see a function raised to a power, do

Answer 5

the CHAIN RULE
Ex: (x+2)⁴

Question 6

What step to do first when seeing a question like this:

(3t - 1)⁴ (2x+1)⁻³

Answer 6

Do the product rule first

Question 7

When the question asks for f’(x), do

Answer 7

implicit differentiation

y’ whenever taking the derivative of y

Question 8

With implicit differentiation, you must

Answer 8

take the derivative of both sides of the equation

SOLVE FOR Y’

Question 9

(ab)ᵐ

Answer 9

aᵐ * bᵐ

Question 10

(a/b)ᵐ

Answer 10

aᵐ / bᵐ

Question 11

(a/b)⁻ᵐ

Answer 11

bᵐ / aᵐ = (b/a)ᵐ

Question 12

a⁻ᵐ / b⁻ᵖ

Answer 12

bᵖ / aᵐ

Question 13

If function fails horizontal line test, then

Answer 13

it’s NOT one-to-one

Question 14

logₐ X = y is equivalent to

Answer 14

aʸ = X

Question 15

logₐ (XY) =

Answer 15

logₐ X + logₐ Y

Question 16

logₐ (X/Y) =

Answer 16

logₐ X – logₐ Y

Question 17

logₐ Xᴿ

Answer 17

R logₐ X

Question 18

logₐ X

Answer 18

log₆ₐₛₑ X / log₆ₐₛₑX

Question 19

lim of ln x as x-> infinity =

Answer 19

infinity

Question 20

lim of ln x as x-> 0 from pos side =

Answer 20

-infinity

Question 21

ln e

Answer 21

1

Question 22

derivative of ln x

Answer 22

1 / x

Question 23

derivative of ln u

Answer 23

1/ u * u’

Question 24

derivative of logₐ X

Answer 24

1/ x ln a

Question 25

derivative of eᵏˣ

Answer 25

k * eᵏˣ

Question 26

derivative of eᶠ⁽ˣ⁾

Answer 26

f’(x) * eᶠ⁽ˣ⁾

Question 27

derivative of Cᵘ
c - constant (1,2,3)
u - variable

Answer 27

Cᵘ * ln (c) * u’

Question 28

When the equation looks like this, (2-x)ˣ = y

Answer 28

bring the x down / take the LN of both sides
PRODUCT RULE & implicit differentiation
remember to plug in the y

Question 29

arcsin range

Answer 29

-pi/2, pi/2

Question 30

arccos range

Answer 30

0, pi

Question 31

arctan range

Answer 31

-pi/2, pi/2

Question 32

arcsin equals

Answer 32

1/sqrt(1-x²)

Question 33

arccos equals

Answer 33

– 1/sqrt(1-x²)

Question 34

arctan equals

Answer 34

1 / 1 + x²

Question 35

even function

Answer 35

f(-x) = f(x)

Question 36

odd function

Answer 36

f(-x) = -f(x)

Question 37

HA occurs when

Answer 37

f’(x) = 0

Question 38

VA occurs when

Answer 38

where the DENOM of function = 0
if not then f is continuous everywhere

Question 39

Inverse formula

Answer 39

1 / f’ [f⁻⁻¹(x)]

Question 40

linearization formula

Answer 40

L(x) =f’(a) (x-a) + f(a)

Question 41

To find the letter “a” in linearization

Answer 41

chose the number closest to the X in L(X)

Question 42

The X in the L(x) is next to the words

Answer 42

Estimate

Question 43

Ex: sin(arccos (1/3))

Answer 43

set Cos y = 1/3
do pythag thrm to find Sin