2.1-2.5 quiz

Question 1

sin x ‘

Answer 1

cos x

Question 2

cos x’

Answer 2

-sin x

Question 3

tan x’

Answer 3

sec^2 x

Question 4

cot x

Answer 4

-csc^2 x

Question 5

sec x’

Answer 5

sec x(tan x)

Question 6

csc x’

Answer 6

-csc x(cot x)

Question 7

product rule

h(x) = f(x)(g(x))

Answer 7

h’(x) = f x g’ + g x f’

Question 8

quotient rule

h(x) = f(x)/g(x)

Answer 8

h’(x) = g x f’ - f x g’ / g^2

Question 9

chain rule

h(x) = f(g(x))

Answer 9

think like peeling an onion

h’(x) = f’(g(x)) x g’(x)

Question 10

sin 2x

Answer 10

2sinxcosx

Question 11

consider the curve : formula

show that dy/dx equals formula

Answer 11

basically dy/dx is the formula of the curve

Question 12

what is differentiation

Answer 12

where the slope equals the tangent line

Question 13

the process of finding a derivative is called

Answer 13

differentiation

Question 14

what does differentiability imply

Answer 14

continuity

Question 15

the derivative doesn’t exist at

Answer 15

discontinuities
cusps
when the tangent becomes vertical(straight up and down line part)

Question 16

power rule

d/dx[x^n] =

Answer 16

nx^n-1

Question 17

power rule on constants

d/dx 5

Answer 17

0

Question 18

find equation of tangent line to graph of f(x) at the point(x,y)

how to solve

Answer 18

find derivative of f(x) to solve for the slope at any point on the function

then plug in (x,y) to find the slope for that particular point

Question 19

implicit differentiation procedure

Answer 19

differentiate each term with respect to x (dy/dx)

nivel all terms contains dy/dx to the left side of the equation and all other terms to the right side

factor out dy/dx on left side of equation

solve for dy/dx

Question 20

product vs chain tule for trig functions:

Answer 20

if the angle theta is just x it is product rule; if it is 2x however or (2x-1) whether it has parentheses or not it is chain rule

Question 21

what happens to the “y” with implicit differentiation

Answer 21

it becomes dy/dx

with respect to x

Question 22

what happens to x in implicit differentiation

Answer 22

nothing with respect to x becomes 1 so the x is left alone

Question 23

L’hospitsl rule

How to wrote AP TESTA ND FRQ

Answer 23

1) show f(x) and g(x) are continuos and differentiable
(Write f(x) and g(x) separately!)

2) show lim f(x) = 0 and lim g(x) = 0
x->a x-> a
3) using L’Hospitals rule
lim f(x)/ g(x) = lim f’(x)/g’(x) plug in number
x->a x-> a

Question 24

L’hospital rule idea

Answer 24

If in indeterminate form find derivative of og equation and replug in pts, if still indeterminate keep finding derivatives

Question 25

finding second derivative

Answer 25

determine what is f(x) and g(x) and f’(x) and g’(x) [still use implicit differentiation plug in dy/dx] then solve for the dy/dx (it equals previous derivative D’)

Question 26

derivative of “xy”

Answer 26

use product rule (x*dy/dx + y*1) keep subtraction sign outside of parentheses and distribute once finished differentiating

Question 27

constant on right side of equal sign

Answer 27

don’t forget to differentiate it!! equals 0

Question 28

how to write second differentiation in (dy/dx) format

Answer 28

d^2y/dx^2 ??

Question 29

when solving any typa derivative problem

Answer 29

first identify and write down f(x), f’(x), g(x), g’(x) then identify what rule to use and write that down write down formula for rule too if needed

Question 30

differentiation of 4xy

Answer 30

group (4x) as f(x) and (y) as g(x) becuase 4 is only multiplied to the y(not distributed to the whole thing)

Question 31

when doing implicit differentiation only put derivatives on y when

Answer 31

differentiating for y for example in product rule where f(x)*g’(x) if f(x) = y DONT put dy/dx becuase u aren’t supposed to find the derivative of f(x) differentiating is ONLY used when u need to find the derivative of THAT PARTICULAR variable

Question 32

is cos^2(x) composite

Answer 32

yes inner function: cos(x) outer function: x^2

Question 33

derivative of sec(4x)

Answer 33

sec(4x)tan(4x)*4

Question 34

trig function inside of trig function means how many parts of an onion

Answer 34

3 layers of onion

Question 35

``` lim f(x+h) -f(x)/ x-h x->h ``` identify derivative of f(x) @ any point

Answer 35

derivative of f(x) @ any point is the derivative upper right part of the fraction f’(x)

Question 36

instantaneous rate of change

Answer 36

find derivative | then plug in point

Question 37

average rate of change

Answer 37

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