Unit 2- Differentiation: Definition & Fundamental Properties

Question 1

Avg. RoC on interval [a,b]

Answer 1

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

Question 2

If the x-value of point a = a and x-value of point b = a +h, then what will be the formula used for avg. RoC

Answer 2

f(a+h) - f(a) / h

Question 3

The slope of the secant line can be modeled algebraically by

Answer 3

the avg. RoC formula

Question 4

The slope of the secant line can be modeled algebraically by

Answer 4

taking the lim of the RoC formula as it approaches 0

lim f(a+h) - f(a) / h
h->0

lim f(x) - f(a) / x-a
x->a

Question 5

Dertivative

Answer 5

Expression that calculates the instan. RoC of a function at any given x-value

Question 6

Derivative notation

Answer 6

Can be notated as
f’(x) “f prime of x”
y’ “y prime”
dx/dy “derivative of x with respect to y”

Question 7

Official definition of derivative involving limits

Answer 7

Limit gives an expression that calculates instantaneous RoC of f(x) at any given point

lim f(x+h) - f(x) / h
h->0

Question 8

Equation of tangent line

Answer 8

Line tangent to curve of f(x) at x=a can be represented in point-slope form

y-y1 = m(x-x1) where
y1 = y-value of the point at a
x1 = x-value of point at a
m = slope of the tangent line found by derivative

Question 9

Differentiability

Answer 9

Derivative exists for each point in domain

Graph must be smooth line or curve for derivative to exist

Question 10

Local linearity

Answer 10

Looks like a line when zoomed in

Question 11

Derivative fails to exist at

Answer 11

Discontinuities -> jump or removable
Corner/ cusp -> sharp corners
Vertical tangent -> slope is undefined

Question 12

Differentiability implies

Answer 12

continuity but continuity does not imply differentiability

Question 13

Power rule

f(x) = x^n

Answer 13

f’(x) = n*x^n-1

Question 14

Parallel tangent line

Answer 14

Tangent lines that has the same slope

Question 15

To find parallel tangent lines on two different functions,

Answer 15

set the derivatives of the functions equal to each other and solve for 0

Question 16

Derivative of a constant is

Answer 16

ALWAYS zero

Question 17

Derivative rule: Constant multiple

d/dx cu =

Answer 17

c du/cx
Exp -> y = 3x^2
y’ = 3 * 2x = 6x

Question 18

Derivative rule: Sum

d/dx (u + v) =

Answer 18

du/dx + dv/dx

Question 19

Derivative rule: Difference

d/dx (u - v) =

Answer 19

du/dx - dv/dx

Question 20

Horizontal tangent lines

Answer 20

Slope is always equal to zero

Question 21

To determine where a function has a horizontal tangent,

Answer 21

Set derivative to zero and solve

Question 22

Normal lines

Answer 22

Passes through the same point the tangent line does but it is perpendicular to tangent line

Question 23

To find a normal line,

Answer 23

Perform steps as needed to determine a tangent line

Convert the slope of the tangent line to its negative reciprocal

Question 24

To determine continuity at a point,

Answer 24

Set the left side and right side of the function at that point equal to each other
Substitute same point into left and right side
If sum at left is equal to sum at right, then the function is continous at that point

Question 25

To determine differentability at a point

Answer 25

Set the derivative of the left and right sides of a function equal to each other
If evaluated to the same sum, then the function is differentiable at that point

Question 26

d/dx cos(x) =

Answer 26

-sin(x)

Question 27

d/dx sin(x) =

Answer 27

cos(x)

Question 28

ln (1) =
ln (0) =
e^0 =
e^lna =
ln^ea=

Answer 28

0
undefined
1
a
a

Question 29

d/dx a^x =

Answer 29

a^xln(a)

Question 30

d/dx e^x =

Answer 30

e^x

Question 31

d/dx loga(x) =

Answer 31

1/x * 1/ln(a)

Question 32

d/dx ln(x) =

Answer 32

1/x

Question 33

Product rule

h(x) = f*g

Answer 33

h’(x) = [f’(x)g(x)] + [[f(x)g’(x)]

Question 34

Quotient rule

h(x) = f/g

Answer 34

h’(x) = [f’(x)g(x)] - [[f(x)g’(x)] / g^2

Question 35

d/dx tan(x) =

Answer 35

sec^2(x)

Question 36

d/dx cot(x) =

Answer 36

-csc^2(x)

Question 37

d/dx sec(x) =

Answer 37

sec(x)tan(x)

Question 38

d/dx csc(x) =

Answer 38

-csc(x)cot(x)

Question 39

To evaluate a derivative with a calculator,

Answer 39

MATH -> nDeriv(function, x, specific point)

Question 40

If a point is not in the domain of f, then

Answer 40

it is not in the domain of f’.