Unit 2 Differentiation

Question 1

Average Rate of Change

Answer 1

F(x) - f(a)
————-
X - a

Question 2

Average rate of change is the slope of the _________________ line.

Answer 2

Secant

Question 3

Instantaneous Rate of Change

Answer 3

Lim [f(x+h) - f(x)] / h
h ->0

Question 4

Instantaneous rate of change is the slope of the ______________ line.

Answer 4

Tangent

Question 5

Limit Definition of Derivative

Answer 5

F’(x) = lim [(f(x+h) - f(x)] / h
h->0

Question 6

A ________________ gives the slope of the tangent line

Answer 6

Derivative

Question 7

Equation of tangent line at x=a

Answer 7

Y - f(a) = f ‘(a) (x - a)

Question 8

Three way a function is not differential:

Answer 8

1) discontinuity
2) corner/cusp
3) vertical tangent

Question 9

Definition of differentiability

Answer 9

1) derivative exists for each point in domain; smooth line or curve
2) graph looks like a line if zoomed in (local linearity)

Question 10

The Power Rule

Derivative of f(x) = x^n

Answer 10

F’(x) = nx^(n-1)

Question 11

What is true of parallel tangent lines?

Answer 11

Derivatives (slopes) will be equal

Question 12

Constant Derivative Rule

dy/dx C =

Answer 12

0

Question 13

Constant Multiple Rule

dy/dx 3x^4 =

Answer 13

3 * 4x^3 = 12x^3

Question 14

Sum/Difference Derivative Rule

dy/dx (3x^5 + 6x^2 - 7x - 9) =

Answer 14

15x^4 + 12x - 7

Question 15

Horizontal tangent lines have a slope of ___________ because it’s an max/min point.

Answer 15

0

Question 16

_______________________ go through the same point as tangent line, but is perpendicular to tangent line.

Answer 16

Normal lines

Question 17

Two things to check for differentiability of piecewise functions

Answer 17

1) continuity - to make sure the function is continuous
2) equal derivatives - to make sure there is not a corner/cusp in the continuous function

Question 18

Derivative of sin x

Answer 18

Cos x

Question 19

Derivative of cos x

Answer 19

• sin x

Question 20

Derivative of a^x

Answer 20

a^x lna

Question 21

Derivative of e^x

Answer 21

e^x

Question 22

Derivative of log base a of x

Answer 22

(1/x)(1/lna)

Question 23

Derivative of ln x

Answer 23

1/x

Question 24

Product Rule

If h(x)=f(x) g(x), then h’(x) =

Answer 24

h’(x) = f’(x) g(x) + f(x) g’(x)

Question 25

Quotient Rule

If h(x)=f(x)/g(x), then h’(x) =

Answer 25

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

Question 26

Derivative of tan x

Answer 26

sec^2 x

Question 27

Derivative of cot x

Answer 27

-csc^2 x

Question 28

Derivative of sec x

Answer 28

(sec x)(tan x)

Question 29

Derivative of csc x

Answer 29

(-csc x)(cot x)