derivates memory test 2

Question 1

if derivate exists at a point

Answer 1

1. not discontinuous
2. smooth graph
3. no corner
4. no cusp
5. no vertical tangent

Question 2

average rate of change on [a,b]

Answer 2

b-a

Question 3

definition of derivative

Answer 3

    h

Question 4

notation for instantaneous rate of change (derivative at a point):

Answer 4

d
— [f(x)]
dx

Question 5

derivatives of constant & linear functions

Answer 5

d
— [c] =0 or c
dx

Question 6

sin x

Answer 6

cos x

Question 7

cos x

Answer 7

• sin x

Question 8

tan x

Answer 8

sec ^2 x

Question 9

cot

Answer 9

• csc ^2 x

Question 10

sec x

Answer 10

sec x tan x

Question 11

csc x

Answer 11

• csc x cot x

Question 12

e^ x

Answer 12

e^ x

Question 13

ln x

Answer 13

x

Question 14

ln (x+d)

Answer 14

x+d

Question 15

writing tangent line

Answer 15

slope = f’(a)
point f(a)
y-f(a)=f’(a)(x-a)

Question 16

writing normal line

Answer 16

point: f(a)
slope: -1
—-
f’(a)

y-f(a)=-1
—- (x-a)
f’(a)

Question 17

finding linear approximation

Answer 17

write tangent line
plug in x-value to approximation and find y

Question 18

justifications for linear approximation estimates

Answer 18

a linear approximation (tangent line) is an overestimate if the curve is concave down

a linear approximation (tangent line) is an underestimate if the curve is concave up

Question 19

justification for horizontal tangent lines

Answer 19

dy
—- = 0
dx

Question 20

justification for vertical tangent lines

Answer 20

dy
—- = undefined
dx

Question 21

sin ^-1 x

Answer 21

square rt 1-x^2

Question 22

cos ^-1 x

Answer 22

• ## 1square rt 1-(kx)^2