Calc Unit 2 Exam

Derivatives and graphs

when f(x) is increasing

f’(x) is positive

when f(x) is concave up

f’(x) is increasing and f’’(x) is positive

when f(x) is decreasing

f’(x) is negative

when f(x) is concave down

f’(x) is decreasing and f’’(x) is negative

when f(x) has a inflection point

f’(x) has a local min/max and f’’(x) has a zero

when f(x) has a vertical tangent

f’(x) is undefined

if f(x) has a sharp corner or cusp

f’(x) is undefined

when f(x) is discontinuous

f’(x) is undefined

where f(x) has local min/max

f’(x) is 0

where f(x) has an inflection point

f’(x) has a local max/min

inflection point

when the graph changes direction

acceleration

f’’

velocity

f’

position

f

f’(sin x) =

cos x

f’(cos x) =

-sin x

f’(sec x) =

sec x tan x

quotient rule

f'(x)g(x)-g'(x)f(x)/g(x)²

product rule

f’(x)g(x)+f(x)g’(x)

In e

1

d/dx e^x

e^x

exponential function

d/dx a^x = ln (a)(a^x)

f(x) has a maximum

f’(x) switches from positive to negative at a 0

f(x) has a minimum

f’(x) has a 0 and changes from negative to positive