ap calc ab review

when is a function increasing

f’(x) is positive/above the x axis

when is a function decreasing

f’(x) is negative/below the x axis

when is a function moving to the right

v(t)>0 (f’(x) above the x-axis)

when is a function moving to the left

v(t)<0 (f’(x) below the x-axis)

when is a function at rest

v(t)=0 (f’(x) x-intercepts)

when is a function’s rate of change increasing (speeding up?)

v(t) and a(t) have the same signs

when is a function’s rate of change decreasing (slowing down?)

v(t) and a(t) have different signs

when is a function concave up

• f’(x) slope is going up

• f’’(x) is positive/above the x axis

when is a function concave down

• f’(x) slope is going down

• f’’(x) is negative/below the x axis

when does a function have an inflection point

• the point f’(x) has a change in slope

• f’’(0) and actually crosses the x-axis

when does a function have a relative max

• the point f’(x) crosses from positive to negative

when does a function have an relative min

the point f’(x) is crosses from negative to positive

derivative of sin(x)?

cos(x)

derivative of cos(x)?

-sin(x)

derivative of sec(x)?

sec(x)tan(x)

derivative of tan(x)?

sec(x)²

derivative of csc(x)?

-csc(x)cot(x)

derivative of cot(x)?

-csc(x)²

how to find HORIZONTAL tangents (with implicit)

• set NUMERATOR to 0

• solve for x or y

• plug in 0s into original equation for other coordinate

how to find VERTICAL tangents (with implicit)

• set DENOMINATOR to 0

• solve for x or y

• plug in 0s into original equation for other coordinate

inverse derivative of sin(x)?

u’/(sqrt 1-u²)

inverse derivative of cot(x)?

-u’/ (1+u²)

inverse derivative of cos(x)?

-u’/(sqrt 1-u²)

inverse derivative of csc(x)?

u’/ (|u| sqrt u²-1)

inverse derivative of tan(x)?

u’/ (1+u²)

inverse derivative of sec(x)?

-u’/ (|u| sqrt u²-1)

template to answer rate of change problems in the context of the problem

At (time), (description of function) is (increasing or decreasing) at a rate of (absolute value of answer)(units)

ex. At H’(t) = 6 years, the height of the tree is increasing at a rate of 5/2 meters/year

squeeze theorem

intermediate value theorem

mean value theorem (for derivatives)

rolle’s theorem

L’Hopital

Fundamental Theorem of Calculus (pt 1)

Fundamental Theorem of Calculus (pt 2)

Mean Value Theorem (for integrals)

washer rotation formula

square cross section

rectangle cross section

triangle w leg cross section

triangle w hypotenuse cross section

equilateral triangle cross section

circle cross section