CALCULUS 1 FINAL FORMULAS TO
KNOW EXAM QUESTIONS AND
ANSWERS
disk method - ANSWER-integral from a to b pi(f(x)^2) dx
shell method - ANSWER-integral from a to b, 2pi(r)*f(x) dx
position, then velocity, then acceleration - ANSWER-if given acceleration, go
backwards. Initial rate is C for velocity, and initial position is C for position
4 - ANSWER-lim x²-4/x-2
x->2
0 - ANSWER-lim 10x³-3x+1000/3x⁴-4x+50
x->∞
4000 - ANSWER-lim 8000/2+4.1e^-.53x
x->∞
3/2√x - ANSWER-use limit definition of derivative.
f(x)=3√x
15/4x²(5x³+1)^-3/4 - ANSWER-find the derivative.
⁴√5x³+1
(7x-2)⁵(3x-8)³(210x-312) - ANSWER-find the derivative.
(7x+2)⁶(3x-8)⁴
x^x³(3x²lnx+x²) - ANSWER-use logarithmic differentiation.
y=x^x³
$22,666.67 - ANSWER-Find the point of diminishing returns for R(x)=4000+68x²-x³
r=22.5cm h=22.5cm - ANSWER-A cylindrical box will be tied up with ribbon. You have
300cm of ribbon and you need 30cm for a bow. what are the dimensions of the box of
max volume?
IDK YET - ANSWER-what size square should you cut out of the corners of a piece of
cardboard that measures 22x25 inches that when the sides are folded up to make a box
will result in an open topped box of max volume?
, 750X625 ft - ANSWER-a rectangular fence is to be built with one side along a wall so
that side will not require fencing. if the fencing costs $5 per foot for the two sides and
$12 per foot for the side opposite the wall, what are the dimensions of the largest area
inside the enclosure that will use up a $15,000 budget?
r=2.17in h=4.34in - ANSWER-The bitty moore company requires that its corned beef
hash cans have a capacity of 64 cubic inches. determine the radius and height of the
container that requires the least amount of material (surface area). U CAN LOOK AT
FORMULAS
2000 specialty bikes 4000 scooters - ANSWER-R(x,y)=12xy+5-2y² and C(x,y)=y²+4x³
how many units of each will maximize the profit?
x=250 y=1000 - ANSWER-f(x,y)=4x^1/4 y^3/4
where x=labor and y=capital. labor costs are $600 per unit and capital costs are $450
per unit. how many of each should they allocate to max the productivity if their budget is
$600,000?
A function is not differentiable at places where there is... - ANSWER-1) A discontinuity
2) a sharp corner or cusp
3) a vertical tangent line
A derivative tells you... - ANSWER-The instantaneous rate of change- the slope of the
tangent line
A function is increasing when... - ANSWER-f'(x) > 0 (positive derivative), the graph of
f'(x) is above the x axis
A function is decreasing when... - ANSWER-f'(x) < 0 (negative derivative), the graph of
f'(x)
What does it mean when f'(x) = 0? - ANSWER-1) The tangent line to f(x) is horizontal
2) The function is neither increasing or decreasing
3) the graph of f'(x) has an x intercept
4) this is a "turn around point" meaning that if a function is going to go from increasing
to decreasing, it must stop increasing, stop and start heading the other direction
the derivative of a constant is... - ANSWER-0
KNOW EXAM QUESTIONS AND
ANSWERS
disk method - ANSWER-integral from a to b pi(f(x)^2) dx
shell method - ANSWER-integral from a to b, 2pi(r)*f(x) dx
position, then velocity, then acceleration - ANSWER-if given acceleration, go
backwards. Initial rate is C for velocity, and initial position is C for position
4 - ANSWER-lim x²-4/x-2
x->2
0 - ANSWER-lim 10x³-3x+1000/3x⁴-4x+50
x->∞
4000 - ANSWER-lim 8000/2+4.1e^-.53x
x->∞
3/2√x - ANSWER-use limit definition of derivative.
f(x)=3√x
15/4x²(5x³+1)^-3/4 - ANSWER-find the derivative.
⁴√5x³+1
(7x-2)⁵(3x-8)³(210x-312) - ANSWER-find the derivative.
(7x+2)⁶(3x-8)⁴
x^x³(3x²lnx+x²) - ANSWER-use logarithmic differentiation.
y=x^x³
$22,666.67 - ANSWER-Find the point of diminishing returns for R(x)=4000+68x²-x³
r=22.5cm h=22.5cm - ANSWER-A cylindrical box will be tied up with ribbon. You have
300cm of ribbon and you need 30cm for a bow. what are the dimensions of the box of
max volume?
IDK YET - ANSWER-what size square should you cut out of the corners of a piece of
cardboard that measures 22x25 inches that when the sides are folded up to make a box
will result in an open topped box of max volume?
, 750X625 ft - ANSWER-a rectangular fence is to be built with one side along a wall so
that side will not require fencing. if the fencing costs $5 per foot for the two sides and
$12 per foot for the side opposite the wall, what are the dimensions of the largest area
inside the enclosure that will use up a $15,000 budget?
r=2.17in h=4.34in - ANSWER-The bitty moore company requires that its corned beef
hash cans have a capacity of 64 cubic inches. determine the radius and height of the
container that requires the least amount of material (surface area). U CAN LOOK AT
FORMULAS
2000 specialty bikes 4000 scooters - ANSWER-R(x,y)=12xy+5-2y² and C(x,y)=y²+4x³
how many units of each will maximize the profit?
x=250 y=1000 - ANSWER-f(x,y)=4x^1/4 y^3/4
where x=labor and y=capital. labor costs are $600 per unit and capital costs are $450
per unit. how many of each should they allocate to max the productivity if their budget is
$600,000?
A function is not differentiable at places where there is... - ANSWER-1) A discontinuity
2) a sharp corner or cusp
3) a vertical tangent line
A derivative tells you... - ANSWER-The instantaneous rate of change- the slope of the
tangent line
A function is increasing when... - ANSWER-f'(x) > 0 (positive derivative), the graph of
f'(x) is above the x axis
A function is decreasing when... - ANSWER-f'(x) < 0 (negative derivative), the graph of
f'(x)
What does it mean when f'(x) = 0? - ANSWER-1) The tangent line to f(x) is horizontal
2) The function is neither increasing or decreasing
3) the graph of f'(x) has an x intercept
4) this is a "turn around point" meaning that if a function is going to go from increasing
to decreasing, it must stop increasing, stop and start heading the other direction
the derivative of a constant is... - ANSWER-0
