IB Precalculus – Introduction to Calculus
CLASSWORK
WORD PROBLEMS
1.
VOLUME:
A closed box has a square base
with side lengths, x, and height, h.
h
x
x
a)
Write the function, S, that represents the SURFACE AREA of the box.
b)
Write the function, V, that represents the volume of the box.
(Volume = length ● width ● height)
c)
Each box must have a volume of 12 cm . Express the HEIGHT in terms of x.
2
V = x h

3
Solve for h.
d)
Express the SURFACE AREA function in terms of x.
(from part a)
S =
e)
Determine the dimensions (side lengths & height) of the box that would MINIMIZE
the SURFACE AREA.
1)
Find the derivative of S(x) 
2)
Set equal to ZERO

3)
Solve for x.

(Use the "zero" function on your graphing calculator.)
f)
Find the surface area.
(Plug the x-value for part e into S(x))
VELOCITY:
2.
A ball is thrown from the top of a 100 meter building so that its height in feet above the
2
ground after t seconds is: h(t) = 100 + 49t – 4.9t .
a)
Find the average velocity (slope) over the interval from t = 0 seconds to t = 2 seconds.
h(2) =
h(0) =
Find slope:
b)
What is the "instantaneous" velocity (derivative) at time t?
2
Find the derivative of h(t) = 100 + 49t – 4.9t
h'(t) =
c)
What is the "instantaneous" velocity (derivative) at 1 second after the ball is thrown?
h'(1) =
d)
What is the ball's MAXIMUM height above the ground?
1)
Set the derivative equal to ZERO
to find the TIME when the ball
is at the MAXIMUM height
h'(t) = 49 – 4.9t
0 = 49 – 4.9t

Solve for t.
2)
Plug this value into the height function.
2
h(t) = 100 + 49t – 4.9t
3.
The cost of producing x items is given by:
2
C(x ) = 7x – 3000 –
x
10,000
dollars.
If each item sells for $30, find:
4.
a)
the revenue function, R( x )
b)
the profit function, P( x )
c)
the marginal profit function, P'( x )
d)
the marginal profit when x = 50.
e)
Define marginal profit.
An arrow is shot upward from the bottom of a canyon that is
400 feet below the edge of a cliff. If the initial velocity of the arrow is 160 ft/s, then
2
s(t) = 160t – 16t gives the arrow's height, in feet, above the bottom of the canyon t
seconds after the arrow is shot.
a)
In how many seconds does the arrow first pass the cliff's edge?
b)
How high does the arrow travel before it starts to descend?
c)
If it lands on the cliff's edge, how long was the arrow in the air?
d)
At what velocity does the arrow hit the cliff's edge?
e)
What is the acceleration (2nd derivative) of the arrow when it hit the cliff's edge?