The crusaders ventured into the Temple of Mathematics and ran into a bit of a problem.

1. THE CALCULUS CRUSADERS
Accumulation Functions: The Beetles Question

2. .:. THE QUESTION .:.
E(t) is the rate of which the beetles rush into the
chamber, whereas L(t) is the rate of which the
beetles rush out of the chamber. Both E(t) and L(t)
are measured in beetles per minute.

3. .:. THE QUESTION .:.
At t = 0, Jamie hears the noise of the
beetles. The beetles start rushing in and out
of the chamber at t = 1.84.
However, Bench estimates that everyone
must escape the chamber until t = 15.17.
(After t = 15.17, the beetles would have
filled up the chamber completely.)

4. .:. THE QUESTION .:.
a) How many beetles have entered the
chamber at t = 10?
b) With all these beetles filling up the
chamber, Bench, Jamie, and Zeph have limited space.
3 m3 of the chamber is filled up for every beetle that
enters the chamber until t = 10. After t = 10, 5 m3 of
the chamber is filled up for every beetle that enters the
chamber. How many cubic metres of the chamber
would be filled up with beetles at t = 15.17?

5. .:. THE QUESTION .:.
c) Let H(t) for 1.84 ≤ t ≤ 15.17. Determine
H’(10) and explain the meaning of H’(10).
d) At what time, during 1.84 ≤ t ≤ 15.17, will
H(t) reach a maximum?

6. PART A
How many beetles have entered the chamber at t = 10?

7. .:. THE SOLUTION .:.
E(t) is measured in beetles per minute. To obtain an
answer in beetles, we multiply beetles per minute
by a change in time. This is the definition of an
integral. This way of thinking is called a unit
analysis.

8. .:. THE SOLUTION .:.
We know the domain is 1.84 ≤ t ≤ 15.17. We know
the upper limit of what we are integrating is t = 10.

9. PART B
With all these beetles filling up the chamber, Bench,
Jamie, and Zeph have limited space. 3 m3 of the
chamber is filled up for every beetle that enters the
chamber until t = 10. After t = 10, 5 m3 of the chamber
is filled up for every beetle that enters the chamber.
How many cubic metres of the chamber would be filled
up with beetles at t = 15.17?

10. .:. THE SOLUTION .:.
NOTE THAT:
“3 m3 of the chamber is filled up for every beetle
that enters the chamber until t = 10. After t = 10,
5 m3 of the chamber is filled up for every beetle
that enters the chamber.”

11. .:. THE SOLUTION .:.
Because of the statement written in the question, the
rates of the chamber filling up with beetles are two
different rates. Therefore, we integrate the function E(t)
from the intervals where beetles would accumulate at
the rate of 3m3 and 5m3.

12. PART C
Let H(t) for 1.84 ≤ t ≤ 15.17. Determine H’(10) and
explain the meaning of H’(10).

13. .:. THE SOLUTION .:.
H(t) is defined as the integral of the
difference of L(x) and E(x).
(Integrating beetles per minutes gives us beetles as
discussed in Part A.)

14. .:. THE SOLUTION .:.
We are to determine H’(t). In this case, we are
differentiating an anti derivative. (Note the quot;∫quot;.)
Differentiation and anti differentiation can be seen
as inverse processes of each other; The derivative
of x2 is 2x; an antiderivative of 2x is x2.

15. .:. THE SOLUTION .:.
Not only are we differentiating an
antiderivative, we’re differentiating an
accumulation function, a function that
measures the accumulating area under a
graph.

16. .:. THE SOLUTION .:.
The derivative of an accumulation function is the
original function, by The Second Fundamental
Theorem of Calculus.
The Second Fundamental Theorem of Calculus:
If f is continuous in a closed interval, A’(x) = f(x),
where A(x) is the accumulation function and f(x)
is the original function.

17. .:. THE SOLUTION .:.
We can see there is a function within a
function in s(t). (Note there are two
variables, t and x.)
To differentiate a function within a
function, we use The Chain Rule.

18. .:. THE SOLUTION .:.
The Chain Rule: [fg]’(x) = f’(g(x))g’(x)

19. .:. THE SOLUTION .:.
Since H’(t) is a transcendental function, a function
that contains an exponential function and a
trigonometric function, we cannot apply the algebra
we know to solve for the roots of v’(t), so we have
to use our calculator and solve numerically.

20. .:. THE SOLUTION .:.
H’(10) is the rate at which the number
of beetles in the chamber is changing.
The number of beetles in the chamber
is increasing at approximately beetles
per minute.

21. PART D
At what time, during 1.84 ≤ t ≤ 15.17, will H(t) reach a
maximum?

22. .:. THE SOLUTION .:.
By The First Derivative Test, the critical
point of a derivative indicates the
original function has a local extrema.

23. .:. THE SOLUTION .:.
The First Derivative Test
 If c is a critical number and if f’ changes sign at
x = c, then
 f has a local minimum at x = c if f’ is negative to
the left of c and positive to the right of c;
 f has a local maximum at c if f’ is positive to the
left of c and negative to the right of c.

24. .:. THE SOLUTION .:.
Using our calculator’s features to
determine roots and intersections, we
find that t = 1.8400082 minutes.

25. .:. THE SOLUTION .:.
By The Extreme Value Theorem, the endpoints are
considered local extrema too. (In this case, the
critical number found previously is also an
endpoint.)
The Extreme Value Theorem
 If the function f is continuous on the interval
[a, b], then there exist numbers c and d in [a,b]
such that for all x in [a, b], f(c) ≤ f(x) and f(d) ≥
f(x).

26. .:. THE SOLUTION .:.
Looking at all the local extrema, we
find that H(15.17) yields the largest
number, the absolute maximum.

27. AAAAH!!!! BEETLES!