Math 125 Final Exam Fall 2023
1. Progressive Insurance examined the driving records of 900 drivers over the past year and found the following
data:
Age
16-25 26-75 76-92
at least one accident 30 110 40
no accidents 105 560 55
(a) Find the probability that a randomly selected driver was at least 26 years old and had no accidents.
(b) Find the probability that a driver was less than 76 years old or had no accidents.
(c) Find the probability that a driver who had at least one accident was age 16 to 25.
2. Let E and F be events in a sample space with P (E) = 0.30, P (E ∪ F ) = 0.58, and P (E ∩ F ) = 0.12.
(a) Find: P (F )= P (E 0 |F )=
(b) Are E and F independent? Justify your answer mathematically.
3. There are twenty math SIs– 8 work for Terri and 12 work for Chris.
(a) In how many ways can six of the twenty SIs be assigned to proctor exams in six different classrooms?
(b) Eight of the 20 SIs will form a committee to plan the math gala. How many possible gala committees
include Nell or Nidhi (2 of the SIs)?
(c) Find the probability that a committee of eight SIs includes at least one of Terri’s SIs.
4. Twenty-nine percent of adults have heart disease, while 8% of adults are diagnosed with diabetes. Adults
who have diabetes are three times more likely to have heart disease than adults who do not have diabetes.
Find the probability that an adult does not have heart disease and is not diabetic.
5. Sketch the graph of a function f (x) that is defined for ALL values of x and that satisfies ALL of the following:
• f (x) is continuous everywhere except x = 0;
• f 0 (−2) does not exist;
• f 00 (x) < 0 only when x > 0;
• f 0 (x) < 0 only when −2 < x < −1;
• lim f (x) = −∞;
x→0+
• lim = −2.
x→±∞
√
6. The cost to produce x bobble-heads is given by C(x) = 5 x + 10 dollars.
(a) Determine the average rate of change in the cost function (with units!) when production increases from
6 bobble-heads to 15 bobble-heads.
(b) Use the limit definition of the derivative (that is, the four-step process) to determine the instantaneous
rate of change when 6 bobble-heads are produced. Be sure to include correct units!
7. Differentiate. Final answers should look nice– at least simplify within each term.
(a) y = 2x sin5 (x) cos(5x3 )
3−x 4
(b) y =
2x + 5
5 √
5
(c) y = x − 3(5x ) + 3e5 − x3 − ln(5x3 )
e
8. (a) The first derivative of a function f (x) is given by f 0 (x) = 4x5 e2x − 9x3 e2x . At what values of x do
the relative extrema of f (x) occur? Classify each as a relative min or relative max.
(b) A continuous function g(x) has first derivative g 0 (x) = 2x ln x − 8x. Over what interval(s) is g(x)
concave down?
9. A pipeline is to be run from a source on the edge of a lake to a small resort community on an island 5 miles
offshore. It will cost 3 million dollars per mile to build the pipeline across land and 5 million dollars per mile
to build the pipeline under water. How much of the pipeline should be constructed on land to minimize the
total cost?
::::
island
:::
Q
6 Q
Q
QQ
5 miles QQ
Q
Q
QQ
? QQ
- source
10 mi
10. A spotlight is on the ground 30 feet from a wall. A 5-foot tall woman walks away from the wall and toward
the light. Determine the rate at which the height of the woman’s shadow on the wall is changing when she
is 12 feet from the wall and walking 3 ft per second.
dy
11. Find : 5x − 3x5 y 2 = 4y + y cos(x) + 1
dx
12. (a) Let f (x) = ln(3x). Write the third-degree Taylor polynomial for f (x) at x = 2.
2(−3)n n
(b) A function f (x) has a Taylor polynomial given by Pn (x) = 2 − 6x + 9x2 − 9x3 + . . . + x .
n!
Determine the value of f (23) (0). (You do not need to simplify the answer.)
2
13. (a) Solve the difference equation: an = 10 − (an−1 ) , a0 = 8.
3
(b) Solve the difference equation: an = an−1 − 5, a0 = 2.
(c) Find the fixed points of the difference equation. Classify each of them as stable/unstable/semi-stable.
1
an = (an−1 )2 + 2 (an−1 )
3
Have a great break!
