Math 117 - Fall 2021 - Common Final Exam, version A Solutions
1. The graph below shows P, the population of a region in thousands of people, as a function of t, years
since 2000.
15,000
10,000
5,000
0
0 5 10 15 20
t, years since 2000
∆P
(a) (3 points) Estimate the average rate of change ∆t over the interval 0 ≤ t ≤ 10.
∆P 10,000− 3,000
Solution: ∆t ≈ 10− 0 = 700
1 pt Evidence of using ∆ f / ∆ t
1 pt At least some progress toward correct solution
1 pt Correct solution
∆P
(b) (2 points) Give an interval of t values over which the average rate of change ∆t is negative.
(b) 6<t<8
Solution: Any interval with a negative average rate of change is OK. 2 pts for a correct answer, 0
otherwise
(c) (2 points) Circle the interval below over which ∆∆Pt is
greatest.
0 ≤ t ≤ 10 6 ≤ t ≤ 10 8 ≤ t ≤ 10
Solution: Correct choice: 8 ≤ t ≤ 10. 2 points for correct choice, 0 otherwise.
, (d) (2 points) Circle any intervals below over which the graph of P is concave up.
2≤t≤4 7≤t≤9 12 ≤ t ≤ 14
Solution: Correct choice: 7 ≤ t ≤ 9. 2 points for correct choice, 0 otherwise.
2. A metal casting is being heat-treated in an industrial oven. The table below gives values for h(t), the
temperature of the casting t minutes after it is placed in the oven.
t (minutes) 0 8 16 24
h(t) (◦ F) 1200 800 600 500
(a) (3 points) Find the average rate of change ∆∆ht over the interval 0 ≤ t ≤ 8. Give correct units
and explain the meaning of this quantity in a sentence.
Solution: ∆ ht = 8008−−1200
∆
0 = − 50 degrees F per minute. Over this interval, the casting is
cooling at an average rate of 50 degrees per minute.
1 pt Correct average rate of change
1 pt Correct units
1 pt Substantially correct interpretation, at least referencing the interval somehow, or
that this is an averagerate of change.
(b) (3 points) Does h appear to be an increasing or decreasing function? Explain your answer
in a sentence.
Solution: h appears to be decreasing, because the values of h decrease as t increases, or
because the average rates of change are all negative.
1 pt Correctly identify function as decreasing
2 pt Correct explanation referencing at least one piece of evidence.
(c) (3 points) Does h appear to be a concave up or concave down function? Explain your answer in a
sentence with some work to back it up.
Solution: h appears to be concave up, because the average rate of change over successive
intervals is increasing.
1 pt Correctly identify concavity
1 pt Answer is consistent with statement about increasing or decreasing average rate of change.
1. The graph below shows P, the population of a region in thousands of people, as a function of t, years
since 2000.
15,000
10,000
5,000
0
0 5 10 15 20
t, years since 2000
∆P
(a) (3 points) Estimate the average rate of change ∆t over the interval 0 ≤ t ≤ 10.
∆P 10,000− 3,000
Solution: ∆t ≈ 10− 0 = 700
1 pt Evidence of using ∆ f / ∆ t
1 pt At least some progress toward correct solution
1 pt Correct solution
∆P
(b) (2 points) Give an interval of t values over which the average rate of change ∆t is negative.
(b) 6<t<8
Solution: Any interval with a negative average rate of change is OK. 2 pts for a correct answer, 0
otherwise
(c) (2 points) Circle the interval below over which ∆∆Pt is
greatest.
0 ≤ t ≤ 10 6 ≤ t ≤ 10 8 ≤ t ≤ 10
Solution: Correct choice: 8 ≤ t ≤ 10. 2 points for correct choice, 0 otherwise.
, (d) (2 points) Circle any intervals below over which the graph of P is concave up.
2≤t≤4 7≤t≤9 12 ≤ t ≤ 14
Solution: Correct choice: 7 ≤ t ≤ 9. 2 points for correct choice, 0 otherwise.
2. A metal casting is being heat-treated in an industrial oven. The table below gives values for h(t), the
temperature of the casting t minutes after it is placed in the oven.
t (minutes) 0 8 16 24
h(t) (◦ F) 1200 800 600 500
(a) (3 points) Find the average rate of change ∆∆ht over the interval 0 ≤ t ≤ 8. Give correct units
and explain the meaning of this quantity in a sentence.
Solution: ∆ ht = 8008−−1200
∆
0 = − 50 degrees F per minute. Over this interval, the casting is
cooling at an average rate of 50 degrees per minute.
1 pt Correct average rate of change
1 pt Correct units
1 pt Substantially correct interpretation, at least referencing the interval somehow, or
that this is an averagerate of change.
(b) (3 points) Does h appear to be an increasing or decreasing function? Explain your answer
in a sentence.
Solution: h appears to be decreasing, because the values of h decrease as t increases, or
because the average rates of change are all negative.
1 pt Correctly identify function as decreasing
2 pt Correct explanation referencing at least one piece of evidence.
(c) (3 points) Does h appear to be a concave up or concave down function? Explain your answer in a
sentence with some work to back it up.
Solution: h appears to be concave up, because the average rate of change over successive
intervals is increasing.
1 pt Correctly identify concavity
1 pt Answer is consistent with statement about increasing or decreasing average rate of change.
