SI Math 1350
5-1 and 5-2
Instructor: Kathryn
5-1
1. Copy graph from board
a. Identify intervals on which 𝑓(𝑥) is increasing
b. Identify intervals on which 𝑓(𝑥) is decreasing
c. Identify the x coordinates where 𝑓(𝑥) has a local maximum
d. Identify the x coordinates where 𝑓(𝑥) has a local minimum
For 2-4 find
a. 𝑓′(𝑥)
b. The critical values of 𝑓
c. The partition numbers for 𝑓′
2. 𝑓(𝑥) = 𝑥 3 − 12𝑥 + 8
1
3. 𝑓(𝑥) = (𝑥 + 5)3
6
4. 𝑓(𝑥) = 𝑥+2
For 5-10
a. Find the intervals on which 𝑓(𝑥) is increasing
b. Find the intervals on which 𝑓(𝑥) is decreasing
c. Find the local extrema
d. Sketch the graph
5. 𝑓(𝑥) = 2𝑥 2 − 4𝑥
6. 𝑓(𝑥) = −2𝑥 2 − 16𝑥 − 25
7. 𝑓(𝑥) = 3𝑥 4 − 4𝑥 3 + 5
8. 𝑓(𝑥) = 10𝑙𝑛𝑥 + 𝑥 2
9. 𝑓(𝑥) = 10 − 12𝑥 + 6𝑥 2 − 𝑥 3
10. 𝑓(𝑥) = 𝑥 4 − 18𝑥 2
For 11, sketch the graph of 𝑓 using the given information
11. 𝑓(−1) = 2, 𝑓(0) = 0, 𝑓(1) = 2
𝑓 ′ (−1) = 0, 𝑓 ′ (1) = 0, 𝑓 ′ (0) 𝑖𝑠 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
𝑓 ′ (𝑥) > 0 𝑜𝑛 (−∞, −1) 𝑎𝑛𝑑 (0,1)
𝑓 ′ (𝑥) < 0 𝑜𝑛 (−1,0)𝑎𝑛𝑑 (1, ∞)
12. A manufacturer incurs the following costs in producing x water ski vests in one day,
for 0 < x < 150: fixed costs, $320; unit production cost $20/vest; equipment
maintenance and repairs, 0.05𝑥 2 dollars.
a. Write the equation for the cost of manufacturing x vests in one day
b. What is the average cost per vest if x vests are produced in one day?
c. Find the critical values the average cost, the intervals on which the average
cost per vest is decreasing and increasing, and the local extrema. (Do not
graph).
SI Math 1350
5-1 and 5-2
Instructor: Kathryn
5-2
1. Use graph from board to identify
a. Intervals on which the graph of 𝑓 is concave upward
b. Intervals on which the graph of 𝑓 is concave downward
c. Intervals on which 𝑓 ′ (𝑥) is increasing
d. Intervals on which 𝑓 ′ (𝑥) is decreasing
e. The x coordinates of inflection points
f. The x coordinates of the local extrema for 𝑓′(𝑥)
2. Sketch the basic graph for each indicated function
a. 𝑓 ′ (𝑥) > 0 𝑎𝑛𝑑𝑓 ′′ (𝑥) > 0
b. 𝑓 ′ (𝑥) > 0 𝑎𝑛𝑑 𝑓 ′′ (𝑥) < 0
c. 𝑓 ′ (𝑥) < 0 𝑎𝑛𝑑 𝑓 ′′ (𝑥) > 0
d. 𝑓 ′ (𝑥) < 0 𝑎𝑛𝑑 𝑓 ′′ (𝑥) < 0
For 3-6 find the second derivative
3. 𝑓(𝑥) = 2𝑥 6 + 4𝑥 3 − 5𝑥 2 − 6𝑥 + 3
4. 𝑓(𝑥) = −6𝑥 −2 + 12𝑥 −3
5. 𝑓(𝑥) = (𝑥 2 − 16)5
ln 𝑥
6. 𝑓(𝑥) = 𝑥 2
For 7-10, find the intervals on which the graph is concave upward, the intervals on which
the graph is concave downward, and the inflection points
7. 𝑓(𝑥) = 𝑥 4 + 6𝑥 2
8. 𝑓(𝑥) = 𝑥 3 − 4𝑥 2 + 5𝑥 − 2
9. 𝑓(𝑥) = −𝑥 4 + 12𝑥 3 − 12𝑥 + 24
10. 𝑓(𝑥) = 8𝑒 𝑥 − 𝑒 2𝑥
For 11-15, summarize the pertinent information obtained by applying the graphing
strategy and sketch the graph of the equation
11. 𝑓(𝑥) = (𝑥 − 3)(𝑥 2 − 6𝑥 − 3)
12. 𝑓(𝑥) = 0.25𝑥 4 − 2𝑥 3
13. 𝑓(𝑥) = (𝑥 2 − 1)(𝑥 2 − 5)
14. 𝑓(𝑥) = 2𝑒 0.5𝑥 + 𝑒 −0.5𝑥
15. 𝑓(𝑥) = 5 − 3 ln 𝑥