1. Unit 4:
curve sketching

2. WARM-UP
1. Solve 𝑦 3 +4 𝑦 2 +𝑦−6=0.
(A) 𝑦=−1, −3, −2
(B) 𝑦=1, −3, −2
(C) 𝑦=−1, 3, −2
(D) 𝑦=1, −3, 2

3. WARM-UP
2. Solve 5 3−𝑥 ≤3𝑥−1.
(A) 𝑥≤−2
(B) 𝑥≥−2
(C) 𝑥≥2
(D) 𝑥≤2

4. WARM-UP
3. Solve 𝑥 2 +3𝑥−4>0.
(A) 𝑥>1
(B) 𝑥<−4 and 𝑥>1
(C) 𝑥<−4
(D) −4<𝑥<1

5. WARM-UP
4. Evaluate lim 𝑥→3 𝑥 3 −27 𝑥−3
(A) 27
(B) 0
(C) 3
(D) 9

6. WARM-UP
5. Divide ( 𝑥 2 −5𝑥+4)÷(𝑥+3), and then write
the answer in the form 𝑎𝑥+𝑏+ 𝑟 𝑞(𝑥) .
(A) 𝑥−2− 2 𝑥+3
(C) 𝑥−2+ 10 𝑥+3
(B) 𝑥−8− 20 𝑥+3
(D) 𝑥−8+ 28 𝑥+3

7. WARM-UP
6. State the vertical asymptote(s) of 𝑦= 5𝑥 2 −20 3 𝑥 2 −16𝑥+5
(A) 𝑥=4
(B) 𝑦=4
(C) 𝑥= and 𝑥=5
(D) 𝑥=−2 and 𝑥=2

8. WARM-UP
7. State the horizontal asymptote(s) of 𝑦= 5𝑥 2 −20 3 𝑥 2 −16𝑥+5
(A) 𝑥=0
(B) 𝑦=0
(C) 𝑦= 5 3
(D) 𝑥=−2 and 𝑥=2

9. WARM-UP
8. State the 𝑥-intercept(s) of 𝑦= 5𝑥 2 −20 3 𝑥 2 −16𝑥+5
(A) 𝑥= and 𝑥=5
(B) There are no 𝑥-intercepts
(C) 𝑥=−4
(D) 𝑥=−2 and 𝑥=2

10. WARM-UP
9. State the 𝑦-intercept(s) of 𝑦= 5𝑥 2 −20 3 𝑥 2 −16𝑥+5
(A) 𝑦=−4
(B) 𝑦=4
(C) 𝑦= 4 5
(D) 𝑦= −4 5

11. WARM-UP
10. State the domain of 𝑦= 5𝑥 2 −20 3 𝑥 2 −16𝑥+5
(A) 𝐷={𝑥∈ℝ}
(B) 𝐷= 𝑥∈ℝ 𝑥≠−2, 2
(C) 𝐷= 𝑥∈ℝ 𝑥≠ 1 3 , 5
(D) 𝐷= 𝑥∈ℝ 𝑥≠−4

12. WARM-UP
11. State the range of 𝑦= 5𝑥 2 −20 3 𝑥 2 −16𝑥+5
(A) 𝑅={𝑦∈ℝ}
(B) 𝑅= 𝑦∈ℝ 𝑥≠ 5 3
(C) 𝑅= 𝑦∈ℝ 𝑥≠ 1 3 , 5
(D) 𝑅= 𝑦∈ℝ 𝑥≠−4

13. 4.1 Increasing and Decreasing Functions
A function 𝑓 is increasing on an interval if, for any value of 𝑥 1 < 𝑥 2 in the interval, 𝑓 𝑥 1 <𝑓 𝑥 2 .
A function 𝑓 is decreasing on an interval if, for any value of 𝑥 1 < 𝑥 2 in the interval, 𝑓 𝑥 1 >𝑓 𝑥 2 .

14. 4.1 Increasing and Decreasing Functions
For a function 𝑓 that is continuous and differentiable on an interval 𝐼.
𝑓 𝑥 is increasing on 𝐼 if 𝑓 ′ 𝑥 >0 for all values of 𝑥 in 𝐼
𝑓 𝑥 is decreasing on 𝐼 if 𝑓 ′ 𝑥 <0 for all values of 𝑥 in 𝐼.

15. Using the derivative to reason about intervals of increase/decrease.
Example #1:
Using the derivative to reason about intervals of increase/decrease.
(a) 𝑦= 𝑥 3 +3 𝑥 2 −2
𝑦′= 3𝑥 2 +6𝑥
Value of 𝒙
𝒙<−𝟐
−𝟐<𝒙<𝟎
𝒙>𝟎
Sign of 𝑑𝑦 𝑑𝑥
𝑑𝑦 𝑑𝑥 >0
𝑑𝑦 𝑑𝑥 <0
Slope of tangents
positive
negative
Increasing/decreasing
increasing
decreasing
0= 3𝑥 2 +6𝑥
0=3𝑥(𝑥+2)
𝑥=−2, 0

16. Using the derivative to reason about intervals of increase/decrease.
Example #1:
Using the derivative to reason about intervals of increase/decrease.
(b) 𝑦= 𝑥 𝑥 2 +1
Value of 𝒙
𝒙<−𝟏
−𝟏<𝒙<𝟏
𝒙>𝟏
Sign of 𝑑𝑦 𝑑𝑥
𝑑𝑦 𝑑𝑥 <0
𝑑𝑦 𝑑𝑥 >0
Slope of tangents
negative
positive
Increasing/decreasing
decreasing
increasing
𝑦 ′ = 𝑥 2 +1−𝑥(2𝑥) ( 𝑥 2 +1) 2
𝑦 ′ = −𝑥 ( 𝑥 2 +1) 2
0= −𝑥 ( 𝑥 2 +1) 2
0= −𝑥 2 +1
𝑥 2 =1 ⇒
𝑥=−1 or 𝑥=1

17. Example #2:
Graphing a function given the derivative.
Consider the graph of 𝑓 ′ 𝑥 . Graph 𝑓 𝑥 .

18. In summary …
QUESTIONS: p #4cde, 5-9