Unit 4: curve sketching

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

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

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

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

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

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= 1 3 and 𝑥=5 (D) 𝑥=−2 and 𝑥=2

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

WARM-UP 8. State the 𝑥-intercept(s) of 𝑦= 5𝑥 2 −20 3 𝑥 2 −16𝑥+5 (A) 𝑥= 1 3 and 𝑥=5 (B) There are no 𝑥-intercepts (C) 𝑥=−4 (D) 𝑥=−2 and 𝑥=2

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

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

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

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 .

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.

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

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 2 +1 ( 𝑥 2 +1) 2 0= −𝑥 2 +1 ( 𝑥 2 +1) 2 0= −𝑥 2 +1 𝑥 2 =1 ⇒ 𝑥=−1 or 𝑥=1

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

In summary … QUESTIONS: p.169-171 #4cde, 5-9