a summary of curve sketching - utep mathematics · a summary of curve sketching 3 . note •there...

30
A Summary of Curve Sketching By Tuesday J. Johnson 1

Upload: others

Post on 01-Jun-2020

12 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

A Summary of Curve Sketching

By Tuesday J. Johnson

1

Page 2: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

Suggested Review Topics

• Algebra skills reviews suggested:

– None

• Trigonometric skills reviews suggested:

– None

2

Page 3: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

Applications of Differentiation

A Summary of Curve Sketching

3

Page 4: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

Note

• There is a video that accompanies this section on my website and on YouTube. The link is: https://www.youtube.com/watch?v=bsb9Opn1420

• In this video I talk specifically about one example of using information from the function and its derivatives to sketch a graph without the use of a graphing calculator

4

Page 5: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

Guidelines for Analyzing the Graph of a Function

1. Determine the domain and range of the function.

2. Determine the intercepts, asymptotes, and symmetry of the graph.

3. Locate the x-values for which f’(x) and f”(x) either are zero or do not exist. Use the results to determine relative extrema and points of inflection.

5

Page 6: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

Examples: Analyze and sketch 1. 𝑦 = −2𝑥4 + 3𝑥2

• From Pre-Calculus, we know this graph has both ends facing the same direction and that direction is down. We could use infinite limits in Calculus to conclude the same thing.

• We can find the x-intercepts: 0 = 𝑥2(−2𝑥2 + 3)

𝑥 = 0, ± 3/2

• The y-intercept is at (0,0) and there are no asymptotes.

6

Page 7: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

Examples: Analyze and sketch 1. 𝑦 = −2𝑥4 + 3𝑥2

First Derivative: 𝑦′ = −8𝑥3 + 6𝑥

Critical Numbers: 0 = −2𝑥(4𝑥2 − 3) so 𝑥 = 0

and 𝑥 = ±3

2.

7

Page 8: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

Examples: Analyze and sketch 1. 𝑦 = −2𝑥4 + 3𝑥2

First derivative: 𝑦′ = −8𝑥3 + 6𝑥

Second derivative: 𝑦" = −24𝑥2 + 6

Set to zero: 0 = −6(4𝑥2 − 1) so 𝑥 = ±1

2

8

Page 9: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

Examples: Analyze and sketch 1. 𝑦 = −2𝑥4 + 3𝑥2

• We found the locations of some important points using the derivatives so now we find the corresponding y-values using the original function.

• Maximums: −3

2,

9

8 and

3

2,

9

8

• Minimum: 0,0

• Points of inflection: −1

2,

5

8 and

1

2,

5

8

9

Page 10: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

10

Page 11: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

11

Page 12: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

Examples: Analyze and sketch 2. 𝑓 𝑥 =

2𝑥2−5𝑥+5

𝑥−2

• The numerator is never zero (discriminant is negative) so there is no x-intercepts.

• Vertical asymptote at 𝑥 = 2.

• The y-intercept is at 0, −5

2.

• No horizontal asymptotes as the limit as x approaches infinity does not exist.

• There is a slant asymptote of 𝑦 = 2𝑥 − 1

12

Page 13: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

Examples: Analyze and sketch 2. 𝑓 𝑥 =

2𝑥2−5𝑥+5

𝑥−2

First derivative:

𝑓′ 𝑥 =𝑥 − 2 4𝑥 − 5 − (2𝑥2 − 5𝑥 + 5)(1)

(𝑥 − 2)2

𝑓′ 𝑥 =4𝑥2 − 5𝑥 − 8𝑥 + 10 − 2𝑥2 + 5𝑥 − 5

(𝑥 − 2)2

𝑓′(𝑥) =2𝑥2 − 8𝑥 + 5

(𝑥 − 2)2

13

Page 14: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

Examples: Analyze and sketch 2. 𝑓 𝑥 =

2𝑥2−5𝑥+5

𝑥−2

First derivative: 𝑓′(𝑥) =2𝑥2−8𝑥+5

(𝑥−2)2

Critical numbers: 0 = 2𝑥2 − 8𝑥 + 5 so 𝑥 =4± 6

2

and 𝑥 = 2

14

Page 15: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

Examples: Analyze and sketch 2. 𝑓 𝑥 =

2𝑥2−5𝑥+5

𝑥−2

First derivative: 𝑓′(𝑥) =2𝑥2−8𝑥+5

(𝑥−2)2

Second derivative:

𝑓"(𝑥) =𝑥 − 2 2 4𝑥 − 8 − 2𝑥2 − 8𝑥 + 4 2(𝑥 − 2)

(𝑥 − 2)4

𝑓"(𝑥) =𝑥 − 2 4𝑥 − 8 − 2(2𝑥2 − 8𝑥 + 4)

(𝑥 − 2)3

𝑓"(𝑥) =6

(𝑥 − 2)3

15

Page 16: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

Examples: Analyze and sketch 2. 𝑓 𝑥 =

2𝑥2−5𝑥+5

𝑥−2

First derivative: 𝑓′(𝑥) =2𝑥2−8𝑥+5

(𝑥−2)2

Second derivative: 𝑓"(𝑥) =6

(𝑥−2)3

16

Page 17: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

Examples: Analyze and sketch 2. 𝑓 𝑥 =

2𝑥2−5𝑥+5

𝑥−2

• We found the x-values of key points now we find the y-values from the original function to go with them.

• Maximum: 4− 6

2, 3 − 2 6 ≈ 0.8, −1.9

• Minimum: 4+ 6

2, 3 + 2 6 ≈ (3.2, 7.9)

17

Page 18: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

18

Page 19: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

19

Page 20: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

Examples: Analyze and sketch 3. 𝑦 = 𝑥 9 − 𝑥2

• From the function we can find the domain: 9 − 𝑥2 ≥ 0

So the domain is −3,3 .

• The x-intercepts are at 𝑥 = −3,0,3 and the y-intercept is a 0,0 .

20

Page 21: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

Examples: Analyze and sketch 3. 𝑦 = 𝑥 9 − 𝑥2

First derivative:

𝑦′ = 𝑥1

29 − 𝑥2 −

12 −2𝑥 + 9 − 𝑥2 1

𝑦′ =−𝑥2

9 − 𝑥2+ 9 − 𝑥2

𝑦′ =−𝑥2

9 − 𝑥2+

9 − 𝑥2

9 − 𝑥2=

9 − 2𝑥2

9 − 𝑥2

21

Page 22: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

Examples: Analyze and sketch 3. 𝑦 = 𝑥 9 − 𝑥2

First derivative: 𝑦′ =9−2𝑥2

9−𝑥2

The derivative is zero when 9 − 2𝑥2 = 0 so

when 𝑥 = ±3 2

2. The derivative does not exist at

the endpoints of the domain.

22

Page 23: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

Examples: Analyze and sketch 3. 𝑦 = 𝑥 9 − 𝑥2

First derivative: 𝑦′ =9−2𝑥2

9−𝑥2

Second derivative:

𝑦" =9 − 𝑥2 −4𝑥 − (9 − 𝑥2)

12

9 − 𝑥2 −12(−2𝑥)

9 − 𝑥22

𝑦" =

−4𝑥 9 − 𝑥2 +𝑥(9 − 𝑥2)

9 − 𝑥2

9 − 𝑥2

23

Page 24: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

𝑦" =

−4𝑥 9 − 𝑥2 +𝑥(9 − 𝑥2)

9 − 𝑥2

9 − 𝑥2

𝑦" =

−4𝑥(9 − 𝑥2)

9 − 𝑥2+

𝑥(9 − 𝑥2)

9 − 𝑥2

9 − 𝑥2

𝑦" =−36𝑥 + 4𝑥3 + 9𝑥 − 2𝑥3

(9 − 𝑥2)3/2

𝑦" =2𝑥3 − 27𝑥

(9 − 𝑥2)3/2

24

Page 25: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

Examples: Analyze and sketch 3. 𝑦 = 𝑥 9 − 𝑥2

First derivative: 𝑦′ =9−2𝑥2

9−𝑥2

Second derivative: 𝑦" =2𝑥3−27𝑥

(9−𝑥2)3/2

The second derivative is zero when 𝑥 = 0, ±3 6

2

But these last two are outside of the domain so the only one that we focus on is 𝑥 = 0.

25

Page 26: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

Examples: Analyze and sketch 3. 𝑦 = 𝑥 9 − 𝑥2

First derivative: 𝑦′ =9−2𝑥2

9−𝑥2

Second derivative: 𝑦" =2𝑥3−27𝑥

(9−𝑥2)3/2

26

Page 27: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

Examples: Analyze and sketch 3. 𝑦 = 𝑥 9 − 𝑥2

• We found important locations, now we find the y-values that go with them.

• Maximum: 3 2

2,

9

2

• Minimum: −3 2

2, −

9

2

• Point of inflection: (0,0)

27

Page 28: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

28

Page 29: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

29

Page 30: A Summary of Curve Sketching - UTEP MATHEMATICS · A Summary of Curve Sketching 3 . Note •There is a video that accompanies this section on my website and on YouTube. The link is:

End of Lecture

30