1. Packet #16 Graphing using Calculus
Math 180
Packet #16
Graphing using Calculus

2. How to sketch a curve
1. Find domain. 2. Find/plot intercepts. (if possible) 3. Find/draw asymptotes. 4. Find 𝒇′ and 𝒇′′, and determine when each are 0 or DNE. 5. Do a sign analysis on 𝑓′ and 𝑓′′. 6. Find/plot maxima/minima, and inflection points. 7. Sketch curve.

3. How to sketch a curve
1. Find domain. 2. Find/plot intercepts. (if possible) 3. Find/draw asymptotes. 4. Find 𝒇′ and 𝒇′′, and determine when each are 0 or DNE. 5. Do a sign analysis on 𝑓′ and 𝑓′′. 6. Find/plot maxima/minima, and inflection points. 7. Sketch curve.

4. How to sketch a curve
1. Find domain. 2. Find/plot intercepts. (if possible) 3. Find/draw asymptotes. 4. Find 𝒇′ and 𝒇′′, and determine when each are 0 or DNE. 5. Do a sign analysis on 𝑓′ and 𝑓′′. 6. Find/plot maxima/minima, and inflection points. 7. Sketch curve.

5. How to sketch a curve
1. Find domain. 2. Find/plot intercepts. (if possible) 3. Find/draw asymptotes. 4. Find 𝒇′ and 𝒇′′, and determine when each are 0 or DNE. 5. Do a sign analysis on 𝑓′ and 𝑓′′. 6. Find/plot maxima/minima, and inflection points. 7. Sketch curve.

6. How to sketch a curve
1. Find domain. 2. Find/plot intercepts. (if possible) 3. Find/draw asymptotes. 4. Find 𝒇′ and 𝒇′′, and determine when each are 0 or DNE. 5. Do a sign analysis on 𝑓′ and 𝑓′′. 6. Find/plot maxima/minima, and inflection points. 7. Sketch curve.

7. How to sketch a curve
1. Find domain. 2. Find/plot intercepts. (if possible) 3. Find/draw asymptotes. 4. Find 𝒇′ and 𝒇′′, and determine when each are 0 or DNE. 5. Do a sign analysis on 𝑓′ and 𝑓′′. 6. Find/plot maxima/minima, and inflection points. 7. Sketch curve.

8. How to sketch a curve
1. Find domain. 2. Find/plot intercepts. (if possible) 3. Find/draw asymptotes. 4. Find 𝒇′ and 𝒇′′, and determine when each are 0 or DNE. 5. Do a sign analysis on 𝑓′ and 𝑓′′. 6. Find/plot maxima/minima, and inflection points. 7. Sketch curve.

9. Ex 1. Sketch the graph of 𝑓 𝑥 =3 𝑥 4 +4 𝑥 3

10. Ex 1. Sketch the graph of 𝑓 𝑥 =3 𝑥 4 +4 𝑥 3

11. Ex 1. Sketch the graph of 𝑓 𝑥 =3 𝑥 4 +4 𝑥 3

12. Ex 1. Sketch the graph of 𝑓 𝑥 =3 𝑥 4 +4 𝑥 3

13. Ex 1. Sketch the graph of 𝑓 𝑥 =3 𝑥 4 +4 𝑥 3

14. Ex 1. Sketch the graph of 𝑓 𝑥 =3 𝑥 4 +4 𝑥 3

15. Ex 1. Sketch the graph of 𝑓 𝑥 =3 𝑥 4 +4 𝑥 3

16. Ex 1. Sketch the graph of 𝑓 𝑥 =3 𝑥 4 +4 𝑥 3

17. Review of asymptotes
1. A vertical asymptote 𝑥=𝑎 happens when lim 𝑥→ 𝑎 − 𝑓 𝑥 =±∞ or lim 𝑥→ 𝑎 + 𝑓 𝑥 =±∞.
For rational functions 𝑃 𝑥 𝑄 𝑥 , they occur when 𝑄 𝑥 =0 and 𝑃 𝑥 ≠0.
ex: Find the vertical asymptotes of 𝑓 𝑥 = 𝑥 3 𝑥 4 −16 .
ex: Does 𝑓 𝑥 =𝑥 ln 𝑥 have a vertical asymptote at 𝑥=0?

18. Review of asymptotes
1. A vertical asymptote 𝑥=𝑎 happens when lim 𝑥→ 𝑎 − 𝑓 𝑥 =±∞ or lim 𝑥→ 𝑎 + 𝑓 𝑥 =±∞.
For rational functions 𝑃 𝑥 𝑄 𝑥 , they occur when 𝑄 𝑥 =0 and 𝑃 𝑥 ≠0.
ex: Find the vertical asymptotes of 𝑓 𝑥 = 𝑥 3 𝑥 4 −16 .
ex: Does 𝑓 𝑥 =𝑥 ln 𝑥 have a vertical asymptote at 𝑥=0?

19. Review of asymptotes
1. A vertical asymptote 𝑥=𝑎 happens when lim 𝑥→ 𝑎 − 𝑓 𝑥 =±∞ or lim 𝑥→ 𝑎 + 𝑓 𝑥 =±∞.
For rational functions 𝑃 𝑥 𝑄 𝑥 , they occur when 𝑄 𝑥 =0 and 𝑃 𝑥 ≠0.
ex: Find the vertical asymptotes of 𝑓 𝑥 = 𝑥 3 𝑥 4 −16 .
ex: Does 𝑓 𝑥 =𝑥 ln 𝑥 have a vertical asymptote at 𝑥=0?

20. Review of asymptotes
1. A vertical asymptote 𝑥=𝑎 happens when lim 𝑥→ 𝑎 − 𝑓 𝑥 =±∞ or lim 𝑥→ 𝑎 + 𝑓 𝑥 =±∞.
For rational functions 𝑃 𝑥 𝑄 𝑥 , they occur when 𝑄 𝑥 =0 and 𝑃 𝑥 ≠0.
ex: Find the vertical asymptotes of 𝑓 𝑥 = 𝑥 3 𝑥 4 −16 .
ex: Does 𝑓 𝑥 =𝑥 ln 𝑥 have a vertical asymptote at 𝑥=0?

21. Review of asymptotes
2. A horizontal asymptote 𝑦=𝐿 happens when lim 𝑥→∞ 𝑓 𝑥 =𝐿 or lim 𝑥→−∞ 𝑓 𝑥 =𝐿. For rational functions 𝑃 𝑥 𝑄 𝑥 … …if degree(𝑃) < degree(𝑄), then HA is 𝑦=0. …if degree(𝑃) = degree(𝑄), then HA is 𝑦= leading coefficient 𝑃 leading coefficient 𝑄 . …if degree(𝑃) > degree(𝑄), then no HA. ex: Find the horizontal asymptote of 𝑓 𝑥 = 5 𝑥 2 3 𝑥 ex: Find the horizontal asymptote of 𝑓 𝑥 = 𝑥 3 𝑒 −𝑥 .

22. Review of asymptotes
2. A horizontal asymptote 𝑦=𝐿 happens when lim 𝑥→∞ 𝑓 𝑥 =𝐿 or lim 𝑥→−∞ 𝑓 𝑥 =𝐿. For rational functions 𝑃 𝑥 𝑄 𝑥 … …if degree(𝑃) < degree(𝑄), then HA is 𝑦=0. …if degree(𝑃) = degree(𝑄), then HA is 𝑦= leading coefficient 𝑃 leading coefficient 𝑄 . …if degree(𝑃) > degree(𝑄), then no HA. ex: Find the horizontal asymptote of 𝑓 𝑥 = 5 𝑥 2 3 𝑥 ex: Find the horizontal asymptote of 𝑓 𝑥 = 𝑥 3 𝑒 −𝑥 .

23. Review of asymptotes
2. A horizontal asymptote 𝑦=𝐿 happens when lim 𝑥→∞ 𝑓 𝑥 =𝐿 or lim 𝑥→−∞ 𝑓 𝑥 =𝐿. For rational functions 𝑃 𝑥 𝑄 𝑥 … …if degree(𝑃) < degree(𝑄), then HA is 𝑦=0. …if degree(𝑃) = degree(𝑄), then HA is 𝑦= leading coefficient 𝑃 leading coefficient 𝑄 . …if degree(𝑃) > degree(𝑄), then no HA. ex: Find the horizontal asymptote of 𝑓 𝑥 = 5 𝑥 2 3 𝑥 ex: Find the horizontal asymptote of 𝑓 𝑥 = 𝑥 3 𝑒 −𝑥 .

24. Review of asymptotes
2. A horizontal asymptote 𝑦=𝐿 happens when lim 𝑥→∞ 𝑓 𝑥 =𝐿 or lim 𝑥→−∞ 𝑓 𝑥 =𝐿. For rational functions 𝑃 𝑥 𝑄 𝑥 … …if degree(𝑃) < degree(𝑄), then HA is 𝑦=0. …if degree(𝑃) = degree(𝑄), then HA is 𝑦= leading coefficient 𝑃 leading coefficient 𝑄 . …if degree(𝑃) > degree(𝑄), then no HA. ex: Find the horizontal asymptote of 𝑓 𝑥 = 5 𝑥 2 3 𝑥 ex: Find the horizontal asymptote of 𝑓 𝑥 = 𝑥 3 𝑒 −𝑥 .

25. Ex 2. Sketch the graph of 𝑓 𝑥 = 𝑥 𝑥+1 2

26. Ex 2. Sketch the graph of 𝑓 𝑥 = 𝑥 𝑥+1 2

27. Ex 2. Sketch the graph of 𝑓 𝑥 = 𝑥 𝑥+1 2

28. Ex 2. Sketch the graph of 𝑓 𝑥 = 𝑥 𝑥+1 2

29. Ex 2. Sketch the graph of 𝑓 𝑥 = 𝑥 𝑥+1 2

30. Ex 2. Sketch the graph of 𝑓 𝑥 = 𝑥 𝑥+1 2

31. Ex 2. Sketch the graph of 𝑓 𝑥 = 𝑥 𝑥+1 2

32. Ex 2. Sketch the graph of 𝑓 𝑥 = 𝑥 𝑥+1 2

33. Ex 2. Sketch the graph of 𝑓 𝑥 = 𝑥 𝑥+1 2