MATH1013 Calculus I

Tutorial 10 Curve Sketching

Conavity

Test for Concavity

Point of Inflection

Slant Asymptote

If 𝒚 = 𝒂𝒙 + 𝒃 is a slant asymptote of the curve 𝒚 = 𝒇(𝒙) , Then 𝒂 =

𝐥𝐢𝐦𝒙→∞𝒇(𝒙)

𝒙 and 𝒃 = 𝐥𝐢𝐦𝒙→∞[𝒇(𝒙) − 𝒂𝒙].

Guildlines for sketching a Curve

Example 1 : Use the guidelines to sketch the curve 𝒚 =𝟐𝒙𝟐

𝒙𝟐−𝟏 .

Example 2 : Sketch the graph of the function 𝒇(𝒙) = 𝒙𝟐

𝟑(𝟔 − 𝒙)𝟏

𝟑 .

Example 3 : Sketch the graph of 𝒇(𝒙) =𝒙𝟐

√𝒙+𝟏 .

Example 4 : Sketch the graph of 𝒇(𝒙) = 𝒙 𝒆𝒙 .

Example 5 : Sketch the graph of 𝒚 = 𝒍𝒏(𝟒 − 𝒙𝟐) .

Exercises:

Sketch the graph

For Q1-11 Sketch the graph of 𝒚 = 𝒇(𝒙).

1) 4.5#18 𝒚 =𝒙

𝒙𝟑−𝟏

2) 4.5#20 𝒚 =𝒙𝟑

𝒙−𝟐

3) 4.5#24 𝒚 = √𝒙𝟐 + 𝒙 − 𝒙

4) 4.5#28 𝒚 = √𝒙𝟐 + 𝟏𝟑

5) Review 4#27 𝒚 = 𝒙√𝟐 + 𝒙

6) 4.5#30 𝒚 = 𝒙𝟓

𝟑 − 𝟓𝒙𝟐

𝟑

7) 4.5#31 𝒚 = √𝒙𝟐 − 𝟏𝟑

8) Review 4 #28 𝒚 =𝒙

√𝒙𝟐−𝟏

9) 4.5#42 𝒚 = (𝟏 − 𝒙)𝒆𝒙

10) 4.5#46 𝒚 = 𝒆𝟐𝒙 − 𝒆𝒙

11) 4.5#50 𝒚 = 𝒍𝒏(𝒙𝟐 − 𝟑𝒙 + 𝟐)

Slant Asymptote

12) Find an equation of the slant asymptote. Do not sketch the curve.

(a) 4.5#61 𝒚 =𝒙𝟐+𝟏

𝒙+𝟏

(b) 4.5#62 𝒚 =𝟐𝒙𝟑+𝒙𝟐+𝒙+𝟑

𝒙𝟐+𝟐𝒙

(c) 4.5#64 𝒚 =𝟓𝒙𝟒+𝒙𝟐+𝒙

𝒙𝟑−𝒙𝟐+𝟐

13) 4.5#65

Sketch the curve 𝒚 =𝒙𝟐

𝒙−𝟏 and find an equation of the slant asymptote.

14) 4.5#68

Sketch the curve 𝒚 =𝒙𝟑

(𝒙+𝟏)𝟐 and find an equation of the slant asymptote.