1. Using Derivatives for Curve Sketching And Finding Asymptotes Thanks to Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 1995 Old Faithful Geyser, Yellowstone National Park

2. Using Derivatives for Curve Sketching Curves have local Maximums and Minimums that are like Hills and Valleys They may be like the wild terrain in his Picture !! Photo by Vickie Kelly, 2007 Yellowstone Falls, Yellowstone National Park

3. Using Derivatives for Curve Sketching Mammoth Hot Springs, Yellowstone National Park Our Job is to use our mathematical tools to sketch curves We have a formidable Arsenal of maths tools!!

4. In the past, one of the important uses of derivatives was as an aid in curve sketching. Even though we usually use a calculator or computer to draw complicated graphs, it is still important to understand the relationships between derivatives and graphs. Curve Sketching Tools 1.Inspection 2.Intercepts and roots 3.Gradients 4.2 nd derivative 5.Asymptote analysis 6.Plus Some

5. First derivative: is positive Curve is rising. is negative Curve is falling. is zero Possible local maximum or minimum. Second derivative: is positive Curve is concave up. is negative Curve is concave down. is zero Possible inflection point (where concavity changes).

6. f’(x) f’’(x) f(x)=2x 3 -4x-4 f’(x)=0 f(x) decreasing as f’(x)<0 f(x) increasing as f’(x)>0 f ’’(x)<0 Concave up f ’’(x)>0 Concave down f ‘’(x)=0 Pt Inflection

7. Example: Graph There are roots at and. Set First derivative test: negative positive Possible extreme at. We can use a chart to organize our thoughts.

8. Example: Graph There are roots at and. Set First derivative test: maximum at minimum at Possible extreme at.

9. Example: Graph First derivative test: NOTE: In any Exam, it is not sufficient to simply draw the chart and write the answer. You must give a written explanation! There is a local maximum at (0,4) because for all x in and for all x in (0,2). There is a local minimum at (2,0) because for all x in (0,2) and for all x in.

10. Because the second derivative at x = 0 is negative, the graph is concave down and therefore (0,4) is a local maximum. Example: Graph There are roots at and. Possible extreme at. Or you could use the second derivative test: Because the second derivative at x = 2 is positive, the graph is concave up and therefore (2,0) is a local minimum.

11. inflection point at There is an inflection point at x = 1 because the second derivative changes from negative to positive. Example: Graph We then look for inflection points by setting the second derivative equal to zero. Possible inflection point at. negative positive

12. Make a summary table: rising, concave down local max falling, inflection point local min rising, concave up 

13. Finding Asymptotes

14. xf(x)f(x) 20.5 11 2 0.110 0.01100 0.0011000 xf(x)f(x) -2-0.5 -0.5-2 -0.1-10 -0.01-100 -0.001-1000 As x → 0 –, f(x) → -∞. As x → 0 +, f(x) → +∞. Rational Function A rational function is a function of the form f(x) =, where P(x) and Q(x) are polynomials and f(x) = 0. f(x) = Example: f (x) = is defined for all real numbers except x = 0.

15. As x → 0 +, f(x)→ +∞. As x → 0 –, f(x)→ -∞.

16. x x = a as x → a – f(x) → + ∞ x x = a as x → a – f(x) → – ∞ x x = a as x → a + f(x) → + ∞ x x = a as x → a + f(x) → – ∞ The line x = a is a vertical asymptote of the graph of y = f(x), if and only if f(x) → + ∞ or f(x) → – ∞ as x → a + or as x → a –. Vertical Asymptote

17. Example: Show that the line x = 2 is a vertical asymptote of the graph of f(x) =. xf(x)f(x) 1.516 1.9400 1.9940000 2- 2.0140000 2.1400 2.516 x→2 –, f (x) → + ∞ x→2 +, f (x) → + ∞ This shows that x = 2 is a vertical asymptote. y x 100 0.5 x = 2 x→2 – { f (x) → + ∞ { { x→2 + { f (x) → + ∞

18. Set the denominator equal to zero and solve. Solve the quadratic equation x 2 + 4x –(x – 1)(x + 5) = 0 Therefore, x = 1 and x = -5 are the values of x for which f may have a vertical asymptote. Example 2: Vertical Asymptote A rational function may have a vertical asymptote at x = a for any value of a such that Q(a) = 0. Example: Find the vertical asymptotes of the graph of f(x) =.

19. As x →1 –, f(x) → – ∞ As x →1 +, f(x) → + ∞ As x → -5 –, f(x) → + ∞ As x →-5 +, f(x) → – ∞ x = 1 is a vertical asymptote. x = -5 is a vertical asymptote. Example 1: Vertical Asymptote

20. 1. Find the roots of the denominator. 0 = x 2 – 4 = (x + 2)(x –2) Possible vertical asymptotes are x = -2 and x = +2. 2. Since the (x + 2) will cancel from the numerator and denominator, x = -2 is not a VA. The table on the graphing calculator will show that as x → -2 –, f(x) → -0.25 and as x → -2 +, f(x) → -0.25; another reason why x = -2 is not a vertical asymptote. x → +2 –, f(x) → – ∞ and x →+2 +, f(x) → + ∞. So, x = 2 is a vertical asymptote. f is undefined at x = -2 A hole in the graph of f at (-2, -0.25) shows a removable singularity. x = 2 Example 3: Vertical Asymptote Example: Find the vertical asymptotes of the graph of f(x) =. x y (-2, -0.25)

21. y y = b as x → + ∞ f(x) → b – y y = b as x → – ∞ f(x) → b – y y = b as x → + ∞ f(x) → b + y y = b as x → – ∞ f(x) → b + The line y = b is a horizontal asymptote of the graph of y = f(x) if and only if Horizontal Asymptote

22. xf(x)f(x) 100.1 1000.01 10000.001 0 – -10-0.1 -100-0.01 -1000-0.001 As x becomes unbounded positively, f(x) approaches zero from above; therefore, the line y = 0 is a horizontal asymptote of the graph of f. As f(x) → – ∞, x → 0 –. Example: Show that the line y = 0 is a horizontal asymptote of the graph of the function f(x) =. x y f(x) = y = 0

23. Nobody wants to plot points to determine a horizontal asymptote! Luckily you can determine the horizontal asymptotes by remembering just 3 little things: 1.If the degree of the num  degree of the denom, then y = 0 is a HA. 2. If the degree of the num  degree of the denom, there is NO HA. 3. If the degree of the num  degree of the denom, then the line y = LC of num is the HA. LC of denom Example: has a HA at.

24. y x The degree of the num = the degree of the denom so the graph has a HA at y = 1. Therefore, f(x) has y = 1 as a horizontal asymptote. Example: Determine the horizontal asymptotes of the graph of f(x) =. Example 2: Horizontal Asymptote y = 1 Example 1: Horizontal Asymptote

25. Finding Asymptotes for Rational Functions Check to factor and cancel. Vertical asymptotes occur at x values that make the denominator equal zero. If n > d, then there are no horizontal asymptotes. If n < d, then y = 0 is a horizontal asymptote. If n = d, then y = LC m is a horizontal asymptote. LC n Given a rational function: f (x) = N(x) D(x)

26. Factor the numerator and denominator. Since (x + 1) will cancel, there is a hole in the graph at x = -1. x = 2 will be a vertical asymptote. Since the polynomials have the same degree, y = 3 will be a horizontal asymptote. Example: Find all horizontal and vertical asymptotes of f (x) =. y = 3 x = 2 x y

27. Slant Asymptote A slant asymptote occurs when the degree of the numerator is exactly 1 more than the degree of the denominator. A slant asymptote is found by using long division.

28. The slant asymptote is y = 2x – 5. As x → + ∞, → 0 +. Example: Find the slant asymptote for f(x) =. x y x = -3 y = 2x - 5 Divide : Therefore as x →  ∞, f(x) is more like the line y = 2x – 5. As x → – ∞, → 0 –.