MAT137 Calculus! Lecture 20

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1 official website MAT137 Calculus! Lecture 20 Today: 4.6 Concavity 4.7 Asypmtotes Next: 4.8 Curve Sketching

2 Indeterminate Forms for Limits Which of the following are indeterminate forms for limits?

3 Indeterminate and Non-Indeterminate Forms Indeterminate Forms for Limits Non-Indeterminate Forms for Limits 1 A limit in any of these forms is equal to 0: A limit in any of these forms is :

4 L Hôpital s Rule Example 1 Compute lim (1 + 1 x x x ).

5 Indeterminate Powers: 0 0, 1, 0 When lim x a [f (x)] g(x) has the indeterminate form 1, 0 0 or 0, we can logarithms to transform exponentiation into a product. Let y = [f (x)] g(x), then ln y = g(x) ln f (x). Since e x is continuous lim y = lim x a x a eln y = e lim ln y x a. In all the three cases above, this method leads to computing g(x) ln f (x) which is always indeterminate of type 0. lim x a

6 L Hôpital s Rule Example 2 Compute lim x 0 + x x. Answer: e

7 L Hôpital s Rule Example 3 3x + cos x Compute lim. x x Answer: 1 DNE

8 L Hôpital s Rule Example 9 3x + cos x Compute lim. x x Answer: 1 DNE

9 L Hôpital s Rule Example 9 3x + cos x Compute lim. x x Note that this limit has the form /. Wrong Solution Applying L Hôpital s Rule we have 3x + cos x lim x x L H? = lim lim x 3 sin x 1 DNE L Hôpital s Rule doesn t apply!!! f (x) To reach any conclusion from L Hôpital s Rule lim x a g (x) ±. must exist or be

10 L Hôpital s Rule Example 9 3x + cos x Compute lim. x x Right Solution Note that 3x + cos x lim x x So by the Squeeze Theorem Hence, lim x 3x + cos x x = lim (3 + cos x ) x x 0 cos x 1 0 cos x 1 x x = 3. cos x lim x x = 0

11 Concavity y y B B A A a b x a b x

12 Concavity y y B B A A a b x a b x Definition (Concavity) Let f be a function differentiable on an open interval I. The graph of f is 1 concave up on I if f increases on I ; 2 concave down on I if f decreases on I.

13 What does f tell us about f? Theorem Suppose that f is twice differentiable on an open interval I. 1 If f (x) > 0 for all x I, then f increases on I, and the graph of f is concave up. 2 If f (x) < 0 for all x I, then f decreases on I, and the graph of f is concave down.

14 Concavity Definition (Inflection Point) The point P = (c, f (c)) is called an inflection point IF f is continuous at c, and the graph changes from concave up to concave down or from concave down to concave up at P.

15 Inflection Point Theorem If the point (c, f (c)) is a point of inflection, then f (c) = 0 or f (c) does not exist. True or False. If f (a) = 0, then f has an inflection point at a.

16 Inflection Point WARNING The converse is not true. The curve y = x 4 has no inflection point at x = 0. Even though the second derivative y = 12x 2 is zero there, it does not change sign.

17 Concavity Example 4 Consider the function f (x) = x 1/3. Find the inflection points of the graph of f.

18 Concavity Example 4 Consider the function f (x) = x 1/3. Find the inflection points of the graph of f.

19 Concavity and Local Extrema Theorem Second Derivative Test for Local Extrema Suppose f is continuous on an open interval that contains c. 1 If f (c) = 0 and f (c) < 0, then f has a local maximum at x = c. 2 If f (c) = 0 and f (c) > 0, then f has a local minimum at x = c. 3 If f (c) = 0 and f (c) = 0, then the test fails. The function f may have a local maximum, a local minimum, or neither.

20 Vertical Asymptotes Definition The line x = a is called a vertical asymptote of the curve y = f (x) if at least one of the following statements is true. lim f (x) = x a f (x) = lim lim x a lim f (x) = x a x a f (x) = lim lim f (x) = x a + x a + f (x) = a a x x a a x x a a x x

21 Vertical Asymptotes Example 5 Find the vertical asymptotes of f (x) = 3x + 6 x 2 2x 8.

22 Vertical Asymptotes Example 5 Find the vertical asymptotes of f (x) = 3x + 6 x 2 2x 8. y 2 x

23 Horizontal Asymptotes Definition The line y = L is called a horizontal asymptote of the curve y = f (x) if either lim f (x) = L or lim f (x) = L. x x y y y = L horizontal asympotote x horizontal asympotote y = L x

24 Horizontal Asymptotes Example 6 Find the horizontal asymptotes of the graph of the function f (x) = 2 + sin x x.

25 Horizontal Asymptotes Example 6 Find the horizontal asymptotes of the graph of the function f (x) = 2 + sin x x. Does the graph of f have vertical asymptotes?

26 Horizontal Asymptotes Example 6 Find the horizontal asymptotes of the graph of the function f (x) = 2 + sin x x. Does the graph of f have vertical asymptotes? A curve may cross one of its horizontal asymptotes infinitely often.

27 More about Asymptotic Behaviour Definition Let f and g be continuous function. We say that f behaves like g asymptotically if lim [f (x) g(x)] = 0. x ± In particular, we say that the line y = mx + b is an oblique asymptote or slant asymptote if lim [f (x) (mx + b)] = 0. x ± Remark For rational functions, slant asymptotes occur when the degree of the numerator are more than the degree of the denominator. In such a case, the equation of the slant asymptote can be found by long division.

28 Slant Asymptotes Example 7 Find the slant asymptotes of the graph of f (x) = x 2 + 1

29 Slant Asymptotes Example 7 Find the slant asymptotes of the graph of f (x) = x = x = = -

30 Curve Sketching Example 8 Find the vertical and horizontal asymptotes of the graph of the function f (x) = ex + 1 e x 1.

31 Curve Sketching Example 9 Sketch the graph of the function f (x) = ex + 1 e x 1. I have computed the first two derivatives for you: f 2e x (x) = (e x 1) 2 f (x) = 2ex (e x + 1) (e x 1) 3

32 Curve Sketching Example 8 Sketch the graph of the function f (x) = ex + 1 e x

33 Curve Sketching Example 10 Sketch the graph of the function f (x) = e 1/x I have computed the first two derivatives for you: f (x) = e1/x x 2 f (x) = e1/x (2x + 1) x Beatriz Navarro-Lameda L MAT November 2017

34 Curve Sketching Example 7 Sketch the graph of the function f (x) = 8 x 2 4.