MAT137 Calculus! Lecture 12

Transcription

1 MAT137 Calculus! Lecture 12 Today we will study more curve sketching ( ) and we will make a review Test 2 will be next Monday, June 26. You can check the course website for further information Next class: you will have a new instructor for the second half of the summer. You can take a look at the syllabus to see what is next

2 Guidelines for Curve Sketching A. Domain. B. Intercepts. The y-intercept is the value f(0); the x-intercepts are the solutions of the equation f(x) = 0. C. Symmetry/Periodicity. If f is even (f( x) = f(x)), then the graph of f is symmetric in the y-axis. If f is odd (f( x) = f(x)), then the graph of f is symmetric in the origin. If f is periodic with period p (f(x +p) = f(p) for all x), then f replicates itself on intervals of length p. D. Asymptotes. E. Increasing/Decreasing. F. Local Max and Min. G. Concavity and Inflection points. H. Sketch the Curve.

3 Guidelines for Curve Sketching Example 0 Example Find the vertical and horizontal asymptotes of the graph of the function f(x) = ex +1 e x 1.

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

5 Guidelines for Curve Sketching Example 1 Example Sketch the graph of the function f(x) = ex +1 e x 1.

6 Guidelines for Curve Sketching Example 1 Example Sketch the graph of the function f(x) = ex +1 e x 1. x = 0 y = 1 f(x) = ex +1 e x 1 y = 1

7 Guidelines for Curve Sketching Example 1 Example Sketch the graph of the function f(x) = xe 1/x We have computed the first two derivatives for you: f (x) = e1/x (x 1) x f (x) = e1/x x 3

8 Guidelines for Curve Sketching Example 1 Example Sketch the graph of the function f(x) = xe 1/x We have computed the first two derivatives for you: f (x) = e1/x (x 1) x f (x) = e1/x x 3 x = 0 f(x) = xe 1/x y = x + 1

9 Guidelines for Curve Sketching Homework Homework Sketch the graph of the function f(x) = 8 x 2 4. We have computed the first two derivatives for you: f (x) = 16x (x 2 4) 2 f (x) = 16(3x2 +4) (x 2 4) 3

10 Curve Sketching Sketch the graph of a function f that has ALL of the following properties: f is continuous everywhere. f(0) = 2. f (x) = 0 for x 2. f (x) < 0 for 0 < x < 1. f (x) > 0 for 2 < x < 0 and for x > 1. f (x) > 0 for 2 < x < 0 and for 0 < x < 2. f (x) < 0 for x > 2. lim f(x) = 2. x

11 Curve Sketching Example 2 Example The graph of the derivative of a continuous function f is shown. 1 On what intervals is f increasing? Decreasing? 2 At what values of x does f have a local maximum? Local minimum? 3 On what intervals is f concave upward? Concave downward? 4 Assuming that f(0) = 0, sketch a graph of f y = f (x)

12 Review! Let s remember some things!

13 Review! Example 3 If f(x) = cos(log( x)), then f is given by: 1 sin(ln( x)) 2 sin(ln( x)) 1 x 3 sin(ln( x)) 2 x 4 sin(ln( x)) 1 2x

14 Review! Let s remember some things!

15 Review! Example 3 If f(x) = cos(log( x)), then f is given by: 1 sin(ln( x)) 2 sin(ln( x)) 1 x 3 sin(ln( x)) 2 x 4 sin(ln( x)) 1 2x

16 Review! Example 4: True of False? If f is odd, the f is even. 1 True 2 False

17 Review! Example 4: True of False? If f is odd, the f is even. 1 True 2 False

18 Review! Example 5 Find the equation of the tangent line to the curve e 3x+y = y 2 x 2 +y 3 +2y +1 at (0,0) 1 y = x 2 y = x +1 3 y = 3x 4 y = x

19 Review! Example 5 Find the equation of the tangent line to the curve e 3x+y = y 2 x 2 +y 3 +2y +1 at (0,0) 1 y = x 2 y = x +1 3 y = 3x 4 y = x

20 Review! Example 6 If f is differentiable and f (x) 0 for all x R, then f(0) f(1). 1 True 2 False

21 Review! Example 6 If f is differentiable and f (x) 0 for all x R, then f(0) f(1). 1 True 2 False

22 Review! Example 7 Find arcsin (sin 54 ) π π π 4 3 π 4

23 Review! Example 7 Find arcsin (sin 54 ) π π π 4 3 π 4

24 Review! Example 8 Evaluate the following limit π 2 lim arctanx x 1 x

25 I think you should restate your question since we do not have a definition of continuity at infinity.

26 Review! Example 9 A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into a circle. Where should the wire be cut so that the total area enclosed is a maximum?