Unformatted text preview:

MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Lecture 10 18.01 Fall 2006 Lecture 10: Curve Sketching Goal: To draw the graph of f using the behavior of f� and f��. We want the graph to be qualitatively correct, but not necessarily to scale. Typical Picture: Here, y0 is the minimum value, and x0 is the point where that minimum occurs. x0= critical pointy0Figure 1: The critical point of a function Notice that for x < x0, f�(x) < 0. In other words, f is decreasing to the left of the critical point. For x > x0, f�(x) > 0: f is increasing to the right of the critical point. Another typical picture: Here, y0 is the critical (maximum) value, and x0 is the critical point. f is decreasing on the right side of the critical point, and increasing to the left of x0. x0= critical pointy0f’(x) < 0x > x0Figure 2: A concave-down graph 1Lecture 10 18.01 Fall 2006 Rubric for curve-sketching 1. (Precalc skill) Plot the discontinuities of f — especially the infinite ones! 2. Find the critical points. These are the points at which f�(x) = 0 (usually where the slope changes from positive to negative, or vice versa.) 3. (a) Plot the critical points (and critical values), but only if it’s relatively easy to do so. (b) Decide the sign of f�(x) in between the critical points (if it’s not already obvious). 4. (Precalc skill) Find and plot the zeros of f. These are the values of x for which f (x) = 0. Only do this if it’s relatively easy. 5. (Precalc skill) Determine the behavior at the endpoints (or at ±∞). Example 1. y = 3x − x3 1. No discontinuities. 2. y� = 3 − 3x2 = 3(1 − x2) so, y� = 0 at x = ±1. 3. (a) At x = 1, y = 3 − 1 = 2. (b) At x = −1, y = −3 + 1 = −2. Mark these two points on the graph. 34. Find the zeros: y = 3x − x = x(3 − x2) = 0 so the zeros lie at x = 0, ±√3. 5. Behavior of the function as x → ±∞. As x → ∞, the x3 term of y dominates, so y → −∞. Likewise, as x → −∞, y → ∞. Putting all of this information together gives us the graph as illustrated in Fig. 3) (-√3,0)(√3,0)(-1,-2)(1,2)21-2-13Figure 3: Sketch of the function y = 3x − x . Note the labeled zeros and critical points Let us do step 3b (the sign of f�) to double-check for consistency. y� = 3 − 3x 2 = 3(1 − x 2) y� > 0 when |x| < 1; y� < 0 when |x| > 1. Sure enough, y is increasing between x = −1 and x = 1, and is decreasing everywhere else. 21Example 2. y = . x This example illustrates why it’s important to find a function’s discontinuities before looking at the properties of its derivative. We calculate y� = −x2 1 < 0 Warning: The derivative is never positive, so you might think that y is always decreasing, and its graph looks something like that in Fig. 4. Figure 4: A monotonically decreasing function 1But as you probably know, the graph of looks nothing like this! It actually looks like Fig. 5. In x 1fact, y = is decreasing except at x = 0, where it jumps from −∞ to +∞. This is why we must x watch out for discontinuities. Figure 5: Graph of y = 1. x 3 Lecture 10 18.01 Fall 2006Lecture 10 18.01 Fall 2006 � �Example 3. y = x3 − 3x2 + 3x. y� = 3x 2 − 6x + 3 = 3(x 2 − 2x + 1) = 3(x − 1)2 There is a critical point at x = 1. y� > 0 on both sides of x = 1, so y is increasing everywhere. In this case, the sign of y� doesn’t change at the critical point, but the graph does level out (see Fig. 6. 11horizontal slope(1,1)3Figure 6: Graph of y = y = x − 3x2 + 3x ln xExample 4. y = (Note: this function is only defined for x > 0) x What happens as x decreases towards zero? Let x = 2−n . Then, ln 2−n y =2−n = (−n ln 2)2n → −∞ as n → ∞ In other words, y decreases to −∞ as x approaches zero. Next, we want to find the critical points. y� = ln x � = x( x 1 ) − 1(ln x)=1 − ln x x x2 x2 y� = 0 = ⇒ 1 − ln x = 0 = ⇒ ln x = 1 = ⇒ x = e In other words, the critical point is x = e (from previous page). The critical value is ln e 1 y(x) |x=e = e = e 4Lecture 10 18.01 Fall 2006 Next, find the zeros of this function: y = 0 ln x = 0 ⇔ So y = 0 when x = 1. What happens as x → ∞? This time, consider x = 2+n . ln 2n n ln 2 n(0.7) y = = 2n 2n ≈ 2n So, y → 0 as n → ∞. Putting all of this together gets us the graph in Fig. 7. e11/e(e,1/e)Figure 7: Graph of y = ln xx Finally, let’s double-check this picture against the information we get from step 3b: y� =1 − ln x> 0 for 0 < x < e x2 Sure enough, the function is increasing between 0 and the critical point. 5Lecture 10 18.01 Fall 2006 2nd Derivative Information When f�� > 0, f� is increasing. When f�� < 0, f� is decreasing. (See Fig. 8 and Fig. 9) slope < 0slope = 0slope > 0Figure 8: f is convex (concave-up). The slope increases from negative to positive as x increases. Figure 9: f is concave-down. The slope decreases from positive to negative as x increases. Therefore, the sign of the second derivative tells us about concavity/convexity of the graph. Thus the second derivative is good for two purposes. 1. Deciding whether a critical point is a maximum or a minimum. This is known as the second derivative test. f�(x0) f��(x0) Critical point is a: 0 negative maximum 0 positive minimum 2. Concave/convex “decoration.” 6Lecture 10 18.01 Fall 2006 The points where f�� = 0 are called inflection points. Usually, at these points the graph changes from concave up to down, or vice versa. Refer to Fig. 10 to see how this looks on Example 1. Inflection point (where f” = 0)3Figure 10: Inflection point: y = 3x − x , y�� = −6x = 0, at x = 0.


View Full Document

MIT 18 01 - Curve Sketching

Documents in this Course
Graphing

Graphing

46 pages

Exam 2

Exam 2

3 pages

Load more
Download Curve Sketching
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Curve Sketching and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Curve Sketching 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?