Curve Sketching
description
Transcript of Curve Sketching
![Page 1: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/1.jpg)
Curve
Sketching TJ Krumins
Rebecca Stoddard
![Page 2: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/2.jpg)
Curve Sketching BreakdownCurve Sketching Breakdown
1)1) Find Find interceptsintercepts and and asymptotesasymptotes2)2) Take derivativeTake derivative3)3) Set up Set up sign linesign line4)4) Find Find critical pointscritical points5)5) Take 2Take 2ndnd derivative derivative6)6) Set up sign lineSet up sign line7)7) Find critical pointsFind critical points8)8) Graph Graph
![Page 3: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/3.jpg)
The problem (dun…nun…nuh!)
Y=x^3-3x^2+4
1) Find y’
2) Find y’’
3) Graph
![Page 4: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/4.jpg)
Step 1: First Derivative Y=x^3-3x^2+4 Y’=3x^2-6x
Step 2:Factor Y’=3x(x-2)
Step 3: Sign Line + - + x-2------------------______
3x---________________________________F’(x) 0 2
Therefore, it is increasing when x<0 and increasing when x>2, but decreasing from 0<x<2
Look! No asymptotes!
![Page 5: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/5.jpg)
Don’t forget MAX and MIN
x=0 and x=2
• Step 4:Plug into the function– F(0)=4– F(2)=0
• Therefore! (4,0) and (2,0) are either a max or a min-(4,0) is a max because it is increasing and then
decreasing
-(2,0) is a min because it is decreasing then increasing
![Page 6: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/6.jpg)
Now repeat those steps for the second derivativeNow repeat those steps for the second derivative
Step 5: Second derivativeStep 5: Second derivative Y’= 3x^2-6x Y’= 3x^2-6x Y”= 6x-6Y”= 6x-6
Step 6: FactorStep 6: Factor Y”=6(x-1)Y”=6(x-1)
Step 7: Sign Line for the second derivativeStep 7: Sign Line for the second derivative - +- +(x-1)------------__________(x-1)------------__________ _________________f”(x)_________________f”(x) 11
Therefore, it is concave down when x<1 and concave up when x>1
![Page 7: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/7.jpg)
• Step 8: Plug in x=1 for an inflection pointF(1)=2
• Step 9: What do we now know?
– Max at (0,4)– Min at (2,0)– Inflection point at (2,1)– it is increasing when x<0 and increasing when
x>2, but decreasing from 0<x<2– it is concave down when x<1 and concave up
when x>1
![Page 8: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/8.jpg)
![Page 9: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/9.jpg)
The problem (dun…nun…nuh!)
Y=(x+3)/(x-2)
1) Find y’
3) Graph
![Page 10: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/10.jpg)
• Step 1: First Derivative– Y=(x+3)/(x-2) – Y’=(x-2)(1)-(x-3)(1)/((x-2)^2)– ((x-2)-(x-3))/((x-2)(x-2))
• Step 2: Simplify – -5/(x-2)(x-2)
• Step 3: Sign Line - -
-5------------------------x-2------__________
x-2------______________________________F’(x) 2
Therefore, it is decreasing for all real numbers
Look! Vertical asymptotes at x=2
Horizontal at y=1
X intercept at x=-3
![Page 11: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/11.jpg)
• Step 4: What do we now know?
– Look! Vertical asymptotes at x=2 – Horizontal at y=1– X intercept at x=-3– it is decreasing for all real numbers
![Page 12: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/12.jpg)
![Page 13: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/13.jpg)
The problem (dun…nun…nuh!)
Y=x/(x-1)
1) Find y’
3) Graph
![Page 14: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/14.jpg)
• Step 1: First Derivative– Y=(x)/(x-1) – Y’=(x-1)(1)-(x)(1)/(x-1)^2– -1/((x-1)(x-1))
• Step 2: Sign Line - -
-1------------------------x-1------__________
x-1------______________________________F’(x) 1
Look! Vertical asymptotes at x=1
Horizontal at y=1
X intercept at x=0
![Page 15: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/15.jpg)
![Page 16: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/16.jpg)
Sample Problem:
1st Derivative: 2nd Derivative:
![Page 17: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/17.jpg)
X-Intercepts and Asymptotes
• X-Intercept(s)
no x-intercepts
• Asymptotes– Vertical:
– Horizontal:
![Page 18: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/18.jpg)
Sign Lines
Y’ Y’’
-22x---------------x+3---------x+3---------x-3-------------------------x-3-------------------------
-3 0 3
- + +-
66-------------------x2+3x+3---------x+3---------x+3---------x-3---------------------------x-3---------------------------x-3---------------------------
-3 0 3
- - ++
Sign Line of 1st Derivative (Increasing/Decreasing)
Sign Line of 2nd Derivative (Concavity)
![Page 19: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/19.jpg)
Reading Sign Lines
Y’ Y’’
-22x---------------x+3---------x+3---------x-3-------------------------x-3-------------------------
-3 0 3
- + +-
66-------------------x2+3x+3---------x+3---------x+3---------x-3---------------------------x-3---------------------------x-3---------------------------
-3 0 3
- - ++Plus sign “+” means that the function is increasing
Minus sign “-” means that the function is decreasing
Closed circle because the critical point is not a hole or asymptote
Critical Points: x=-3, x=0, x=3
Open circle because the critical point is either a hole or asymptote
![Page 20: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/20.jpg)
Graph of
![Page 21: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/21.jpg)
1996 AB 1 FRQ
The figure above shows the graph of f’(x), the derivative of a function f. The domain of f is the set of all real numbers x such that -3<x<5.
a]For what values of x does f have a relative maximum? Why?b]For what values of x does f have a relative minimum? Why?c]On what intervals is the graph of f concave upward? Use f’ to justify your answer.d]Suppose that f(1)=0. In the xy-plane provided, draw a sketch that shows the general shape of the graph of the function f on the open interval 0<x<2.
![Page 22: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/22.jpg)
Basic Info From Graph
Found in intervals above x-axis Inflection Points of derivative
[-3,-2) increasing x=-2…maximum(-2,4) decreasing x=4….minimum (4,5) increasing
Found in areas between max’s/min’s of derivative graph[-3,-1) concave down(-1,1) concave up(1,3) concave down(3,5] concave up
![Page 23: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/23.jpg)
Part A
• X=-2….Maximum
• X-intercepts of the derivative are max’s and min’s of the function and x=-2 is the point where the function changes from increasing to decreasing (which is seen through the intervals where the derivative is above or below the x-axis)
![Page 24: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/24.jpg)
Part B
• X=4…minimum
• X-intercepts of the derivative are max’s and min’s of the function.
• x=4 is the point where the function changes from decreasing to increasing– This change from decreasing to increasing is
a minimum because the graph dips down before rising up, similar to a U-shape
![Page 25: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/25.jpg)
Part C
• (-1,1) & (3,5]
• Max’s and min’s of the derivative are points of inflection of the function.
• A minimum in the function occurs in the interval (3,5] & concavity changes at each inflection point.
• Therefore, these intervals are concave up.
![Page 26: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/26.jpg)
Part D
![Page 27: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/27.jpg)
THE END
![Page 28: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/28.jpg)
Intercepts and Asymptotes
• Find where x=0 in the original function
• Do this by factoring (unless already factored)
• Y=x^2+x-6(X+3)(x-2)
X intercepts at x=-3 and x=2
• Vertical asymptotes: where the denominator equals zero or where there is a negative under a radical
• Horizontal Asymptotes:– Power on bottom is bigger
y=0– Power on top is oblique– Powers are equal: Ratio of
the coefficients
![Page 29: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/29.jpg)
Oblique Asymptotes
• Oblique is where power on the top is greater than the power on the bottom
• To solve these use long division (divide the numerator by the denominator).
• The answer will be a line (if done correctly) and will be the oblique asymptote
![Page 30: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/30.jpg)
Setting up a sign line• Draw a line and label it accordingly • List all factors on the left most column• List all critical points underneath the line• Label accordingly • For each critical point draw a circle
– Draw open circles for the factors that are in the denominator– Draw closed circles for the factors that are in the numerator
• Where x is positive in the factor draw ---- leading up to the circle, then a solid line following it. (do the opposite if it is negative)
• For each interval, if the number of --- lines is even draw a + sign over that interval.
• If the interval has a odd amount of ---- lines than draw a negative sign over the interval
• These will tell you either if the graph is increasing/decreasing (first derivative) or if it is concave up/concave down (second derivative)
Parts of a sign line
![Page 31: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/31.jpg)
Critical Points
• Max and min: Found on the first derivative sign line. Once the x is found plug back into the original function to find the y value
• X-intercepts: See intercepts slide• Vertical asymptotes: See asymptotes slide• Inflection points: Found on the sign line of the
second derivative. Once the x value is found plug back into the original function to find the y value.
![Page 32: Curve Sketching](https://reader033.fdocuments.net/reader033/viewer/2022061513/56814bf1550346895db8dad8/html5/thumbnails/32.jpg)
Parts of a Sign Line
- -
-5------------------------
x-2------__________
x-2------______________________________F’(x)
2
Show if the function is negative or positive at this point
Factors Critical Points
The circles are because the factors are in the denominator