# curve sketching

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Curve Sketching. TJ Krumins Rebecca Stoddard. Curve Sketching Breakdown. Find intercepts and asymptotes Take derivative Set up sign line Find critical points Take 2 nd derivative Set up sign line Find critical points Graph. The problem (dunnunnuh!). Y=x^3-3x^2+4 1) Find y - PowerPoint PPT Presentation

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• Curve Sketching

TJ Krumins Rebecca Stoddard

• Curve Sketching BreakdownFind intercepts and asymptotesTake derivativeSet up sign lineFind critical pointsTake 2nd derivativeSet up sign lineFind critical pointsGraph

• The problem (dunnunnuh!)Y=x^3-3x^2+4

1) Find y2) Find y3) Graph

• Step 1: First DerivativeY=x^3-3x^2+4Y=3x^2-6x

Step 2:FactorY=3x(x-2)

Step 3: Sign Line + - + x-2------------------______ 3x---________________________________F(x) 0 2Therefore, it is increasing when x2, but decreasing from 0

• Dont forget MAX and MIN

x=0 and x=2

Step 4:Plug into the functionF(0)=4F(2)=0Therefore! (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

• Now repeat those steps for the second derivative Step 5: Second derivativeY= 3x^2-6x Y= 6x-6

Step 6: FactorY=6(x-1)

Step 7: Sign Line for the second derivative - +(x-1)------------__________ _________________f(x) 1Therefore, it is concave down when x1

• 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 x2, but decreasing from 0

• The problem (dunnunnuh!)Y=(x+3)/(x-2)

1) Find y3) Graph

• Step 1: First DerivativeY=(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 numbersLook! Vertical asymptotes at x=2 Horizontal at y=1X intercept at x=-3

• Step 4: What do we now know?

Look! Vertical asymptotes at x=2 Horizontal at y=1X intercept at x=-3it is decreasing for all real numbers

• The problem (dunnunnuh!)Y=x/(x-1)

1) Find y3) Graph

• Step 1: First DerivativeY=(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=1Horizontal at y=1X intercept at x=0

• Sample Problem:

1st Derivative:2nd Derivative:

• X-Intercepts and Asymptotes

X-Intercept(s)

no x-intercepts

AsymptotesVertical:

Horizontal:

• Sign Lines

YY-22x---------------x+3---------x+3---------x-3-------------------------x-3--------------------------303-++-66-------------------x2+3x+3---------x+3---------x+3---------x-3---------------------------x-3---------------------------x-3----------------------------303--++Sign Line of 1st Derivative (Increasing/Decreasing)Sign Line of 2nd Derivative (Concavity)

• Reading Sign LinesYY-22x---------------x+3---------x+3---------x-3-------------------------x-3--------------------------303-++-66-------------------x2+3x+3---------x+3---------x+3---------x-3---------------------------x-3---------------------------x-3----------------------------303--++Plus sign + means that the function is increasingMinus sign - means that the function is decreasingClosed circle because the critical point is not a hole or asymptoteCritical Points: x=-3, x=0, x=3Open circle because the critical point is either a hole or asymptote

• Graph of

• 1996 AB 1 FRQThe 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
• Basic Info From GraphFound in intervals above x-axisInflection Points of derivative[-3,-2) increasingx=-2maximum(-2,4) decreasingx=4.minimum (4,5) increasing

Found in areas between maxs/mins of derivative graph[-3,-1) concave down(-1,1) concave up(1,3) concave down(3,5] concave up

• Part AX=-2.MaximumX-intercepts of the derivative are maxs and mins 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)

• Part BX=4minimumX-intercepts of the derivative are maxs and mins of the function.x=4 is the point where the function changes from decreasing to increasingThis change from decreasing to increasing is a minimum because the graph dips down before rising up, similar to a U-shape

• Part C(-1,1) & (3,5]Maxs and mins 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.

• Part D

• THE END

• Intercepts and AsymptotesFind where x=0 in the original functionDo this by factoring (unless already factored)Y=x^2+x-6(X+3)(x-2)X intercepts at x=-3 and x=2Vertical asymptotes: where the denominator equals zero or where there is a negative under a radicalHorizontal Asymptotes:Power on bottom is bigger y=0Power on top is obliquePowers are equal: Ratio of the coefficients

• Oblique AsymptotesOblique is where power on the top is greater than the power on the bottomTo 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

• Setting up a sign lineDraw a line and label it accordingly List all factors on the left most columnList all critical points underneath the lineLabel accordingly For each critical point draw a circleDraw open circles for the factors that are in the denominatorDraw 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 intervalThese 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

• Critical PointsMax and min: Found on the first derivative sign line. Once the x is found plug back into the original function to find the y valueX-intercepts: See intercepts slideVertical asymptotes: See asymptotes slideInflection 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.

• Parts of a Sign Line - - -5------------------------x-2------__________ x-2------______________________________F(x) 2 Show if the function is negative or positive at this pointFactorsCritical PointsThe circles are because the factors are in the denominator