11 x1 t12 04 concavity (2013)

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<ul><li><p>Concavity </p></li><li><p>Concavity The second deriviative measures the change in slope with respect to x, </p><p>this is known as concavity </p></li><li><p>Concavity </p><p> up concave is curve the,0 If xf</p><p>The second deriviative measures the change in slope with respect to x, </p><p>this is known as concavity </p></li><li><p>Concavity </p><p> up concave is curve the,0 If xf</p><p> down concave is curve the,0 If xf</p><p>The second deriviative measures the change in slope with respect to x, </p><p>this is known as concavity </p></li><li><p>Concavity </p><p> up concave is curve the,0 If xf</p><p> down concave is curve the,0 If xf</p><p> inflection ofpoint possible ,0 If xf</p><p>The second deriviative measures the change in slope with respect to x, </p><p>this is known as concavity </p></li><li><p>Concavity </p><p> up concave is curve the,0 If xf</p><p>235sketch derivative second at the lookingBy e.g. 23 xxxy</p><p> down concave is curve the,0 If xf</p><p> inflection ofpoint possible ,0 If xf</p><p>The second deriviative measures the change in slope with respect to x, </p><p>this is known as concavity </p></li><li><p>Concavity </p><p> up concave is curve the,0 If xf</p><p>235sketch derivative second at the lookingBy e.g. 23 xxxy</p><p> down concave is curve the,0 If xf</p><p> inflection ofpoint possible ,0 If xf</p><p>3103 2 xxdx</p><p>dy</p><p>The second deriviative measures the change in slope with respect to x, </p><p>this is known as concavity </p></li><li><p>Concavity </p><p> up concave is curve the,0 If xf</p><p>235sketch derivative second at the lookingBy e.g. 23 xxxy</p><p> down concave is curve the,0 If xf</p><p> inflection ofpoint possible ,0 If xf</p><p>3103 2 xxdx</p><p>dy</p><p>The second deriviative measures the change in slope with respect to x, </p><p>this is known as concavity </p><p>1062</p><p>2</p><p> xdx</p><p>yd</p></li><li><p>Concavity </p><p> up concave is curve the,0 If xf</p><p>235sketch derivative second at the lookingBy e.g. 23 xxxy</p><p> down concave is curve the,0 If xf</p><p> inflection ofpoint possible ,0 If xf</p><p>3103 2 xxdx</p><p>dy</p><p>The second deriviative measures the change in slope with respect to x, </p><p>this is known as concavity </p><p>1062</p><p>2</p><p> xdx</p><p>yd</p><p>0 whenup concave is Curve2</p><p>2</p><p>dx</p><p>yd</p></li><li><p>Concavity </p><p> up concave is curve the,0 If xf</p><p>235sketch derivative second at the lookingBy e.g. 23 xxxy</p><p> down concave is curve the,0 If xf</p><p> inflection ofpoint possible ,0 If xf</p><p>3103 2 xxdx</p><p>dy</p><p>The second deriviative measures the change in slope with respect to x, </p><p>this is known as concavity </p><p>1062</p><p>2</p><p> xdx</p><p>yd</p><p>0 whenup concave is Curve2</p><p>2</p><p>dx</p><p>yd</p><p>3</p><p>5</p><p>0106 i.e.</p><p>x</p><p>x</p></li><li><p>Concavity </p><p> up concave is curve the,0 If xf</p><p>235sketch derivative second at the lookingBy e.g. 23 xxxy</p><p> down concave is curve the,0 If xf</p><p> inflection ofpoint possible ,0 If xf</p><p>3103 2 xxdx</p><p>dy</p><p>The second deriviative measures the change in slope with respect to x, </p><p>this is known as concavity </p><p>1062</p><p>2</p><p> xdx</p><p>yd</p><p>0 whenup concave is Curve2</p><p>2</p><p>dx</p><p>yd</p><p>3</p><p>5</p><p>0106 i.e.</p><p>x</p><p>x</p><p>y </p><p>x </p><p>3</p><p>5</p></li><li><p>Concavity </p><p> up concave is curve the,0 If xf</p><p>235sketch derivative second at the lookingBy e.g. 23 xxxy</p><p> down concave is curve the,0 If xf</p><p> inflection ofpoint possible ,0 If xf</p><p>3103 2 xxdx</p><p>dy</p><p>The second deriviative measures the change in slope with respect to x, </p><p>this is known as concavity </p><p>1062</p><p>2</p><p> xdx</p><p>yd</p><p>0 whenup concave is Curve2</p><p>2</p><p>dx</p><p>yd</p><p>3</p><p>5</p><p>0106 i.e.</p><p>x</p><p>x</p><p>y </p><p>x </p><p>3</p><p>5</p></li><li><p>Turning Points </p></li><li><p>Turning Points All turning points are stationary points. </p></li><li><p>Turning Points All turning points are stationary points. </p><p> point turningminimum ,0 If xf</p></li><li><p>Turning Points All turning points are stationary points. </p><p> point turningminimum ,0 If xf</p><p> point turningmaximum ,0 If xf</p></li><li><p>Turning Points All turning points are stationary points. </p><p> point turningminimum ,0 If xf</p><p> point turningmaximum ,0 If xf</p><p>1 of points turning theFind e.g. 23 xxxy</p></li><li><p>Turning Points All turning points are stationary points. </p><p> point turningminimum ,0 If xf</p><p> point turningmaximum ,0 If xf</p><p>1 of points turning theFind e.g. 23 xxxy</p><p>123 2 xxdx</p><p>dy</p></li><li><p>Turning Points All turning points are stationary points. </p><p> point turningminimum ,0 If xf</p><p> point turningmaximum ,0 If xf</p><p>1 of points turning theFind e.g. 23 xxxy</p><p>123 2 xxdx</p><p>dy</p><p>262</p><p>2</p><p> xdx</p><p>yd</p></li><li><p>Turning Points All turning points are stationary points. </p><p> point turningminimum ,0 If xf</p><p> point turningmaximum ,0 If xf</p><p>1 of points turning theFind e.g. 23 xxxy</p><p>123 2 xxdx</p><p>dy</p><p>262</p><p>2</p><p> xdx</p><p>yd</p><p>0 occur when points Stationary dx</p><p>dy</p></li><li><p>Turning Points All turning points are stationary points. </p><p> point turningminimum ,0 If xf</p><p> point turningmaximum ,0 If xf</p><p>1 of points turning theFind e.g. 23 xxxy</p><p>123 2 xxdx</p><p>dy</p><p>262</p><p>2</p><p> xdx</p><p>yd</p><p>0 occur when points Stationary dx</p><p>dy</p><p>0123 i.e. 2 xx</p></li><li><p>Turning Points All turning points are stationary points. </p><p> point turningminimum ,0 If xf</p><p> point turningmaximum ,0 If xf</p><p>1 of points turning theFind e.g. 23 xxxy</p><p>123 2 xxdx</p><p>dy</p><p>262</p><p>2</p><p> xdx</p><p>yd</p><p>0 occur when points Stationary dx</p><p>dy</p><p>0123 i.e. 2 xx</p><p> 0113 xx</p></li><li><p>Turning Points All turning points are stationary points. </p><p> point turningminimum ,0 If xf</p><p> point turningmaximum ,0 If xf</p><p>1 of points turning theFind e.g. 23 xxxy</p><p>123 2 xxdx</p><p>dy</p><p>262</p><p>2</p><p> xdx</p><p>yd</p><p>0 occur when points Stationary dx</p><p>dy</p><p>0123 i.e. 2 xx</p><p> 0113 xx</p><p>1or 3</p><p>1 xx</p></li><li><p>04 </p><p>216,1when 2</p><p>2</p><p>dx</p><p>ydx</p></li><li><p>04 </p><p>216,1when 2</p><p>2</p><p>dx</p><p>ydx</p><p> point turningmaximum a is 1,2-</p></li><li><p>04 </p><p>216,1when 2</p><p>2</p><p>dx</p><p>ydx</p><p> point turningmaximum a is 1,2-</p><p>04 </p><p>23</p><p>16,</p><p>3</p><p>1when </p><p>2</p><p>2</p><p>dx</p><p>ydx</p></li><li><p>04 </p><p>216,1when 2</p><p>2</p><p>dx</p><p>ydx</p><p> point turningmaximum a is 1,2-</p><p>04 </p><p>23</p><p>16,</p><p>3</p><p>1when </p><p>2</p><p>2</p><p>dx</p><p>ydx</p><p>point turningminimum a is 27</p><p>22,</p><p>3</p><p>1</p></li><li><p>Inflection Points </p></li><li><p>Inflection Points A point of inflection is where there is a change in concavity, to see if </p><p>there is a change, check either side of the point. </p></li><li><p>Inflection Points A point of inflection is where there is a change in concavity, to see if </p><p>there is a change, check either side of the point. </p><p>264 of point(s) inflection theFind e.g. 23 xxy</p></li><li><p>Inflection Points A point of inflection is where there is a change in concavity, to see if </p><p>there is a change, check either side of the point. </p><p>264 of point(s) inflection theFind e.g. 23 xxy</p><p>xxdx</p><p>dy1212 2 1224</p><p>2</p><p>2</p><p> xdx</p><p>yd</p></li><li><p>Inflection Points A point of inflection is where there is a change in concavity, to see if </p><p>there is a change, check either side of the point. </p><p>264 of point(s) inflection theFind e.g. 23 xxy</p><p>xxdx</p><p>dy1212 2 1224</p><p>2</p><p>2</p><p> xdx</p><p>yd</p><p>0 occur when inflection of points Possible2</p><p>2</p><p>dx</p><p>yd</p></li><li><p>Inflection Points A point of inflection is where there is a change in concavity, to see if </p><p>there is a change, check either side of the point. </p><p>264 of point(s) inflection theFind e.g. 23 xxy</p><p>xxdx</p><p>dy1212 2 1224</p><p>2</p><p>2</p><p> xdx</p><p>yd</p><p>0 occur when inflection of points Possible2</p><p>2</p><p>dx</p><p>yd</p><p>01224 .. xei</p><p>2</p><p>1x</p></li><li><p>Inflection Points A point of inflection is where there is a change in concavity, to see if </p><p>there is a change, check either side of the point. </p><p>264 of point(s) inflection theFind e.g. 23 xxy</p><p>xxdx</p><p>dy1212 2 1224</p><p>2</p><p>2</p><p> xdx</p><p>yd</p><p>0 occur when inflection of points Possible2</p><p>2</p><p>dx</p><p>yd</p><p>01224 .. xei</p><p>2</p><p>1x</p><p>x </p><p>2</p><p>2</p><p>dx</p><p>yd0 </p><p>2</p><p>1</p><p>2</p><p>1</p><p>2</p><p>1</p></li><li><p>Inflection Points A point of inflection is where there is a change in concavity, to see if </p><p>there is a change, check either side of the point. </p><p>264 of point(s) inflection theFind e.g. 23 xxy</p><p>xxdx</p><p>dy1212 2 1224</p><p>2</p><p>2</p><p> xdx</p><p>yd</p><p>0 occur when inflection of points Possible2</p><p>2</p><p>dx</p><p>yd</p><p>01224 .. xei</p><p>2</p><p>1x</p><p>x </p><p>2</p><p>2</p><p>dx</p><p>yd0 </p><p>2</p><p>1</p><p>2</p><p>1</p><p>(0) </p><p>(12) </p><p>2</p><p>1</p></li><li><p>Inflection Points A point of inflection is where there is a change in concavity, to see if </p><p>there is a change, check either side of the point. </p><p>264 of point(s) inflection theFind e.g. 23 xxy</p><p>xxdx</p><p>dy1212 2 1224</p><p>2</p><p>2</p><p> xdx</p><p>yd</p><p>0 occur when inflection of points Possible2</p><p>2</p><p>dx</p><p>yd</p><p>01224 .. xei</p><p>2</p><p>1x</p><p>x </p><p>2</p><p>2</p><p>dx</p><p>yd0 </p><p>2</p><p>1</p><p>2</p><p>1</p><p>(-1) (0) </p><p>(-12) (12) </p><p>2</p><p>1</p></li><li><p>Inflection Points A point of inflection is where there is a change in concavity, to see if </p><p>there is a change, check either side of the point. </p><p>264 of point(s) inflection theFind e.g. 23 xxy</p><p>xxdx</p><p>dy1212 2 1224</p><p>2</p><p>2</p><p> xdx</p><p>yd</p><p>0 occur when inflection of points Possible2</p><p>2</p><p>dx</p><p>yd</p><p>01224 .. xei</p><p>2</p><p>1x</p><p>x </p><p>2</p><p>2</p><p>dx</p><p>yd0 </p><p>2</p><p>1</p><p>2</p><p>1</p><p>(-1) (0) </p><p>(-12) (12) </p><p>concavityin change a is there</p><p>2</p><p>1</p></li><li><p>Inflection Points A point of inflection is where there is a change in concavity, to see if </p><p>there is a change, check either side of the point. </p><p>264 of point(s) inflection theFind e.g. 23 xxy</p><p>xxdx</p><p>dy1212 2 1224</p><p>2</p><p>2</p><p> xdx</p><p>yd</p><p>0 occur when inflection of points Possible2</p><p>2</p><p>dx</p><p>yd</p><p>01224 .. xei</p><p>2</p><p>1x</p><p>x </p><p>2</p><p>2</p><p>dx</p><p>yd0 </p><p>2</p><p>1</p><p>2</p><p>1</p><p>(-1) (0) </p><p>(-12) (12) </p><p>concavityin change a is there</p><p>2</p><p>1</p><p>inflection ofpoint a is 32</p><p>1</p><p> ,</p></li><li><p>Inflection Points A point of inflection is where there is a change in concavity, to see if </p><p>there is a change, check either side of the point. </p><p>264 of point(s) inflection theFind e.g. 23 xxy</p><p>xxdx</p><p>dy1212 2 1224</p><p>2</p><p>2</p><p> xdx</p><p>yd</p><p>0 occur when inflection of points Possible2</p><p>2</p><p>dx</p><p>yd</p><p>01224 .. xei</p><p>2</p><p>1x</p><p>x </p><p>2</p><p>2</p><p>dx</p><p>yd0 </p><p>2</p><p>1</p><p>2</p><p>1</p><p>(-1) (0) </p><p>(-12) (12) </p><p>concavityin change a is there</p><p>2</p><p>1</p><p>inflection ofpoint a is 32</p><p>1</p><p> ,</p><p>Horizontal Point of Inflection; 0dx</p><p>dy0</p><p>2</p><p>2</p><p>dx</p><p>yd0</p><p>3</p><p>3</p><p>dx</p><p>yd</p></li><li><p>Alternative Way of Finding </p><p>Inflection Points </p></li><li><p>Alternative Way of Finding </p><p>Inflection Points </p><p>0 occur when inflection of points Possible2</p><p>2</p><p>dx</p><p>yd</p></li><li><p>Alternative Way of Finding </p><p>Inflection Points </p><p>0 occur when inflection of points Possible2</p><p>2</p><p>dx</p><p>yd</p><p>3 5 7</p><p>3 5 7</p><p>If the first non-zero derivative is of odd order,</p><p>i.e 0 or 0 or 0 etc</p><p>then it is a point of inflection</p><p>d y d y d y</p><p>dx dx dx </p></li><li><p>Alternative Way of Finding </p><p>Inflection Points </p><p>0 occur when inflection of points Possible2</p><p>2</p><p>dx</p><p>yd</p><p>3 5 7</p><p>3 5 7</p><p>If the first non-zero derivative is of odd order,</p><p>i.e 0 or 0 or 0 etc</p><p>then it is a point of inflection</p><p>d y d y d y</p><p>dx dx dx </p><p>4 6 8</p><p>4 6 8</p><p>If the first non-zero derivative is of even order,</p><p>i.e 0 or 0 or 0 etc</p><p>then it is not a point of inflection</p><p>d y d y d y</p><p>dx dx dx </p></li><li><p>24..3</p><p>3</p><p>dx</p><p>ydge</p></li><li><p>24..3</p><p>3</p><p>dx</p><p>ydge</p><p>024,2</p><p>1when </p><p>3</p><p>3</p><p>dx</p><p>ydx</p></li><li><p>24..3</p><p>3</p><p>dx</p><p>ydge</p><p>024,2</p><p>1when </p><p>3</p><p>3</p><p>dx</p><p>ydx</p><p>concavityin change a is there</p></li><li><p>24..3</p><p>3</p><p>dx</p><p>ydge</p><p>024,2</p><p>1when </p><p>3</p><p>3</p><p>dx</p><p>ydx</p><p>inflection ofpoint a is 32</p><p>1</p><p> ,</p><p>concavityin change a is there</p></li><li><p>24..3</p><p>3</p><p>dx</p><p>ydge</p><p>024,2</p><p>1when </p><p>3</p><p>3</p><p>dx</p><p>ydx</p><p>inflection ofpoint a is 32</p><p>1</p><p> ,</p><p>Exercise 10E; 1, 2bc, 3, 6ac, </p><p>7bd, 8, 10, 12, 14, 16, 18 </p><p>concavityin change a is there</p></li></ul>