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11 x1 t09 02 first principles (2012)

by Nigel Simmons, Teacher at Baulkham Hills High School on Jun 04, 2012

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11 x1 t09 02 first principles (2012)Webinar Transcript

• The Slope of a Tangent to a Curve
• The Slope of a Tangent to a Curve y y  f  x x
• The Slope of a Tangent to a Curve y y  f  x P x k
• The Slope of a Tangent to a Curve y y  f  x Q P x k
• The Slope of a Tangent to a Curve y y  f  x Slope PQ is an estimate Q for the slope of line k. P x k
• The Slope of a Tangent to a Curve y y  f  x Slope PQ is an estimate Q for the slope of line k. P Q: Where do we position Q x to get the best estimate? k
• The Slope of a Tangent to a Curve y y  f  x Slope PQ is an estimate Q for the slope of line k. P Q: Where do we position Q x to get the best estimate? A: As close to P as possible. k
• The Slope of a Tangent to a Curve y y  f  x Slope PQ is an estimate Q for the slope of line k. P Q: Where do we position Q x to get the best estimate? A: As close to P as possible. k Q P
• The Slope of a Tangent to a Curve y y  f  x Slope PQ is an estimate Q for the slope of line k. P Q: Where do we position Q x to get the best estimate? A: As close to P as possible. k Q P  x, f  x 
• The Slope of a Tangent to a Curve y y  f  x Slope PQ is an estimate Q for the slope of line k. P Q: Where do we position Q x to get the best estimate? A: As close to P as possible. k  x  h, f  x  h  Q P  x, f  x 
• The Slope of a Tangent to a Curve y y  f  x Slope PQ is an estimate Q for the slope of line k. P Q: Where do we position Q x to get the best estimate? A: As close to P as possible. k  x  h, f  x  h  f  x  h  f  x Q mPQ  xhx f  x  h  f  x  h P  x, f  x 
• The Slope of a Tangent to a Curve y y  f  x Slope PQ is an estimate Q for the slope of line k. P Q: Where do we position Q x to get the best estimate? A: As close to P as possible. k  x  h, f  x  h  f  x  h  f  x Q mPQ  xhx f  x  h  f  x  hTo find the exact value of the slope of k, we P  x, f  x calculate the limit of the slope PQ as h getscloser to 0.
• f  x  h  f  xslope of tangent = lim h0 h
• f  x  h  f  x slope of tangent = lim h0 hThis is known as the “derivative of y with respect to x” and issymbolised;
• f  x  h  f  x slope of tangent = lim h0 hThis is known as the “derivative of y with respect to x” and issymbolised; dy dx
• f  x  h  f  x slope of tangent = lim h0 hThis is known as the “derivative of y with respect to x” and issymbolised; dy , y dx
• f  x  h  f  x slope of tangent = lim h0 hThis is known as the “derivative of y with respect to x” and issymbolised; dy , y , f   x  dx
• f  x  h  f  x slope of tangent = lim h0 hThis is known as the “derivative of y with respect to x” and issymbolised; dy , y , f   x  , d  f  x  dx dx
• f  x  h  f  x slope of tangent = lim h0 hThis is known as the “derivative of y with respect to x” and issymbolised; dy , y , f   x  , d  f  x  the derivative dx dx measures the rate of something changing
• f  x  h  f  x slope of tangent = lim h0 hThis is known as the “derivative of y with respect to x” and issymbolised; dy , y , f   x  , d  f  x  the derivative dx dx measures the rate f  x  h  f  x of something f   x   lim h0 h changing
• f  x  h  f  x slope of tangent = lim h0 hThis is known as the “derivative of y with respect to x” and issymbolised; dy , y , f   x  , d  f  x  the derivative dx dx measures the rate f  x  h  f  x of something f   x   lim h0 h changingThe process is called “differentiating from first principles”
• f  x  h  f  x slope of tangent = lim h0 hThis is known as the “derivative of y with respect to x” and issymbolised; dy , y , f   x  , d  f  x  the derivative dx dx measures the rate f  x  h  f  x of something f   x   lim h0 h changingThe process is called “differentiating from first principles”e.g.  i  Differentiate y  6 x  1 by using first principles.
• f  x  h  f  x slope of tangent = lim h0 hThis is known as the “derivative of y with respect to x” and issymbolised; dy , y , f   x  , d  f  x  the derivative dx dx measures the rate f  x  h  f  x of something f   x   lim h0 h changingThe process is called “differentiating from first principles”e.g.  i  Differentiate y  6 x  1 by using first principles. f  x  6x 1
• f  x  h  f  x slope of tangent = lim h0 hThis is known as the “derivative of y with respect to x” and issymbolised; dy , y , f   x  , d  f  x  the derivative dx dx measures the rate f  x  h  f  x of something f   x   lim h0 h changingThe process is called “differentiating from first principles”e.g.  i  Differentiate y  6 x  1 by using first principles. f  x  6x 1f  x  h  6 x  h 1  6 x  6h  1
• f  x  h  f  x slope of tangent = lim h0 hThis is known as the “derivative of y with respect to x” and issymbolised; dy , y , f   x  , d  f  x  the derivative dx dx measures the rate f  x  h  f  x of something f   x   lim h0 h changing The process is called “differentiating from first principles” e.g.  i  Differentiate y  6 x  1 by using first principles. dy f  x  h  f  x f  x  6x 1  lim dx h0 hf  x  h  6 x  h 1  6 x  6h  1
• f  x  h  f  x slope of tangent = lim h0 hThis is known as the “derivative of y with respect to x” and issymbolised; dy , y , f   x  , d  f  x  the derivative dx dx measures the rate f  x  h  f  x of something f   x   lim h0 h changing The process is called “differentiating from first principles” e.g.  i  Differentiate y  6 x  1 by using first principles. dy f  x  h  f  x f  x  6x 1  lim dx h0 h 6 x  6h  1   6 x  1f  x  h  6 x  h 1  lim h0 h  6 x  6h  1
• f  x  h  f  x slope of tangent = lim h0 hThis is known as the “derivative of y with respect to x” and issymbolised; dy , y , f   x  , d  f  x  the derivative dx dx measures the rate f  x  h  f  x of something f   x   lim h0 h changing The process is called “differentiating from first principles” e.g.  i  Differentiate y  6 x  1 by using first principles. dy f  x  h  f  x f  x  6x 1  lim dx h0 h 6 x  6h  1   6 x  1f  x  h  6 x  h 1  lim h0 h  6 x  6h  1  lim 6h h 0 h
• f  x  h  f  x slope of tangent = lim h0 hThis is known as the “derivative of y with respect to x” and issymbolised; dy , y , f   x  , d  f  x  the derivative dx dx measures the rate f  x  h  f  x of something f   x   lim h0 h changing The process is called “differentiating from first principles” e.g.  i  Differentiate y  6 x  1 by using first principles. dy f  x  h  f  x f  x  6x 1  lim dx h0 h 6 x  6h  1   6 x  1f  x  h  6 x  h 1  lim h0 h  6 x  6h  1  lim 6h h 0 h  lim 6 h0
• f  x  h  f  x slope of tangent = lim h0 hThis is known as the “derivative of y with respect to x” and issymbolised; dy , y , f   x  , d  f  x  the derivative dx dx measures the rate f  x  h  f  x of something f   x   lim h0 h changing The process is called “differentiating from first principles” e.g.  i  Differentiate y  6 x  1 by using first principles. dy f  x  h  f  x f  x  6x 1  lim dx h0 h 6 x  6h  1   6 x  1f  x  h  6 x  h 1  lim h0 h  6 x  6h  1  lim 6h h 0 h  lim 6 h0 6
•  ii  Find the equation of the tangent to y  x 2  5 x  2 at the point 1, 2  .
•  ii  Find the equation of the tangent to y  x 2  5 x  2 at the point 1, 2  . f  x   x2  5x  2
•  ii  Find the equation of the tangent to y  x 2  5 x  2 at the point 1, 2  . f  x   x2  5x  2f  x  h   x  h  5 x  h  2 2
•  ii  Find the equation of the tangent to y  x 2  5 x  2 at the point 1, 2  . f  x   x2  5x  2f  x  h   x  h  5 x  h  2 2  x 2  2 xh  h 2  5 x  5h  2
•  ii  Find the equation of the tangent to y  x 2  5 x  2 at the point 1, 2  . f  x   x2  5x  2f  x  h   x  h  5 x  h  2 2  x 2  2 xh  h 2  5 x  5h  2 dy f  x  h  f  x  lim dx h0 h
•  ii  Find the equation of the tangent to y  x 2  5 x  2 at the point 1, 2  . f  x   x2  5x  2f  x  h   x  h  5 x  h  2 2  x 2  2 xh  h 2  5 x  5h  2 dy f  x  h  f  x  lim dx h0 h x 2  2 xh  h 2  5 x  5h  2  x 2  5 x  2  lim h0 h
•  ii  Find the equation of the tangent to y  x 2  5 x  2 at the point 1, 2  . f  x   x2  5x  2f  x  h   x  h  5 x  h  2 2  x 2  2 xh  h 2  5 x  5h  2 dy f  x  h  f  x  lim dx h0 h x 2  2 xh  h 2  5 x  5h  2  x 2  5 x  2  lim h0 h 2 xh  h  5h 2  lim h0 h
•  ii  Find the equation of the tangent to y  x 2  5 x  2 at the point 1, 2  . f  x   x2  5x  2f  x  h   x  h  5 x  h  2 2  x 2  2 xh  h 2  5 x  5h  2 dy f  x  h  f  x  lim dx h0 h x 2  2 xh  h 2  5 x  5h  2  x 2  5 x  2  lim h0 h 2 xh  h  5h 2  lim h0 h  lim 2 x  h  5 h0
•  ii  Find the equation of the tangent to y  x 2  5 x  2 at the point 1, 2  . f  x   x2  5x  2f  x  h   x  h  5 x  h  2 2  x 2  2 xh  h 2  5 x  5h  2 dy f  x  h  f  x  lim dx h0 h x 2  2 xh  h 2  5 x  5h  2  x 2  5 x  2  lim h0 h 2 xh  h  5h 2  lim h0 h  lim 2 x  h  5 h0  2x  5
•  ii  Find the equation of the tangent to y  x 2  5 x  2 at the point 1, 2  . f  x   x2  5x  2f  x  h   x  h  5 x  h  2 2  x 2  2 xh  h 2  5 x  5h  2 dy f  x  h  f  x  lim dx h0 h x 2  2 xh  h 2  5 x  5h  2  x 2  5 x  2  lim h0 h 2 xh  h  5h 2  lim h0 h  lim 2 x  h  5 h0  2x  5 dy when x  1,  2 1  5 dx  3
•  ii  Find the equation of the tangent to y  x 2  5 x  2 at the point 1, 2  . f  x   x2  5x  2f  x  h   x  h  5 x  h  2 2  x 2  2 xh  h 2  5 x  5h  2 dy f  x  h  f  x  lim dx h0 h x 2  2 xh  h 2  5 x  5h  2  x 2  5 x  2  lim h0 h 2 xh  h  5h 2  lim h0 h  lim 2 x  h  5 h0  2x  5 dy when x  1,  2 1  5 dx  3  the slope of the tangent at 1, 2  is  3
• y  2  3  x  1
• y  2  3  x  1y  2  3 x  3y  3 x  1
• y  2  3  x  1y  2  3 x  3y  3 x  1 Exercise 7B; 1, 2adgi, 3(not iv), 4, 7ab i,v, 12 (just h approaches 0)