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12.3 Tangent To A CurveRevelation 21:4 "He will wipe every tear from their eyes. Therewill be no more death, or mourning or crying or pain, for the oldorder of things has passed away."
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Consider this graph:Blue line: the functionGreen line: the secant The slope of the secant line is the Average Rate of Change of the function on the interval
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Δy f ( x + h) − f ( x)Slope of Secant: or Δx hWhile this is the average rate of change ofthe function, it isn’t really a very accuratemeasure! How can we get it better? Let hbe only half what it is now ...
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Watch this animation as h gets smaller andthe affect it has on the secant line:
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Watch this animation as h gets smaller andthe affect it has on the secant line:
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What will be best is if we allow h to approachzero as a limit. This then gives us theInstantaneous Rate of Change of the functionat that speciﬁc point! f ( x + h) − f ( x) lim h→0 h
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What will be best is if we allow h to approachzero as a limit. This then gives us theInstantaneous Rate of Change of the functionat that speciﬁc point! f ( x + h) − f ( x) lim h→0 hSince the secant line becomes the tangentline to the function at that point, this limit isthe slope of that tangent line ... and we callthis value the derivative of the function at thatpoint. And ... it is the instantaneous rate ofchange of that function at that point.
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I’m going to do some problems on the board,but I’ll also have the solutions in these slidesso you can review them at a later date.
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1. Find the slope of the line tangent to the graph of the function at P. 2 f ( x ) = x − 5x + 8 P (1, 4 ) end slide
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1. Find the slope of the line tangent to the graph of the function at P. 2 f ( x ) = x − 5x + 8 P (1, 4 )2. Find the equation of the tangent line to the curve at P. Use Point-Slope form. 2 f ( x ) = 2x − 3 P (1,−1) end slide
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1. Find the slope of the line tangent to the graph of the function at P. 2 f ( x ) = x − 5x + 8 P (1, 4 )2. Find the equation of the tangent line to the curve at P. Use Point-Slope form. 2 f ( x ) = 2x − 3 P (1,−1)3. Find the equation of the tangent line to the curve at P. Use Point-Slope form. f ( x) = x + 6 P ( 3, 3) end slide
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1. Find the slope of the line tangent to the graph of the function at P. 2 f ( x ) = x − 5x + 8 P (1, 4 ) f ( x + h) − f ( x) lim h→0 h 2 ( x + h) − 5 ( x + h ) + 8 − ( x − 5x + 8 ) 2 lim h→0 h 2 2 2 x + 2xh + h − 5x − 5h + 8 − x + 5x − 8 lim h→0 h 2 2xh + h − 5h lim h→0 h
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1. Find the slope of the line tangent to the graph of the function at P. (continued) 2 f ( x ) = x − 5x + 8 P (1, 4 ) 2 2xh + h − 5h lim h→0 h h ( 2x + h − 5 ) lim h→0 h lim ( 2x + h − 5 ) h→0 2x − 5 at (1, 4 ) : 2 (1) − 5 −3
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2. Find the equation of the tangent line to the curve at P. Use Point-Slope form. 2 f ( x ) = 2x − 3 P (1,−1) f ( x + h) − f ( x) lim h→0 h 2 2 ( x + h ) − 3 − ( 2x − 3) 2 lim h→0 h 2 2 2 2x + 4xh + 2h − 3 − 2x + 3 lim h→0 h 2 4xh + 2h lim h→0 h
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2. Find the equation of the tangent line to the curve at P. Use Point-Slope form. 2 f ( x ) = 2x − 3 P (1,−1) 2 4xh + 2h lim h→0 h h ( 4x + 2h ) lim h→0 h lim ( 4x + 2h ) h→0 4x at (1,−1) : 4 y + 1 = 4 ( x − 1)
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3. Find the equation of the tangent line to the curve at P. Use Point-Slope form. f ( x) = x + 6 P ( 3, 3) f ( x + h) − f ( x) lim h→0 h x+h+6 − x+6 lim h→0 h x+h+6 − x+6 x+h+6 + x+6 lim g h→0 h x+h+6 + x+6 x+h+6− x−6 lim h→0 h ( x+h+6 + x+6 )
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3. Find the equation of the tangent line to the curve at P. Use Point-Slope form. f ( x) = x + 6 P ( 3, 3) x+h+6− x−6 lim h→0 h ( x+h+6 + x+6 ) 1 lim h→0 x+h+6 + x+6 1 2 x+6
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3. Find the equation of the tangent line to the curve at P. Use Point-Slope form. f ( x) = x + 6 P ( 3, 3) 1 1 1 at P : → → 2 x+6 2 3+ 6 6 1 y − 3 = ( x − 3) 6
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HW #3"I hear and I forget. I see and I remember. I do and I understand." -- Chinese Proverb.“DO PROBLEMS!” -- Mr. Wright
Clipping is a handy way to collect and organize the most important slides from a presentation. You can keep your great finds in clipboards organized around topics.
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