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Find slope of the tangent to f(x) at P and then find equation of the tangent line at P....
Find slope of the tangent to f(x) at P and then find equation of the tangent line at P. Is the derivative of the equation of the tangent line at p?
Let P(1,5) to be a point through f(x)=6x-x^2
-Let Q(x, 6x-x^2) be on the graph. Find slope of the secant line PQ for x=3, 2, 1.5, 1.01, 1.001.
-Use your aswer to guess the slope of the tangentto f(x) at P.
-Then find the equation of the tangent line at P.
2 Answers | add yours
The line that passes through PQ is written applying the formula:
(xQ-xP)/(x-xP) = (yQ-yP)/(y-yP)
The slope of the line is m = (yQ-yP)/(xQ-xP)
y-yP = (yQ-yP)(x-xP)/(xQ-xP)
We'll substitute the coordinates of P and Q into the formula above:
y-5 = (6x-x^2-5)(x-1)/(x-1)
We'll find slope of the secant line PQ for x=3
m = (6*3-3^2-5)/(3-1)
m = (18 - 9 - 5)/2
m = 4/2
m = 2
We'll find slope of the secant line PQ for x = 2
m = (6*2-2^2-5)/(2-1)
m = (12-4-5)/1
m = 3
We'll find slope of the secant line PQ for x = 1.5
m = (6*1.5 - 1.5^2 - 5)/(1.5-1)
m = (9-2.25) / (0.5)
m = 1.75/0.5
m = 3.5
The tangent line to the function f(x), in the point P is the value of derivative of the function f(x), in that point.
We'll calculate the derivative of f(x).
f'(x) = (6x-x^2)
f'(x) = (6x)' - (x^2)'
f'(x) = 6 - 2x
f'(1) = 6 - 2*1
f'(1) = 4
Posted by giorgiana1976 on September 26, 2010 at 1:28 AM (Answer #1)
High School Teacher
f(x) = 6x-x^2.
P(1,5) is point on the curve as f(1) = 6*1*1^2 = 5 verifies.
The slope of the curve at (1,5) is:
dy/dx = (6x-x^2)' = 6-2x.
So (dy/dx at x= 1 ) is 6-2*1 = 4.
So the equation of tangent at (x1,y1) is y-t1 = (dy/dx)(x-x1)
But (x1,y1) = (1,5) and (dy/dx at x = 1) = 4.
So the equation of tangent at (1,5) is:
y-5 = 4(x-1) .
4x-y -4+5 = 0
4x-y+1 = 0.
The slope secant value PQ :
slope = (Py- Qy)/(Px- Qx).
Qx = 3, 2, 1.5, 1.01, 1.001
Qy =9, 8, 6.75, 5.0399, 5.00399.
The value of the slope :
Slope : (Py-Qy)/(Px-Qx)
(5-9)/(1-3) = 2
(5-6.75)/(1-1.5) = 3.5
(5-5.0399)/(1-1.01) = 3.99
(5-5.003999)/(1-1.001) = 3,999
The process shows that limit of the secant PQ as Q tends P is dy/dx = 4.
Posted by neela on September 26, 2010 at 2:53 AM (Answer #3)
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