the tangent problem find an equation of the tangent line to the curve at the point (2,3)
TRANSCRIPT
The Tangent Problem
Find an equation of the tangent line to the curve at the point (2,3)3623 xxxy
First we need to estimate the slope at the point (2,3).
4 2 2 4
15
10
5
5
10
15
We are going to estimate the slope using secant lines
1 2 3 4
5
5
10
15
From the left: Using points (2,3) and (1,-1)
41
31
21
)2()1(sec
ff
m
1 2 3 4
5
5
10
15
20
25
From the left:Using points (2,3) and (1.5, -0.375)
75.65.
3375.0
25.1
)2()5.1(sec
ff
m
1 2 3 4
5
5
10
15
20
25
From the left:Using points (2,3) and (1.9, 2.069)
31.91.
3069.2
29.1
)2()9.1(sec
ff
m
1 2 3 4
5
5
10
15
20
25
From the left:Using (2,3) and (1.99,2.9007)
1 2 3 4
5
5
10
15
20
25
93.901.
39007.2
299.1
)2()99.1(sec
ff
m
From the right:Using points (2,3) and (3,21)
1 2 3 4
5
5
10
15
20
25
181
321
23
)2()3(sec
ff
m
From the right:Using points (2,3) and (2.5,9.875)
1 2 3 4
5
5
10
15
20
25
75.135.
3875.9
25.2
)2()5.2(sec
ff
m
From the right:Using points (2,3) and (2.1, 4.071)
1 2 3 4
5
5
10
15
20
25
71.101.
3071.4
21.2
)2()1.2(sec
ff
m
From the right:Using (2,3) and (2.01, 3.1007)
1 2 3 4
5
5
10
15
20
25
07.1001.
31007.3
201.2
)2()01.2(sec
ff
m
Summary of Results
Point (x,y)
Slope of secant line between (2,3) and point (x,y) Point (x,y)
Slope of secant line between (2,3) and point (x,y)
(1, -1) 4 (3, 21) 18
(1.5, -0.375) 6.75 (2.5, 9.875) 13.75
(1.9, 2.069) 9.31 (2.1, 4.071) 10.71
(1.99, 2.9007) 9.9301 (2.01, 3.1007) 10.0701
(1.999, 2.99001) 9.993 (2.001, 3.01001) 10.007
From both the left and the right sides, the slopes of the secant lines are approaching 10, so we can estimate that the slope of the tangent line to the point (2,3) is 10.
• Original Question: Find an equation of the tangent line to the curve at the point (2,3)• We know that m = 10 and we have a point (2,3), so
3623 xxxy
1710
20103
)2(103
)( 11
xy
xy
xy
xxmyy