Finding Tangent Line Equation
y = x^3 at x = 2
L(x) = 12x-16
Using: L(x) = 12x-16
Find: x = 1.99
L(1.99) = 7.88
Using: L(x) = 12x-16
At: x = 1.99. L(x) = 7.88
Find the error
l 7.8805 - 7.88 l = .00005
What is this line called?
Tangent Line
What is Lienar Approximation
Approximating the slope of a zoomed-in point on a curve.
y = √x at x = 4
L(x) = 2 + 1/4 (x-4)
Using: L(x) = 2 + 1/4 (x-4)
Find: x = 3.99
L(3.99) = 1.499
Using: L(x) = 2 + 1/4 (x-4)
At: x = 3.99. L(x) = 1.499
Find the error
l 1.997 - 1.499 l = .498
What are the x and y values of this point?
(5,2)
What does a convex graph look like?
*drawing on slides*
f(x) = 1/x-1 at x = -1
L(x) = -1/2 -1/4 (x+1)
Using L(x) = -1/2 -1/4 (x+1)
Find: f(-1.2)
L(-1.2) = -0.45
Using L(x) = -1/2 -1/4 (x+1)
At x=-1.2 L(x) = -0.45
Find the error
l -0.4545 - (-.45) l = 0.0045
lErrorl < 10^-2
What does this point represent?
Exact value
What does a concave graph look like?
*drawing on slides*
y= sinx at x=π/3, corresponding ordered pair: (1/2, √3/2)
f(x) ≈ L(x) = √3/2 + 1/2 (x - π/3)
Using: f(x) ≈ L(x) = √3/2 + 1/2 (x - π/3)
Find: x=1.05
L(x) ≈ 0.867426
Using: f(x) ≈ L(x) = √3/2 + 1/2 (x - π/3)
at x=1.05 L(x) ≈ 0.867426
Find the error
l f(π/3)-L(1.05) l = .0014005962
What is this point called?
Point of Tangency
What is Local Linearity
The zoomed-in point on a curve where there can be a tangent line that can approximate the slope.
f(x) = 2^x at x = 3
f(x) = (8ln2)x - 24ln2 + 8
Using: f(x) = (8ln2)x - 24ln2 + 8
Find: x=2.97
L(2.97) = 7.83364
Using: f(x) = (8ln2)x - 24ln2 + 8
x=2.97 L(2.97) = 7.83364
Find the error percentage
l 7.835-7.833/7.835 l x 100 = .0255% error
What does this point represent?
Linear Approximation Value
What does the visual between a curve and tangent line on a graph represent when finding local linearity?
The error.