Modelling and Change: In-Depth Investigation of Two Functions

Modelling and Change 1
Math 3001: Modelling and Change (Advanced Level), Assignment 2: In-Depth Investigation of
two functions
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Modelling and Change 2
Part 1
a. Purpose of the presentation.
To explore the use of tangent lines, tangent planes and Taylor polynomials for
approximate integration.
b.
c. Equation of the tangent line
The gradient of the tangent line
f ' (x)=sin 2 x
Equation of the tangent line at x= π
2
f ( x )=1
d. area under f(x) between x = 0 and x = π
=π/2 square units
area under the tangent line at x = π/ 2
π/ 2
The two areas are similar
The area under the curve is greater than the area under the tangent line at x = π/ 2
e. Importance of the area under the tangent line
This area provides a linear approximation of the area under the curve. It reveals
information on the areas that is covered by the function if we use a line that passes
through the point only once
f.
T 2=x2

Modelling and Change 3
T 3=0
g.
Blue-the tangent line of f(x) at x=pi/2
Red= f(x)
h.
∫
0
π
T 2=¿ 10.33542 ¿
∫
0
π
T 3=0
i. The use of Polynomial approximation of a single variable function is poor method of
approximating integration of single-variable function in this case because the areas using
using this method is way larger than the actual area.
j.

Modelling and Change 4
k. g ( x , y )= xy
x2+ y2
Volume=∫∫ xy
x2+ y2 dR=0.34657 cubic units
l. At point (1,1), the equation of a plane that is tangential to g(x,y)=
f ( x , y )=1
m. volume between the tangent plane and xy-plane, for R = [0, 1] × [0, 1].
1 cubic units
n. The volume under the tangent plane reveals a linear approximation of the volume being
evaluated.
o. Equation of the second-degree Taylor polynomial G(x, y) of g(x, y) at the point (1, 1).
G ( x , y )= 1
2 + ( 1
4 ) [−x2 +2 xy− y2 ]
p.
q. Present the numerical value of the double integral
=11/24
r. we know that the level curve g(x, y) = 0 exists is known to exist because for the
coordinates x=0,y=0, g(x,y)=0