Chapter 4: Partial Differentiation
Section 4.7: Approximations
Example 4.7.9
Show that the plane tangent to fx,y=3 x2+5 x y−7 y2+4 at x,y=2,3 is the first-degree Taylor polynomial constructed at the point of contact. Show further that approximating Δf by the total differential df amounts to a tangent-plane approximation.
Solution
Mathematical Solution
To obtain the tangent plane from first principles, compute N, a normal at P:2,3,f2,3=2,3,−17:
N=−fx(2,3)−fy(2 ,3)1=−27321
then implement the vector form of the equation of a plane:
0
=R−P·N
=xyz−23−17·−27321
= −27(x−2+32y−3+z+17
= −27 x+32 y+z+54−96+17
= −27 x+32 y+z−25
which, when rearranged, can be written as z=25+27 x−32 y.
Approximating f by its first-degree Taylor polynomial
f2,3+fx2,3⋅x−2+fy2,3⋅y−3
means
f
≐−17+27x−2−32y−3
=27 x−32 y+96−54−17
=27 x−32 y+25
Finally, from Δf=f2+dx,3+dy−f2,3≐df, obtain z=f2+dx,3+dy≐f2,3+df, or
z
≐f2,3+fx2,3⋅dx+fy2,3⋅dy
=−17+27x−2−32y−3
since 2+dx=x⇒dx=x−2, and 3+dy=y⇒dy=y−3.
Maple Solution - Interactive
Initialize
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Context Panel: Assign Function
fx,y=3 x2+5 x y−7 y2+4→assign as functionf
Obtain the tangent plane from first principles
Write a sequence of the point of contact on the surface and a normal at that point. Press the Enter key.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Plane
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation
2,3,f2,3,−∂∂ x fx,y,−∂∂ y fx,y,1x=a|f(x)x=2,y=3
→make plane
<< Plane 2 >>
→representation
−27x+32y+z=25
Obtain the first-degree Taylor polynomial
Use the tutor to obtain the first-degree Taylor polynomial. See Figure 4.7.9(a).
Alternatively, launch the tutor by typing fx,y and pressing the Enter key.
Context Panel: Student Multivariate Calculus≻Tutors≻Taylor Approximation Set x,y=2,3 and set the Degree to 1
Figure 4.7.9(a) The Taylor Approximation tutor
Obtain the first-degree Taylor polynomial from first principles
Set fx and fy as Atomic Identifiers.
Calculus palette: Partial derivative operator
f__xx,y=∂∂ x fx,y→assign as functionf__x
f__yx,y=∂∂ y fx,y→assign as functionf__y
Setting fx and fy as Atomic Identifiers, write the appropriate formula and press the Enter key.
f2,3+f__x2,3⋅x−2+f__y2,3⋅y−3
27x−32y+25
Obtain z=f2+dx,3+dy≐f2,3+df
z≐f2,3+f__x2,3⋅x−2+f__y2,3⋅y−3
z ≐ 27x−32y+25
Maple Solution - Coded
Define the function f.
f≔x,y→3 x2+5 x y−7 y2+4:
Obtain the tangent plane at 2,3
Use the TangentPlane command in the Student VectorCalculus package.
Student:-VectorCalculus:-TangentPlanefx,y,x=2,y=3
Obtain the first-degree Taylor polynomial approximation
Use the TaylorApproximation command in the Student MultivariateCalculus package.
Student:-MultivariateCalculus:-TaylorApproximationfx,y,x,y=2,3,1
Use the differential operator D to obtain the partial derivatives in df.
z≐f2,3+D1f2,3⋅x−2+D2f2,3⋅y−3
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