How do you find the quadratic taylor polynomial q(x,y) approximating #f(x,y)= e^(x) cos (5y)# about (0,0)?

Answer 1

Refer to explanation

The quadratic Taylor approximation is

#q(x,y)=f(0,0) + (df)/dx(0,0) x + (df)/dy(0,0) y +(1/(2!)) *[(d^2f)/(d^2x)(0,0) x^2 + 2 (d^2f)/(dxdy)(0,0) *xy + (d^2f)/(d^2y)(0,0) y^2]#

Hence we have that

#f(0,0)=e^0*cos(0)=1#
# (df)/dx(0,0)=e^0*cos0=1#
#(df)/dy(0,0) = -5*e^0*sin0=0#
#(d^2f)/(d^2x)(0,0)=e^0*cos0=1#
#(d^2f)/(dxdy)(0,0)=-5e^0sin0=0#
#(d^2f)/(d^2y)(0,0)=-25e^0cos0=-25#

Hence we have that

#q(x,y)=1+x+(1/(2!))*[x^2-25y^2]#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the quadratic Taylor polynomial ( q(x,y) ) approximating ( f(x,y) = e^x \cos(5y) ) about the point ( (0,0) ), follow these steps:

  1. Calculate the partial derivatives of ( f(x,y) ) with respect to ( x ) and ( y ) at ( (0,0) ).
  2. Use these partial derivatives to compute the coefficients of the quadratic terms in the Taylor polynomial.
  3. The quadratic Taylor polynomial ( q(x,y) ) is given by the following expression:

[ q(x,y) = f(0,0) + f_x(0,0) \cdot x + f_y(0,0) \cdot y + \frac{1}{2} f_{xx}(0,0) \cdot x^2 + f_{xy}(0,0) \cdot xy + \frac{1}{2} f_{yy}(0,0) \cdot y^2 ]

  1. Substitute the calculated values of ( f(0,0) ), ( f_x(0,0) ), ( f_y(0,0) ), ( f_{xx}(0,0) ), ( f_{xy}(0,0) ), and ( f_{yy}(0,0) ) into the expression to obtain the quadratic Taylor polynomial ( q(x,y) ).
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7