Free Printable Legendre Polynomials

1.

Legendre equation is defined as $\left(1-x^2\right)\frac{\text{d}^2y}{\text{d}x^2}-2x\frac{\text{d}y}{\text{d}x}+k\left(k+1\right)y=0$

a)

Yes

b)

No

2.

$P_2\left(x\right)=\frac{1}{2}\left(ax^2-1\right)$

a is (a) .

3.

$P_4\left(x\right)=\frac{1}{2^mm!}\frac{\text{d}^n}{\text{d}x^n}\left(x^2-1\right)^n$

Use Rodrigue's formula, the setup $P_4\left(x\right)$ is shown as above. Find the correct values of m and n.

a)

m=4; n=5

b)

m=4; n=4

c)

m=5; n=4

d)

m=5; n=5

4.

$Expand\ \left(x^2-y^2\right)^4$ using Pascal's triangle.

a)

$x^6-4x^5y^2-6x^4y^4+4x^2y^5-y^6$

b)

$x^6-4x^5y^2+6x^4y^4-4x^2y^5+y^6$

c)

$x^8-4x^6y^2-6x^4y^4+4x^2y^6-y^8$

d)

$x^8-4x^6y^2+6x^4y^4-4x^2y^6+y^8$

5.

$\left(1-2xt+t^2\right)^m$

The expression above is Legendre generating function. What is the value of m?

a)

$-\frac{1}{2}$

b)

$\frac{1}{2}$

c)

-1

d)

1

6.

What is the correct recurrence formula to derive P_k(x)?

a)

b)

c)

d)

7.

where (m,n) are

a)

(3,3)

b)

(7,3)

c)

(7,5)

d)

(6,5)

8.

$If\int_{-1}^1P_m\left(x\right)P_n\left(x\right)dx=b\ \ for\ \ m\ne n,$

what is the value of b?

a)

$\frac{2k+1}{2}$

b)

$\frac{2}{2k+1}$

c)

0

d)

1

9.

$Evaluate\ \int_{-1}^1\left(P_5\left(x\right)\right)^2dx.$

a)

0

b)

$\frac{2}{11}$

c)

$\frac{2}{10}$

d)

$\frac{11}{2}$

10.

$f\left(x\right)=2+x^2-10x^3$

Which is the correct series Legendre polynomial of f(x)?

a)

$A_0P_0\left(x\right)+A_1P_1\left(x\right)$

b)

$A_0P_0\left(x\right)+A_1P_1\left(x\right)+A_2P_2\left(x\right)$

c)

$A_0P_0\left(x\right)+A_1P_1\left(x\right)+A_2P_2\left(x\right)+A_3P_3\left(x\right)$

Legendre Polynomials

Total questions: 10

Worksheet time: 34mins

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