Department of HEAS(MATHEMATICS), RTU, KOTATopic: Differential Caluculus (Taylor Series and Leibnitz Theorem)

Tutorial Sheet (DC-T1)

In Q.No. 1 to 2, find the Taylor polynomials of order 0, 1, 2 and 3 generated by the followingfunctions f at a :

1. f(x) = ln(1 + x), a = 0 2. f(x) = tanx, a = π/4

In Q.No. 3 to 4, fnd the Maclaurin series for the following functions:

3.2 + x

1− x 4. coshx =ex + e−x

2

In Q.No. 5 to 7, find the Taylor series generated by the following functions f at the pointx = a :

5. f(x) = x3 − 2x+ 4, a = 2

6. f(x) = 2x, a = 1

7. f(x) = cos(2x+ (π/2)), a = π/4

8. Find the first three nonzero terms of Maclaurin series for the function f(x) = (sinx) ln(1+x).

9. Let f(x) have derivatives through order n at x = a. Show that the Taylor polynomial of ordern and its first n derivatives have the same values that f and its first n derivatives have atx = a.

In Q.No. 10 to 12, use substitution to find the Taylor series at x = 0 of the following functions:

10. e−5x

11. cos(x2/3/√

2)

12.1

1 +3

4x3

13. Estimate the error if P3(x) = x− (x3/6) is used to estimate the value of sinx at x = 0.1.

14. Show that if the graph of a twice differentiable function f(x) has an inflection point at x = a,then the linearization of f at x = a is also the quadratic approximation of f at x = a.

15. Suppose that f(x) =∞∑n=0

anxn converges for all x in an open interval (−R,R).Show that

(i). If f is even, then a1 = a3 = a5 = ... = 0, i.e., the Taylor series for f at x = 0 containsonly even powers of x.(ii). If f is odd, then a0 = a2 = a4 = .... = 0, i.e., the Taylor series for f at x = 0 containsonly odd powers of x.

16. If y = xn−1. log x, prove that yn =(n− 1)!

x.

17. If y = sin(m sin−1 x), prove that (1− x2)yn+2 − (2n+ 1)x.yn+1 + (m2 − n2)yn = 0.

Differential Calculus(DC-T1) Department of HEAS (Mathematics) Page 2 of 2

18. If y = a cos(log x) + b sin(log x), then show that(i). x2y2 + xy1 + y = 0.(ii). x2yn+2 + (2n+ 1)xyn+1 + (n2 + 1)yn = 0.

19. Find the nth differential coefficient of ex. log x.

20. If y1/m + y−1/m = 2x, prove that (x2 − 1)yn+2 + (2n+ 1)xyn+1 + (n2 −m2)yn = 0.

21. If y = (sin−1 x)2, prove that(i) (1− x2)y2 − xy1 − 2 = 0(ii) (1− x2)yn+2 − (2n+ 1)xyn+1 − n2yn = 0.

22. If y = x log(1 + x), prove that yn =

[(−1)n−2(n− 2)!(x+ n)

](1 + x)n

.

Answers

1. P0(x) = 0, P1(x) = x,

P2(x) = x− x2

2, P3(x) = x− x2

2+x3

3

2. P0(x) = 1, P1(x) = 1 + 2(x− π/4),P2(x) = 1 + 2(x− π/4) + 2(x− π/4)2,

P3(x) = 1 + 2(x− π4

) + 2(x− π4

)2 +8

3(x− π

4)3

3. 2 +

∞∑n=1

3xn

4.x∑

n=0

x2n

(2n)!

5. 8 + 10(x− 2) + 6(x− 2)2 + (x− 2)3

6.

∞∑n=0

2(ln 2)n(x− 1)n

n!

7.∞∑n=0

(−1)n22n

(2n)!(x− π/4)2n

8. x2 − 1

2x3 +

1

6x4 − ......

10.∞∑n=0

(−1)n5nxn

n!

11. 1− x3

2.2!+

x6

22.4!− x9

23.6!+ ...

12. 1− 3

4x3 +

9

16x6 − 27

64x9 + ...

13. error ≤ 4.2× 10−6

19. ex[log x+

n

x− n(n− 1)

2x2+ ..+

(−1)n−1(n− 1)!

xn

].