DOC

# Applications of Differential Calculus Exercise

By Mark Garcia,2014-04-02 19:24
7 views 0
Applications of Differential Calculus Exercise

Application of Differential Calculus

A-Level Pure Mathematics

Chapter 5 Application of Differential Calculus

(L’Hospital’s Rule) Exercise 5A

Date : Name : ________________

xsinx1. Evaluate lim3x0x

2. Evaluate the following limits:

2xsinnxsin1cosx (a) (b) (c) lim limlim2x0x0x0xsinxx

3. Evaluate the following limits:

2sinxsecx1sinx (a) (b) (c) limxlimlim22;xx00x0xx

1

Application of Differential Calculus

4. Evaluate the following limits:

111sinxxtanx??lim (a) (b) (c) limlim?3x0x0xtanxcosxxx?2

5. Evaluate the following limits:

x1tanxx2??xx (a) lim (b) (c) lim(e;x)limcos?2;；x0xx0xxsinx?

1111121Ans: 1. 2. , , 3. 1, , 1 4. 0, 0, 5. , , ne062233

2

Application of Differential Calculus

A-Level Pure Mathematics

Chapter 5 Application of Differential Calculus

(L’Hospital’s Rule) Exercise 5B

Date : Name : ________________ 1 Evaluate the following limits:

11

xlnx (a) (b) lim[ln(x;e)]lim(sinx);x0x0

2. Evaluate:

1x (a) (b) limlim(secxtanx)5x11xx2

[HKAL 1994]

3

Application of Differential Calculus 3. Evaluate:

1xxx??3;2e1sinxe? (a) (b) limlim2?0x0x5x?

[HKAL 1998]

1351e5Ans: 1. , 2. , 3. , e0ee22

4

Application of Differential Calculus

A-Level Pure Mathematics

Chapter 5 Application of Differential Calculus

(Monotonic Functions) Exercise 5C

Date : Name : ________________

3x is strictly increasing for . 1. Show that the function ()tan0xfxxx32

12. Show that the function is strictly increasing for . f(x);2x3x1x

2x3. Determine the interval for which the function f(x) is increasing. 21;x

4. Determine the interval in [0,2] for which the function f(x)x;2sinx is decreasing.

24Ans: 3. [1, 1] 4. . [,]33

5

Application of Differential Calculus

A-Level Pure Mathematics

Chapter 5 Application of Differential Calculus

(Monotonic Functions) Exercise 5D

Date : Name : ________________

x11. Prove for . tanxxx021;x

32xx0ln(1)2. Prove that for x0;x212;x

6

Application of Differential Calculus

lnx3. Let . f(x)x

(a) Show that is strictly increasing on the interval . f(x)(0,e)

(b) Hence, show that if , 0ab(e

ba ab

4. Let . f(x)ln(1;x)x

By finding the greatest value of , prove that . f(x)ln(1;x)(x

7

Application of Differential Calculus

5. (a) Show that for 0x1,

(i) ln(1;x)x

(ii) ln(1x)x

(b) Let be a positive integer greater than . Deduce from (a) that 1n

1 ln(n;1)lnnlnnln(n1)n

Hence show that

3n;11113n ln();;;;ln()nnn;13nn1

(c) Use the above results to evaluate the limit

111??lim; (Ans:) ;;;ln3?nnn13n;?

8

Application of Differential Calculus

A-Level Pure Mathematics

Chapter 5 Application of Differential Calculus

(Maxima and Minima) Exercise 5E

Date : Name : ________________

22x2(1x)1. Find the maximum or minimum points of and . yy'2221;x(1;x)

xxxxxx

f'(x)

f(x)

Maximum point =

Minimum point =

32. Find the maximum or minimum points of . yx(x1)

2

33. Find the maximum or minimum points of . y3(x4)2

9

Application of Differential Calculus

2x(ln)4. Find the maximum or minimum points of . yx

5. Find the maximum or minimum points of yx(x4)

333Ans: 1. Max. pt () Min. pt () 2. Min. pt () 1,11,1,248

22 3. Min. pt () 4. Max. pt () Min. pt () 4,2e,4e1,0

5. Min. pt ( 0, 0 )

10

Report this document

For any questions or suggestions please email
cust-service@docsford.com