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Numerical Derivatives and Integrals Worksheet

Numerical Derivatives and Integrals Worksheet
  • Page 1
 1.  
Find f '(x), if f (x) = 8x2 + 10x + 19.
a.
10
b.
16x - 10
c.
16x
d.
16x + 10


Solution:

f(x) = 8x2 + 10x + 19

f ' (x) = limh0f(x + h) - f(x)h
[Use the definition of f'(x).]

= limh0 [8(x + h)2 +10(x + h) + 19] - [8x2 +10x + 19] h

= limh0[8x2 + 8h2 + 16xh + 10x + 10h + 19 - 8x2 - 10x - 19]h

= limh08h2 + 16xh + 10hh = limh0h[8h + 16x + 10]h

= limh0 8h + 16x + 10
[As h ≠ 0, hh = 1.]

= 16x + 10


Correct answer : (4)
 2.  
Find f ′(x), if f (x) = 8x3.
a.
8 x2
b.
24x2
c.
24x3
d.
- 24x2


Solution:

f(x) = 8x3

f '(x) = limh0f(x + h) - f(x)h
[Use the definition of f ′(x).]

= limh08(x + h)3 - 8(x)3h

= limh08x3+24x2h+24xh2+8h3-8x3h
[Use (a+b)3 =a3+3a2b+3ab2+b3.]

= limh0[24x2+24hx+8h2]
[Simplify.]

= 24x2


Correct answer : (2)
 3.  
Find g ′(8), if g(x) = 1x2.
a.
- 1 4
b.
- 16
c.
- 1 256
d.
1 64


Solution:

g(x) = 1x2

g ′(8) = limh0g(8 + h) - g(8)h
[Use the definition of g ′(8).]

= limh0 1h[1(8 + h)2 -182]

= limh0 1h[64 -(8 + h)264(8 + h)2]

= limh0 1h[- (h2 + 16h)64(8 + h)2]

= limh0 [- (h + 16)64(8 + h)2]
[As h ≠ 0, hh = 1.]

= - 1 / 256


Correct answer : (3)
 4.  
Find g ′(4), if g (x) = x3 + x2.
a.
- 56
b.
56
c.
69
d.
13


Solution:

g(x) = x3 + x2

g ′(4) = limh0g(4 + h) - g(4)h
[Use the definition of g ′(x).]

= limh0 1h[(4 + h)3 +(4 + h)2 - 64 -16]

= limh0 1h[h3+12h2+48h+h2+8h]

= limh0 1h[h3 + 13h2 + 56h]
[As h ≠ 0, hh = 1.]

= limh0[h2 + 13h + 56]

= 56


Correct answer : (2)
 5.  
Find f ′(4), if f(x) = 2x - x2.
a.
- 8
b.
- 6
c.
6


Solution:

f(x) = 2x - x2

f '(4) = limh0f(4 + h) - f(4)h
[Use the definition of f ′(x).]

= limh0 1h[2(4 + h)-(4 + h)2 - 8 + 16]

= limh0 1h[8 + 2h - 16 -h2 - 8h - 8 + 16]

= limh0[- 6 - h]
[As h ≠ 0, hh = 1.]

= - 6


Correct answer : (2)
 6.  
Find f '(π), if f(x) = sin 18x.
a.
- 18
b.
18
c.
1


Solution:

f(x) = sin 18x

f '(π) = limh0f(π + h) - f(π)h
[Use the definition of f '(π).]

= limh0 1h[sin 18(π + h) - sin 18π]

= limh0 1 / h[sin (18π + 18h) - sin 18π]

= limh0 sin 18hh
[Use sin(x + h) = sin xcos h + cos xsin h.]

= 18
[Use limθ0  sin kθθ = k.]


Correct answer : (2)
 7.  
Find f ′(π2), if f (x) = 42cos x.
a.
- 42
b.
Does not exist
c.
42


Solution:

f(x) = 42 cos x

f ′(π2) = limh0f(π2 + h) - f(π2)h
[Use the definiton of f ′(π2).]

= limh0 1h[42cos (π2 + h) - 42cos(π2)]

= limh0 (- 42) sin hh

= (- 42) limh0 sin hh

= - 42.
[Use limθ0  sin θθ = 1.]


Correct answer : (1)
 8.  
Find NDER f(3), if f (x) = 1 + 3x2, by calculating the symmetric difference quotient with h = 0.001.
a.
14
b.
18
c.
19
d.
16


Solution:

f(x) = 1 + 3x2

NDER f(3) = f(3 + 0.001) - f(3 - 0.001)0.002
[Use NDER f(a) = f(a + h) - f(a - h)2h.]

= f(3.001) - f(2.999)0.002

= [1 + 3(3.001)2] - [1 + 3(2.999)2]0.002

= 18.
[Use a calculator and simplify.]


Correct answer : (2)
 9.  
Find NDER f(- 1), if f (x) = 4 + x3 by calculating the symmetric difference quotient with h = 0.001.


Solution:

f(x) = 4 + x3

NDER f(-1) = f(- 1 + 0.001) - f(- 1 - 0.001)0.002
[Use NDER f(a) = f(a + h) - f(a - h)2h.]

= f(-0.999) - f(- 1.001)0.002

= [4 +(-0.999)3] - [4 +(- 1.001)3]0.002

= 3.000001.
[Use a calculator and simplify.]


Correct answer : (0)
 10.  
Find NDER f(π4), if f(x) = sin x by calculating the symmetric difference quotient with h = 0.001.
a.
0.707
b.
1
c.
-0.7
d.
-1


Solution:

f(x) = sin x

NDER f(π4)= f(π4 + 0.001) -  f(π4 - 0.001)0.002
[Use NDER f (a) = f(a + h) - f(a - h)2h.]

= sin(π4 + 0.001) -  sin(π4 - 0.001)0.002

= 0.707.
[Use a calculator and simplify.]


Correct answer : (1)

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