(a) Find a simplified form of the difference quotient and (b) complete the following table. f(x) = - 4x + 7

a) -4

b)

x    h                   [f(x+h) - f(x)] / h

3    2                  -4

3    1                   -4

3    0.1                -4

3    0.01              -4

Step-by-step explanation:

a) The difference quotient is:

[f(x+h) - f(x)] / h

Our function is -4x + 7, then:

f(x+h) = -4(x+h) + 7 = -4x -4h + 7

f(x) = -4x + 7

f(x+h) - f(x) = -4x -4h + 7 - (-4x + 7 )= -4x -4h + 7 +4x - 7 = -4h

Finally:

[f(x+h) - f(x)] / h =  -4h/h = -4

b)

First point in the table:

x = 3

h = 2

f(x+h) = -4(3+2) + 7 = -13

f(x) = -4(3) + 7 = -5

f(x+h) - f(x) = -13 -(-5) = -8

[f(x+h) - f(x)] / h = -8 / 2 = -4

Second point in the table:

x = 3

h = 1

f(x+h) = -4(3+1) + 7 = -9

f(x) = -4(3) + 7 = -5

f(x+h) - f(x) = -9 -(-5) = -4

[f(x+h) - f(x)] / h = -4 / 1 = -4

Third point in the table:

x = 3

h = 0.1

f(x+h) = -4(3+0.1) + 7 = -5.4

f(x) = -4(3) + 7 = -5

f(x+h) - f(x) = -5.4 -(-5) = -0.4

[f(x+h) - f(x)] / h = -0.4 / 0.1 = -4

Fourth point in the table:

x = 3

h = 0.01

f(x+h) = -4(3+0.01) + 7 = -5.04

f(x) = -4(3) + 7 = -5

f(x+h) - f(x) = -5.04 -(-5) = -0.04

[f(x+h) - f(x)] / h = -0.04 / 0.01 = -4