Question

(a) Find a simplified form of the difference quotient and (b) complete the following table.

f(x) = - 4x + 7​

Answer 1

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

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