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A function, f, is defined by f(x) = (3 - 2x)^4, where x ∈ ℝ - Scottish Highers Maths - Question 5 - 2023

Question 5

A function, f, is defined by f(x) = (3 - 2x)^4, where x ∈ ℝ. Calculate the rate of change of f when x = 4.

Worked Solution & Example Answer:A function, f, is defined by f(x) = (3 - 2x)^4, where x ∈ ℝ - Scottish Highers Maths - Question 5 - 2023

Step 1

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Answer

To find the rate of change of the function f(x) = (3 - 2x)^4, we will apply the chain rule of differentiation.

Step 2

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Answer

Using the chain rule, the derivative f'(x) is given by:

$f'(x) = 4(3 - 2x)^3 imes (-2)$

This simplifies to:

$f'(x) = -8(3 - 2x)^3$

Step 3

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Answer

Now, we need to evaluate f'(x) at x = 4:

$f'(4) = -8(3 - 2(4))^3$ $= -8(3 - 8)^3$ $= -8(-5)^3$ $= -8(-125)$ $= 1000$

Thus, the rate of change of f when x = 4 is 1000.

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