a) Differentiate $f(x) = 2x^2 + 4x$ with respect to $x$, from first principles. b) A rectangle is expanding in area. Its width is $x$ cm, where $x \in \mathbb{R}$ and $x > 0$. Its length is always four times its width. Find the rate of change of the area of the rectangle with respect to its width, $x$, when the area of the rectangle is 225 cm².

Leaving Cert Mathematics Differentiation: a) Differentiate $f(x) = 2x^2 +

a) Differentiate $f(x) = 2x^2 + 4x$ with respect to $x$, from first principles - Leaving Cert Mathematics - Question 6 - 2022

Question 6

a) Differentiate $f(x) = 2x^2 + 4x$ with respect to $x$, from first principles. b) A rectangle is expanding in area. Its width is $x$ cm, where $x \in \mathbb{R}$ a... show full transcript

Worked Solution & Example Answer:a) Differentiate $f(x) = 2x^2 + 4x$ with respect to $x$, from first principles - Leaving Cert Mathematics - Question 6 - 2022

Step 1

Differentiate $f(x) = 2x^2 + 4x$ from first principles

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Answer

To differentiate $f(x)$ from first principles, we use the definition:

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Calculate $f(x+h)$ :

$f(x+h) = 2(x+h)^2 + 4(x+h) = 2(x^2 + 2xh + h^2) + 4x + 4h = 2x^2 + 4xh + 2h^2 + 4x + 4h$
Substitute into the limit:

$f'(x) = \lim_{h \to 0} \frac{(2x^2 + 4xh + 2h^2 + 4x + 4h) - (2x^2 + 4x)}{h}$

This simplifies to:

$f'(x) = \lim_{h \to 0} \frac{4xh + 2h^2 + 4h}{h} = \lim_{h \to 0} (4x + 2h + 4)$

Evaluating the limit as $h$ approaches 0 gives:

$f'(x) = 4x + 4$

Step 2

Find the rate of change of area when $A = 225$ cm²

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Answer

Given that the area $A$ of the rectangle is:

$A = l \times w = 4x \times x = 4x^2$

We know:

When $A = 225$ , we have:

$4x^2 = 225 \Rightarrow x^2 = \frac{225}{4} \Rightarrow x = \frac{15}{2} = 7.5$
To find the rate of change of area with respect to width:

First, we differentiate $A$ :

$\frac{dA}{dx} = 8x$
Substitute $x = 7.5$ :

$\frac{dA}{dx} = 8 \left(\frac{15}{2}\right) = 60 \text{ cm}^2/ ext{cm}$

Thus the rate of change of the area with respect to the width $x$ when the area of the rectangle is 225 cm² is (60, \text{cm}^2/\text{cm}).

a) Differentiate $f(x) = 2x^2 + 4x$ with respect to $x$, from first principles - Leaving Cert Mathematics - Question 6 - 2022

Worked Solution & Example Answer:a) Differentiate $f(x) = 2x^2 + 4x$ with respect to $x$, from first principles - Leaving Cert Mathematics - Question 6 - 2022

Differentiate $f(x) = 2x^2 + 4x$ from first principles

Find the rate of change of area when $A = 225$ cm²

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