A-Level Edexcel Maths Pure Combinations of Transformations: The curve C has equation y = \f

The curve C has equation y = \frac{(x+3)(x-8)}{x}, \quad x > 0 (a) Find \frac{dy}{dx} in its simplest form - Edexcel - A-Level Maths Pure - Question 8 - 2010 - Paper 2

Question 8

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The curve C has equation y = \frac{(x+3)(x-8)}{x}, \quad x > 0 (a) Find \frac{dy}{dx} in its simplest form. (b) Find an equation of the tangent to C at the point ... show full transcript

Worked Solution & Example Answer:The curve C has equation y = \frac{(x+3)(x-8)}{x}, \quad x > 0 (a) Find \frac{dy}{dx} in its simplest form - Edexcel - A-Level Maths Pure - Question 8 - 2010 - Paper 2

Step 1

Find \frac{dy}{dx} in its simplest form.

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Answer

To find ( \frac{dy}{dx} ), we start with the equation for ( y ):

y = \frac{(x+3)(x-8)}{x}

We can simplify it further:

y = \frac{x^2 - 5x - 24}{x} = x - 5 + \frac{-24}{x}$$ Now we differentiate this expression:

\frac{dy}{dx} = 1 + \frac{24}{x^2}

Thus, the simplest form of \( \frac{dy}{dx} \) is: $$\frac{dy}{dx} = 1 + \frac{24}{x^2}$$

Step 2

Find an equation of the tangent to C at the point where x = 2.

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Answer

To find the equation of the tangent line at ( x = 2 ), we first evaluate ( \frac{dy}{dx} ) at this point:

\frac{dy}{dx} \bigg|_{x=2} = 1 + \frac{24}{2^2} = 1 + \frac{24}{4} = 1 + 6 = 7

Now we need to find the value of ( y ) at ( x = 2 ):

y = \frac{(2+3)(2-8)}{2} = \frac{5 \cdot (-6)}{2} = -15

With the slope ( m = 7 ) and the point ( (2, -15) ), we can use point-slope form to find the equation of the tangent line:

y - (-15) = 7(x - 2)$$ Simplifying this gives:

y + 15 = 7x - 14 \Rightarrow y = 7x - 29$$

Thus, the equation of the tangent is:

$y = 7x - 29$

The curve C has equation y = \frac{(x+3)(x-8)}{x}, \quad x > 0 (a) Find \frac{dy}{dx} in its simplest form - Edexcel - A-Level Maths Pure - Question 8 - 2010 - Paper 2

Worked Solution & Example Answer:The curve C has equation y = \frac{(x+3)(x-8)}{x}, \quad x > 0 (a) Find \frac{dy}{dx} in its simplest form - Edexcel - A-Level Maths Pure - Question 8 - 2010 - Paper 2

Find \frac{dy}{dx} in its simplest form.

Find an equation of the tangent to C at the point where x = 2.

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