Moderate -0.8 This is a straightforward differentiation and tangent line question requiring standard techniques: rewrite terms as powers, differentiate using power rule, substitute x=4 to find gradient and y-coordinate, then use y-y₁=m(x-x₁). All steps are routine with no problem-solving insight needed, making it easier than average but not trivial due to the algebraic manipulation required.
14 In this question you must show detailed reasoning.
The equation of a curve is \(y = 16 \sqrt { x } + \frac { 8 } { x }\).
Determine the equation of the tangent to the curve at the point where \(x = 4\).
## Question 14:
$y = 16x^{\frac{1}{2}} + 8x^{-1}$ | **B1** | AO 3.1a | May be implied by correct derivative
$\frac{dy}{dx} = 8x^{-\frac{1}{2}} - 8x^{-2}$ | **M1** | AO 1.1 | At least one term of the form $\alpha x^{-\frac{1}{2}}$ or $\beta x^{-2}$ obtained
— | **A1** | AO 1.1 | All correct
$x = 4, \frac{dy}{dx} = \frac{7}{2}$ | **B1FT** | AO 1.1 | FT their $\frac{dy}{dx}$, dep on **M1**
$x = 4, y = 34$ | **B1** | AO 1.1 |
$y - \text{their } 34 = \left(\text{their } \frac{7}{2}\right)(x - 4)$ oe | **M1FT** | AO 1.1 | Their 7/2 must come from substituting $x = 4$ into their derivative
$y = \frac{7}{2}x + 20$ oe | **A1** | AO 3.2a | All correct. Accept any form: $7x - 2y + 40 = 0$ or $y - 34 = \frac{7}{2}(x-4)$ oe. NOTE: Final answer can be obtained from incorrect working — check their derivative.
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14 In this question you must show detailed reasoning.
The equation of a curve is $y = 16 \sqrt { x } + \frac { 8 } { x }$.\\
Determine the equation of the tangent to the curve at the point where $x = 4$.
\hfill \mbox{\textit{OCR MEI AS Paper 2 2023 Q14 [7]}}