AQA AS Paper 2 2021 June — Question 8 4 marks

Exam BoardAQA
ModuleAS Paper 2 (AS Paper 2)
Year2021
SessionJune
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicDifferentiation from First Principles
TypeFirst principles: polynomial with multiple terms
DifficultyModerate -0.3 This is a straightforward application of the first principles formula to a simple polynomial. While it requires careful algebraic manipulation and is worth 4 marks, the technique is standard and the polynomial (two terms, low powers) makes the algebra manageable. Slightly easier than average since it's a routine textbook exercise with no conceptual surprises.
Spec1.07g Differentiation from first principles: for small positive integer powers of x
8 It is given that \(y = 3 x - 5 x ^ { 2 }\) Use differentiation from first principles to find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) [0pt] [4 marks]
LIH
Question 8:
AnswerMarks Guidance
\(\lim_{h\to 0}\left[\frac{3(x+h) - 5(x+h)^2 - (3x - 5x^2)}{h}\right]\)M1 Substitutes \((x+h)\) into \(f(x+h)-f(x)\); condone one slip
\(\lim_{h\to 0}\left[\frac{3x+3h-5x^2-10xh-5h^2-3x+5x^2}{h}\right]\)A1 Correct expanded expression for \(f(x+h)-f(x)\)
\(\lim_{h\to 0}\left[\frac{3h - 10xh - 5h^2}{h}\right]\)M1 Divides each term by \(h\)
\(\lim_{h\to 0}[3 - 10x - 5h]\)— —
\(\frac{dy}{dx} = 3 - 10x\)R1 Completes rigorous argument; must see \(\lim_{h\to 0}\) 
## Question 8:

$\lim_{h\to 0}\left[\frac{3(x+h) - 5(x+h)^2 - (3x - 5x^2)}{h}\right]$ | M1 | Substitutes $(x+h)$ into $f(x+h)-f(x)$; condone one slip |
$\lim_{h\to 0}\left[\frac{3x+3h-5x^2-10xh-5h^2-3x+5x^2}{h}\right]$ | A1 | Correct expanded expression for $f(x+h)-f(x)$ |
$\lim_{h\to 0}\left[\frac{3h - 10xh - 5h^2}{h}\right]$ | M1 | Divides each term by $h$ |
$\lim_{h\to 0}[3 - 10x - 5h]$ | — | — |
$\frac{dy}{dx} = 3 - 10x$ | R1 | Completes rigorous argument; must see $\lim_{h\to 0}$ |

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8 It is given that $y = 3 x - 5 x ^ { 2 }$

Use differentiation from first principles to find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$\\[0pt]
[4 marks]\\
LIH\\

\hfill \mbox{\textit{AQA AS Paper 2 2021 Q8 [4]}}
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