Moderate -0.8 This is a straightforward integration question requiring only the power rule applied to three terms, followed by using a given point to find the constant of integration. It's routine calculus with no problem-solving or conceptual challenges, making it easier than average but not trivial since it requires careful handling of fractional powers and algebraic simplification.
A curve with equation \(y = \text{f}(x)\) passes through the point \((4, 25)\)
Given that
$$\text{f}'(x) = \frac{3}{8}x^2 - 10x^{-\frac{1}{2}} + 1, \quad x > 0$$
find \(\text{f}(x)\), simplifying each term. [5]
Substitute x = 4, y = 25 (cid:159) 25 = 8 – 40 + 4 + c
Answer
Marks
(cid:159) c =
M1
x3 1
f(x)(cid:32) (cid:16)20x2 (cid:14)x(cid:14)53
Answer
Marks
8
A1
(5)
(5 marks)
Notes:
M1: Attempt to integrate xn (cid:111)xn(cid:14)1
A1: Term in x3 or term in x1 2correct, coefficient need not be simplified, no need for +x nor +c
A1: ALL three terms correct, coefficients need not be simplified, no need for + c
M1: For using x = 4, y = 25 in their f(x) to form a linear equation in c and attempt to find c
x3 1
A1: (cid:32) (cid:16)20x2 (cid:14)x(cid:14)53 cao (all coefficients and powers must be simplified to give this
8
answer- do not need a left hand side and if there is one it may be f(x) or y). Need full
expression with 53. These marks need to be scored in part (a).
Answer
Marks
Guidance
Question
Scheme
Marks
Question 7:
7 | (cid:180)(cid:167)3 1 (cid:183)
(cid:16)
f(x)(cid:32) (cid:181)(cid:168) x2 (cid:16)10x 2 (cid:14)1(cid:184)dx
8
(cid:182)(cid:169) (cid:185)
1
3 x3 x2
xn (cid:111)xn(cid:14)1(cid:159)f(x)(cid:32) (cid:117) (cid:16)10 (cid:14)x((cid:14)c)
8 3 1
2 | M1
A1
A1
Substitute x = 4, y = 25 (cid:159) 25 = 8 – 40 + 4 + c
(cid:159) c = | M1
x3 1
f(x)(cid:32) (cid:16)20x2 (cid:14)x(cid:14)53
8 | A1
(5)
(5 marks)
Notes:
M1: Attempt to integrate xn (cid:111)xn(cid:14)1
A1: Term in x3 or term in x1 2correct, coefficient need not be simplified, no need for +x nor +c
A1: ALL three terms correct, coefficients need not be simplified, no need for + c
M1: For using x = 4, y = 25 in their f(x) to form a linear equation in c and attempt to find c
x3 1
A1: (cid:32) (cid:16)20x2 (cid:14)x(cid:14)53 cao (all coefficients and powers must be simplified to give this
8
answer- do not need a left hand side and if there is one it may be f(x) or y). Need full
expression with 53. These marks need to be scored in part (a).
Question | Scheme | Marks
A curve with equation $y = \text{f}(x)$ passes through the point $(4, 25)$
Given that
$$\text{f}'(x) = \frac{3}{8}x^2 - 10x^{-\frac{1}{2}} + 1, \quad x > 0$$
find $\text{f}(x)$, simplifying each term. [5]
\hfill \mbox{\textit{Edexcel P1 2018 Q7 [5]}}