Moderate -0.3 This is a straightforward integration problem using reverse chain rule to find f(x) from f'(x), then applying the initial condition at point A to find the constant. The reverse chain rule pattern is recognizable (linear function raised to a power), and finding the y-intercept requires only substituting x=0. Slightly easier than average due to being a standard technique with clear steps.
A curve with equation \(y = \mathrm{f}(x)\) passes through the point \(A(3, 1)\) and crosses the \(y\)-axis at \(B\). It is given that \(\mathrm{f}'(x) = (3x - 1)^{-\frac{1}{3}}\). Find the \(y\)-coordinate of \(B\). [6]
Equate derivative to zero and obtain the given equation
A1
Total:
3
Answer
Marks
Guidance
4(ii)
Sketch a relevant graph, e.g. y = ln x
B1
3
Sketch a second relevant graph, e.g. y = 1 + , and justify the given statement
Answer
Marks
x
B1
Total:
2
Answer
Marks
4(iii)
3+x
Use iterative formula x = correctly at least once
n+1
lnx
Answer
Marks
n
M1
Obtain final answer 4.97
A1
Show sufficient iterations to 4 d.p.to justify 4.97 to 2 d.p. or show there is a sign
Answer
Marks
Guidance
change in the interval (4.965, 4.975)
A1
Total:
3
Question
Answer
Marks
Question 4:
--- 4(i) ---
4(i) | Use the quotient or product rule | M1
Obtain correct derivative in any form | A1
Equate derivative to zero and obtain the given equation | A1
Total: | 3
--- 4(ii) ---
4(ii) | Sketch a relevant graph, e.g. y = ln x | B1
3
Sketch a second relevant graph, e.g. y = 1 + , and justify the given statement
x | B1
Total: | 2
--- 4(iii) ---
4(iii) | 3+x
Use iterative formula x = correctly at least once
n+1
lnx
n | M1
Obtain final answer 4.97 | A1
Show sufficient iterations to 4 d.p.to justify 4.97 to 2 d.p. or show there is a sign
change in the interval (4.965, 4.975) | A1
Total: | 3
Question | Answer | Marks
A curve with equation $y = \mathrm{f}(x)$ passes through the point $A(3, 1)$ and crosses the $y$-axis at $B$. It is given that $\mathrm{f}'(x) = (3x - 1)^{-\frac{1}{3}}$. Find the $y$-coordinate of $B$. [6]
\hfill \mbox{\textit{CAIE P3 2018 Q4 [6]}}