Question
A curve has equation \(y=x^3e^{0.2x}\) where \(x\geq0\). At the point P on the curve, the gradient of the curve is 15.
(a) Show that the x-coordinates of P satisfies the equation \(x=\sqrt{75e^{0.2x}}{15+x}\).
(b) Use the equation in part (a) to show by calculation that the x-coordinate of P lies between 1.7 and 1.8
(c) Use an iterative formula, based on the equation in part (a), to find the x-coordinate of P correct to 4 significant figures. Gives the result of each iteration to 6 significant figures.
Answer/Explanation
Ans:
(a) Differentiate using the product rule
(b) Obtain \(3x^2e^{0.2x}+0.2x^3e^{0.2x}\)
Equate first derivative to 15 and rearrange to x = …..
Confirm \(x=\sqrt{\frac{75e^{-0.2x}}{15+x}}\)
(b) Consider sign of \(x-\sqrt{\frac{75e^{-0.2x}}{15+x}}\) or equivalent for 1.7 and 1.8
Obtain -0.08…. and 0.03….. or equivalent for 1.7 and 1.8
Obtain -0.08… and 0.03… or equivalents and justify conclusion
(c) Use iterative process correctly at least once
Obtain final answer 1.771
Show sufficient iterations to 6 sf to justify answer or show a sign change in the interval [1.7705, 1.7715]
Question
(a) Given that 2 In(x+1) + In x = In(x+9), show that \(x=\sqrt{\frac{9}{x+2}}\).
(b) It is given that the equation \(x=\sqrt{\frac{9}{x+2}}\) has single root
Show by calculation that this root lies between 1.5 and 2.0
(c) Use an iterative formula, based on the equation in part (b), to find the root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
Answer/Explanation
Ans:
- Use the power law correctly
Use correct process to obtain equation with no logarithms
Confirm \(x=\sqrt{\frac{9}{x+2}}\) - Consider sign of \(x-\sqrt{\frac{9}{x+2}}\)
Obtain -0.1…. and 0.5 or equivalents and justify conclusion - Use iteration process correctly at least once
Obtain final answer 1.58
Show sufficient iterations to 5 sf to justify answer or show a sign change in interval [1.575, 1.585]
Question
The diagram shows part of the curve with equation \(y=\frac{5x}{4x^3+1}\). The shaded region is bounded by the curve and the lines x=1, x=3 and y=0.
(a) Find \(\frac{dy}{dx}\) and hence find the the x-coordinate of the maximum point.
(b) Use the trapezium rule with two intervals to find an approximation to the area of the shaded region. Give your answer correct to 2 significant figures.
(c) State, with a reason, whether your answer to part (b) is an over-estimate or under-estimate of the exact area of the shaded region.
Answer/Explanation
Ans:
- Differentiate using quotient rule (or product rule)
Obtain \(\frac{5(4x^3+1)-60x^3}{(4x^3+1)^2}\)
Equate first derivative to zero and attempt solution
Obtain \(x=\frac{1}{2}\) - Use y values \(\frac{5}{5}, \frac{10}{33}, \frac{15}{109}\) or decimal equavalents
Use correct formula, or equivalent, with h=1
Obtain \(\frac{1}{2}(1+\frac{20}{30}+\frac{15}{109})\) or equivalent and hence 0.87 - State over-estimate with reference to top of each trapezium above curve
Question
The sequence of values given by the iterative formula \(x_{n+1}=\frac{6+8x_{n}}{8+x^{2}_{n}}\) with initial value x1 = 2 converges
to \(\alpha \)
(a) Use the iterative formula to find the value of ! correct to 4 significant figures. Give the result of
each iteration to 6 significant figures. [3]
(b) State an equation satisfied by \(\alpha \) and hence determine the exact value of \(\alpha \) [2]
Answer/Explanation
Ans
(a) Use iteration correctly at least once
Obtain final answer 1.817 A1 Answer required to exactly 4 significant figures
Show sufficient iterations to 6 significant figures to justify answer
or show sign change in interval [1.8165, 1.8175]
(b) State equation \(x=\frac{6+8X}{8+x^{2}}\ or \ equivalent \ using \ \alpha \)
Obtain \(\sqrt[3]{6}\) or exact equivalent