4. (a) Using calculus, find the exact coordinates of the turning points on the curve with equation $y = f(x)$ - Edexcel - A-Level Maths Pure - Question 6 - 2013 - Paper 7
Question 6
4.
(a) Using calculus, find the exact coordinates of the turning points on the curve with equation $y = f(x)$.
(b) Show that the equation $f(x) = 0$ can be written ... show full transcript
Worked Solution & Example Answer:4. (a) Using calculus, find the exact coordinates of the turning points on the curve with equation $y = f(x)$ - Edexcel - A-Level Maths Pure - Question 6 - 2013 - Paper 7
Step 1
Using calculus, find the exact coordinates of the turning points on the curve with equation $y = f(x)$.
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Answer
To find the turning points of the function, we first need to determine the critical points by finding the derivative of the function and setting it to zero:
Compute the first derivative:
f′(x)=50x2ex+25x2ex.
Set the first derivative equal to zero:
50x2ex+25x2ex=0.
Factor out the common terms:
25x2ex(2+1)=0.
Solve for x:
The factor 25x2ex=0 gives x=0.
Substitute x back into the original function:
f(0)=25(0)2e0−16=−16.
Thus, the turning point is at (0,−16).
Step 2
Show that the equation $f(x) = 0$ can be written as $x = \frac{4}{5} e^{x}$.
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Answer
Start with the equation:
25x2ex−16=0
Rearranging gives:
25x2ex=16
Dividing both sides by 25 yields:
x2ex=2516
Taking square roots results in:
xex/2=54
Thus, we can express it as:
x=54ex.
Step 3
Starting with $x_0 = 0.5$, use the iteration formula $x_{n+1} = \frac{4}{5} e^{x_n}$ to calculate the values of $x_1$, $x_2$, and $x_3$, giving your answers to 3 decimal places.
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Answer
Substituting x0=0.5 into the iteration formula:
Calculate x1:
x1=54e0.5≈0.485.
Calculate x2:
x2=54e0.485≈0.492.
Calculate x3:
x3=54e0.492≈0.489.
Thus, we have:
x1≈0.485
x2≈0.492
x3≈0.489.
Step 4
Give an accurate estimate for $\alpha$ to 2 decimal places, and justify your answer.
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Answer
By observing the values calculated previously:
From the iterations, we note that x2≈0.492 and x3≈0.489 are converging.
The root appears close to 0.49.
Thus, the accurate estimate for α to two decimal places is:
α≈0.49.
Justification can be given since x3 is stable around this value.
