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The diagram shows part of the graph of $y = e^{-x^2}$ - AQA - A-Level Maths Pure - Question 7 - 2019 - Paper 3
Question 7
The diagram shows part of the graph of $y = e^{-x^2}$. The graph is formed from two convex sections, where the gradient is increasing, and one concave section, whe... show full transcript
Worked Solution & Example Answer:The diagram shows part of the graph of $y = e^{-x^2}$ - AQA - A-Level Maths Pure - Question 7 - 2019 - Paper 3
Step 1
Find the values of $x$ for which the graph is concave.
Answer
To find the concavity of the graph, we need to determine where the second derivative is negative.
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Begin by calculating the first derivative of the function:
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Next, calculate the second derivative employing the product rule:
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Set the second derivative to less than zero for concavity:
Since is always positive, we can focus on the inequality:
This simplifies to:
oot{2}} < x < rac{1}{ oot{2}}.$$
Hence, the graph is concave when - rac{1}{ oot{2}} < x < rac{1}{ oot{2}}.
Step 2
The finite region bounded by the $x$-axis and the lines $x = 0.1$ and $x = 0.5$ is shaded.
Answer
To estimate the area under the curve using the trapezium rule:
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Identify the function within the given bounds:
extstyle rac{1}{0.1}^{0.5} e^{-x^2}.
We will divide this interval into 4 strips. Each strip will have a width of:
h = rac{0.5 - 0.1}{4} = 0.1. -
Calculate the height at each strip endpoint:
- f(0.1) = e^{-0.01} \\ (0.2) = e^{-0.04} \\ f(0.3) = e^{-0.09} \\ f(0.4) = e^{-0.16} \\ f(0.5) = e^{-0.25}.
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Apply the trapezium rule formula:
ext{Area} = rac{h}{2} [f(a) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(b)]. -
Plug in the values to calculate the estimate:
ext{Area} = rac{0.1}{2} [f(0.1) + 2f(0.2) + 2f(0.3) + 2f(0.4) + f(0.5)].
This should yield an estimate of approximately 0.361 when evaluated.
Step 3
Explain with reference to your answer in part (a), why the answer you found in part (b) is an underestimate.
Answer
In part (a), it was established that the graph is concave on the interval - rac{1}{
oot{2}} < x < rac{1}{
oot{2}}.
Since the region from to is located within the interval where the gradient is decreasing, all trapezoids formed between these points will lie beneath the curve, contributing to an underestimation of the actual area above the x-axis.
Step 4
By considering the area of a rectangle, and using your answer to part (b), prove that the shaded area is 0.4 correct to 1 decimal place.
Answer
Define the rectangle with a height equal to and a width of .
This leads to the area of the rectangle being:
Given that , the area is:
We conclude that since our trapezium rule area estimate (approximately 0.361) indicates that the area must be below 0.4, it verifies that the shaded area is 0.4, correct to 1 decimal place.