Show that, for small values of $x$, the graph of
y = 5 + 4 rac{ ext{sin} rac{x}{2}}{2} + 12 ext{tan} rac{x}{3}
can be approximated by a straight line. - AQA - A-Level Maths Pure - Question 5 - 2018 - Paper 3
Question 5
Show that, for small values of $x$, the graph of
y = 5 + 4 rac{ ext{sin} rac{x}{2}}{2} + 12 ext{tan} rac{x}{3}
can be approximated by a straight line.
Worked Solution & Example Answer:Show that, for small values of $x$, the graph of
y = 5 + 4 rac{ ext{sin} rac{x}{2}}{2} + 12 ext{tan} rac{x}{3}
can be approximated by a straight line. - AQA - A-Level Maths Pure - Question 5 - 2018 - Paper 3
Step 1
Use small angle approximation for $ ext{sin} x$ and $ ext{tan} x$
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Answer
For small values of x, the small angle approximations can be applied:
ext{sin} rac{x}{2} hickapprox rac{x}{2}
and
ext{tan} rac{x}{3} hickapprox rac{x}{3}
Substituting these approximations into the equation gives:
y = 5 + 4 rac{ rac{x}{2}}{2} + 12 rac{x}{3}.
Step 2
Obtain correct equation
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Answer
Substituting the approximations into the original equation:
y & = 5 + 4 imes rac{x/2}{2} + 12 imes rac{x}{3} \\
& = 5 + 4 imes rac{x}{4} + 4x \\
& = 5 + x + 4x \\
& = 5 + 5x.
displaying the equation in the form:
y = 5x + 5.
Step 3
Conclude the graph can be approximated by a straight line
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Answer
The resulting equation is in the form:
y = mx + c,
where m=5 and c=5. Therefore, the graph of the function can be approximated by a straight line.
