The equation of a curve is $$\( y=\mathrm{e}^{{-5x}} \mathrm{tan}^{2}{{x}} \)$$ for $$\( -\frac{1}{{2}} \)$$π < x < $$\( \frac{1}{{2}} \)$$π. Find the x-coordinates of the stationary points of the curve. Give your answers correct to 3 decimal places where appropriate ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ...............................................................................................................................................................
Exam No：9709_s21_qp_32 Year：2021 Question No：8
Answer:
Use correct product (or quotient) rule
Obtain \(\frac{\mathrm{d} y}{\mathrm{~d} x}=-5 \mathrm{e}^{-5 x} \tan ^{2} x+2 \mathrm{e}^{-5 x} \tan x \sec ^{2} x\)
Equate their derivative to zero, use \(\sec ^{2} x=1+\tan ^{2} x\) and obtain an equation in \(\tan x\)
Obtain \(2 \tan ^{2} x-5 \tan x+2=0\)
State answer \(x=0\)
Solve a 3 term quadratic in \(\tan x\) and obtain a value of \(x\)
Obtain answer, e.g. \(0.464\)
Obtain second non-zero answer, e.g. \(1.107\) and no other in the given interval
Alternative method for Question 8
Use correct product (or quotient) rule
Obtain \(\frac{\mathrm{d} y}{\mathrm{~d} x}=-5 \mathrm{e}^{-5 x} \tan ^{2} x+2 \mathrm{e}^{-5 x} \tan x \sec ^{2} x\)
Equate their derivative to zero and obtain an equation in \(\sin x\) and \(\cos x\)
Obtain \(5 \cos x \sin x=2\)
State answer \(x=0\)
Use double angle formula or square both sides and solve for \(x\)
Obtain answer, e.g. \(0.464\)
Obtain second non-zero answer, e.g. \(1.107\) and no other in the given interval
Obtain \(\frac{\mathrm{d} y}{\mathrm{~d} x}=-5 \mathrm{e}^{-5 x} \tan ^{2} x+2 \mathrm{e}^{-5 x} \tan x \sec ^{2} x\)
Equate their derivative to zero, use \(\sec ^{2} x=1+\tan ^{2} x\) and obtain an equation in \(\tan x\)
Obtain \(2 \tan ^{2} x-5 \tan x+2=0\)
State answer \(x=0\)
Solve a 3 term quadratic in \(\tan x\) and obtain a value of \(x\)
Obtain answer, e.g. \(0.464\)
Obtain second non-zero answer, e.g. \(1.107\) and no other in the given interval
Alternative method for Question 8
Use correct product (or quotient) rule
Obtain \(\frac{\mathrm{d} y}{\mathrm{~d} x}=-5 \mathrm{e}^{-5 x} \tan ^{2} x+2 \mathrm{e}^{-5 x} \tan x \sec ^{2} x\)
Equate their derivative to zero and obtain an equation in \(\sin x\) and \(\cos x\)
Obtain \(5 \cos x \sin x=2\)
State answer \(x=0\)
Use double angle formula or square both sides and solve for \(x\)
Obtain answer, e.g. \(0.464\)
Obtain second non-zero answer, e.g. \(1.107\) and no other in the given interval
Knowledge points:
3.3.1 understand the relationship of the secant, cosecant and cotangent functions to cosine, sine and tangent, and use properties and graphs of all six trigonometric functions for angles of any magnitude
3.3.2.1 more contents
3.3.2.2 the expansions of
3.3.2.3 the formulae for sin 2A and tan 2A
3.3.2.4 the expression of
3.4.1 use the derivatives of together with constant multiples, sums, differences and composites (Derivatives of are not required.)
3.4.2 differentiate products and quotients
Solution:
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