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1 of 8 0-day streak

Locate and bisect a circular-equation root

DIFFICULTYintrostandardchallengeexam
QUESTION TYPEMix of allChange
sin with a constant rule, table evidence, positive intervalsin with a constant rule, table evidence, negative intervalsin with a constant rule, monotonicity evidence, positive intervalsin with a constant rule, monotonicity evidence, negative intervalsin with a linear rule, table evidence, positive intervalsin with a linear rule, table evidence, negative intervalsin with a linear rule, monotonicity evidence, positive intervalsin with a linear rule, monotonicity evidence, negative intervalcos with a constant rule, table evidence, positive intervalcos with a constant rule, table evidence, negative intervalcos with a constant rule, monotonicity evidence, positive intervalcos with a constant rule, monotonicity evidence, negative intervalcos with a linear rule, table evidence, positive intervalcos with a linear rule, table evidence, negative intervalcos with a linear rule, monotonicity evidence, positive intervalcos with a linear rule, monotonicity evidence, negative intervalNOWMix of allRecommended
Question 1 of 8

Calculator allowed

The following circular equation is to be solved numerically

\[\sin x=-0.25x - 0.778\]

Write the algebraic function in the equation as \(P(x)=-0.25x - 0.778\)

Let \(F(x)\) be the circular side minus the algebraic side, so \(F(x)=\sin x-\left(-0.25x - 0.778\right)\)

The sign-changing interval under consideration is \(\left[-\frac{9}{10},-\frac{3}{10}\right]\). The table also gives one point just outside each endpoint

The displayed function values are rounded to 3 decimal places; their signs are retained
\(x\)\(F(x)\)
\(-\frac{21}{20}\)\(\approx -0.352\)
\(-\frac{9}{10}\)\(\approx -0.230\)
\(-\frac{3}{10}\)\(\approx 0.407\)
\(-\frac{3}{20}\)\(\approx 0.591\)

On this interval, \(F\) is strictly increasing

Select the two functions, the justified root count and the interval containing the sign change

(a)1 mark

Which pair of functions is being compared in the equation?

(b)1 mark

What root-count conclusion is justified by the supplied evidence?

(c)1 mark

Which interval has an endpoint sign change?