The curve $y^{2} \sin x-2 x=9-\pi$ passes through $(\pi / 2,3)$ . Use local linearization to estimate the value of $y$ at $x=1.67$ . Use $1$ . $57$ for $\pi / 2$ . Round to $2$ d.p.

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Q. The curve

y^{2} \sin x-2 x=9-\pi

passes through

(\pi / 2,3)

. Use local linearization to estimate the value of

y

x=1.67

. Use

1

57

for

\pi / 2

. Round to

2

d.p.

Find Derivative: First, we need to find the derivative of the given curve with respect to $x$ . The curve is defined by the equation $y^2\sin(x) - 2x = 9 - \pi$ . We will differentiate both sides of the equation with respect to $x$ , treating $y$ as a function of $x$ ( $y=y(x)$ ).
Differentiate Equation: Differentiating $y^2\sin(x)$ with respect to $x$ using the product rule and chain rule gives us $2y\frac{dy}{dx}\sin(x) + y^2\cos(x)$ . Differentiating $-2x$ with respect to $x$ gives us $-2$ . The right side of the equation, $9 - \pi$ , is a constant, so its derivative is $0$ .
Set Derivative Equal: Setting the derivative equal to zero, we get $2y\frac{dy}{dx}\sin(x) + y^2\cos(x) - 2 = 0$ . Now we need to solve for $\frac{dy}{dx}$ .
Solve for $\frac{dy}{dx}$ : We plug in the point $(\frac{\pi}{2}, 3)$ into the derivative to find the slope of the tangent line at that point. Since $\sin(\frac{\pi}{2}) = 1$ and $\cos(\frac{\pi}{2}) = 0$ , the equation simplifies to $2\cdot3\cdot\frac{dy}{dx}\cdot1 + 3^2\cdot0 - 2 = 0$ , which simplifies to $6\frac{dy}{dx} - 2 = 0$ .
Substitute Point: Solving for $\frac{dy}{dx}$ , we get $\frac{dy}{dx} = \frac{2}{6} = \frac{1}{3}$ . This is the slope of the tangent line at the point $(\frac{\pi}{2}, 3)$ .
Write Tangent Line: The equation of the tangent line at $(\pi/2, 3)$ can be written in point-slope form: $y - 3 = (1/3)(x - \pi/2)$ . We will use this linear equation to estimate the value of $y$ at $x=1.67$ .
Estimate y Value: We substitute $x=1.67$ and rac{ ext{pi}}{2}=1.57 into the tangent line equation to estimate $y$ : y - 3 = rac{1}{3}(1.67 - 1.57). This simplifies to y - 3 = rac{1}{3}(0.1).
Calculate Result: Calculating the right side of the equation gives us $(\frac{1}{3})(0.1) = 0.0333$ (rounded to $4$ decimal places for intermediate calculation).
Final Answer: Adding $3$ to both sides of the equation to solve for $y$ gives us $y = 3 + 0.0333$ , which equals $3.0333$ .
Final Answer: Adding $3$ to both sides of the equation to solve for $y$ gives us $y = 3 + 0.0333$ , which equals $3.0333$ . Rounding the final answer to two decimal places, we get $y \approx 3.03$ .