Question

a. verify that the given point lies on the curve. b. determine an equation of the line tangent to the curve at the given point. 3 sin y + 5x = y²; (π²/5,π) does the point lie on the curve 3 sin y + 5x = y²? a. yes, because the values found for each side of the equation are equal. b. no, because the values found for each side of the equation are unequal. c. yes, because the value found for y² is greater than the value found for 3 sin y + 5x. d. no, because the value found for y² is less than the value found for 3 sin y + 5x. b. write the equation for the tangent line in slope - intercept form. select the correct choice and, if necessary, fill in the answer boxes to complete your choice. a. the point (π²/5,π) lies on the curve. the equation of the tangent line is y = (type exact answers, using π as needed.)

Explanation:

Step1: Verify the point on the curve

Substitute $x = \frac{\pi^{2}}{5}$ and $y=\pi$ into the equation $3\sin y + 5x=y^{2}$.
Left - hand side: $3\sin(\pi)+5\times\frac{\pi^{2}}{5}=3\times0 + \pi^{2}=\pi^{2}$.
Right - hand side: $y^{2}=\pi^{2}$. Since the left - hand side equals the right - hand side, the point lies on the curve.

Step2: Differentiate the equation implicitly

Differentiate $3\sin y+5x = y^{2}$ with respect to $x$.
Using the chain rule, the derivative of $3\sin y$ with respect to $x$ is $3\cos y\frac{dy}{dx}$, the derivative of $5x$ with respect to $x$ is $5$, and the derivative of $y^{2}$ with respect to $x$ is $2y\frac{dy}{dx}$.
So we have $3\cos y\frac{dy}{dx}+5 = 2y\frac{dy}{dx}$.

Step3: Solve for $\frac{dy}{dx}$

Rearrange the equation $3\cos y\frac{dy}{dx}+5 = 2y\frac{dy}{dx}$ to isolate $\frac{dy}{dx}$.
$3\cos y\frac{dy}{dx}-2y\frac{dy}{dx}=-5$.
Factor out $\frac{dy}{dx}$: $\frac{dy}{dx}(3\cos y - 2y)=-5$.
Then $\frac{dy}{dx}=\frac{-5}{3\cos y - 2y}$.

Step4: Find the slope at the given point

Substitute $y = \pi$ into $\frac{dy}{dx}=\frac{-5}{3\cos y - 2y}$.
$\cos(\pi)=-1$, so $\frac{dy}{dx}=\frac{-5}{3\times(-1)-2\pi}=\frac{5}{3 + 2\pi}$.

Step5: Find the equation of the tangent line

Use the point - slope form $y - y_{1}=m(x - x_{1})$ with $x_{1}=\frac{\pi^{2}}{5}$, $y_{1}=\pi$ and $m=\frac{5}{3 + 2\pi}$.
$y-\pi=\frac{5}{3 + 2\pi}(x-\frac{\pi^{2}}{5})$.
Expand and rewrite in slope - intercept form $y=mx + b$.
$y-\pi=\frac{5}{3 + 2\pi}x-\frac{\pi^{2}}{3 + 2\pi}$.
$y=\frac{5}{3 + 2\pi}x-\frac{\pi^{2}}{3 + 2\pi}+\pi=\frac{5}{3 + 2\pi}x+\frac{3\pi}{3 + 2\pi}$.

Answer:

a. A. Yes, because the values found for each side of the equation are equal.
b. A. The point $(\frac{\pi^{2}}{5},\pi)$ lies on the curve. The equation of the tangent line is $y=\frac{5}{3 + 2\pi}x+\frac{3\pi}{3 + 2\pi}$