Consider the differential equation $\dfrac{dy}{dx} = \dfrac{1}{2}\sin\!\left(\dfrac{\pi}{2}x\right)\sqrt{y+7}$. Let $y = f(x)$ be the particular solution to the differential equation with the initial condition $f(1) = 2$. The function $f$ is defined for all real numbers. (a) A portion of the slope field for the differential equation is given. Sketch the solution curve through the point $(1, 2)$. (b) Write an equation for the line tangent to the solution curve in part (a) at the point $(1, 2)$. Use the equation to approximate $f(0.8)$. (c) It is known that $f''(x) > 0$ for $-1 \leq x \leq 1$. Is the approximation found in part (b) an overestimate or an underestimate for $f(0.8)$? Give a reason for your answer. (d) Use separation of variables to find $y = f(x)$, the particular solution to the differential equation $\dfrac{dy}{dx} = \dfrac{1}{2}\sin\!\left(\dfrac{\pi}{2}x\right)\sqrt{y+7}$ with the initial condition $f(1) = 2$.
Consider the differential equation $\dfrac{dy}{dx} = \dfrac{1}{2}\sin\!\left(\dfrac{\pi}{2}x\right)\sqrt{y+7}$. Let $y = f(x)$ be the particular solution to the differential equation with the initial condition $f(1) = 2$. The function $f$ is defined for all real numbers.
(a) A portion of the slope field for the differential equation is given. Sketch the solution curve through the point $(1, 2)$.
(b) Write an equation for the line tangent to the solution curve in part (a) at the point $(1, 2)$. Use the equation to approximate $f(0.8)$.
(c) It is known that $f''(x) > 0$ for $-1 \leq x \leq 1$. Is the approximation found in part (b) an overestimate or an underestimate for $f(0.8)$? Give a reason for your answer.
(d) Use separation of variables to find $y = f(x)$, the particular solution to the differential equation $\dfrac{dy}{dx} = \dfrac{1}{2}\sin\!\left(\dfrac{\pi}{2}x\right)\sqrt{y+7}$ with the initial condition $f(1) = 2$.