Question
(a) Topic-7.3- Sketching Slope Fields
(b) Topic- 2.10- Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
(c) Topic- 5.6- Determining Concavity of Functions over Their Domains
(d) Topic-7.6- Finding General Solutions Using Separation of Variables
5. Consider the differential equation \(\frac{dy}{dx}=\frac{1}{2}sin\left ( \frac{\pi }{2}x \right )\sqrt{y+7}.\) 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 below. 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 \(\frac{dy}{dx}=\frac{1}{2}sin\left ( \frac{\pi }{2}x \right )\sqrt{y+7}\) with the initial condition f( 1) = 2.
▶️Answer/Explanation
5(a) A portion of the slope field for the differential equation is given below. Sketch the solution curve through the point (1, 2).
5(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).
\(\frac{dy}{dx}|_{(x,y) = (1,2)} = \frac{1}{2}.3 sin(\frac{\pi }{2}) = \frac{3}{2}\)
An equation for the tangent line is\(y=2+\frac{3}{2}(x-1)\).
\(f(0.8) \approx 2+\frac{3}{2}(0.8-1)=1.7\)
5(c) It is known that f ′′(x ) > 0 for −1 ≤ x ≤ 1. Is the approximation found in part (b) an overestimate or an underestimate for f (0.8) ? Give a reason for your answer. Because f ′′( x) > 0, f is concave up on −1 ≤ x ≤ 1, the tangent line lies below the graph of y = f (x ) at x = 0.8, and the approximation for f (0.8) is an underestimate.
5(d) Use separation of variables to find y = f ( x), the particular solution to the differential equation \(\frac{dy}{dx}=\frac{1}{2}sin(\frac{\pi }{2}x)\sqrt{y+7}\) with the initial condition f ( 1) = 2.
