Question

Let f be a function with f(2)=−8 such that for all points (x,y) on the graph of f, the slope is given by y 3x ​ . (a) Write an equation of the line tangent to the graph of f at the point where x=2 and use it to approximate f(1.8). (b) Find an expression for y=f(x) by solving the differential equation dx dy ​ = y 3x 2 ​ with the initial condition f(2)=−8.

Answer 1

To solve the student's problem, we define the slope of the tangent line using the given slope function and solve for the y-intercept to find the tangent line's equation. Next, to find an expression for f(x), we use separation of variables on the differential equation and apply the initial condition after integration.

Step-by-step explanation:

To write the equation of the line tangent to the graph of f at the point where x=2, we first determine the slope of the tangent line using the slope function given, which in this case would be the value of f(2), thus we have a slope of -8. The tangent line at x=2 will then have the form y = mx + b, where m is the slope and b is the y-intercept. We already have m = -8 and since the point (2, -8) lies on the tangent line, we can solve for b, giving us the equation of the tangent as y = -8x + b. To approximate f(1.8), you would then substitute x with 1.8 in the equation and solve for y.

To solve the differential equation dy/dx = y/(3x^2) with the initial condition f(2)=-8, you would perform separation of variables and then integrate both sides of the equation. You would eventually arrive at a general solution for f(x), and apply the initial condition to find the constant of integration and thus find the specific solution for f(x).