Initial Value Problems

Find the curve y = f(x...

Question

Initial Value Problems

Find the curve y = f(x) in the xy-plane that passes through the point (9,4) and whose slope at each point is 3√x.

Accepted Answer

Welcome back everyone. Find the function y equals fx that passes through 0-1 and whose slope at each point is 4 e to the power of 2x. For this problem, let's remember that the slope of the function is basically the derivative of that function. So we can show that fx is equal to 4 e to the power of 2x. We can now find f x by integrating both sides. So that f of x is equal to the integral of 4 e. To the power of 2 x d x. And now we can integrate, right, so f of x is going to be the integral of e to the power of 2 x dx multiplied by 4. We can factor out the constant. Now from here, we are ready to integrate and we can simply use use substitution. Let's suppose that U is 2 X. The deriv a default view with respect to X is going to be 2. And therefore we can show that d x is equal to 1/2 the u. So F of X. is going to be equal to 4 multiplied by the integral of e to the power of u multiplied by dx, which is 1/2 duu. So we can write duu and we can factor out 1/2. So 4 divided by 2 gives us 2, right? Well then, now we have an elementary function to integrate. So we get 2. The integral of e to the power of u is e to the power of u, and we're going to add the constant of integration c because that's an indefinite integral. And now replacing u with 2 x, we get 2 e to the power of 2 x plus c. Our final step is to find C, and we know that F of X. is equal to -1 when x is equal to 0, right? So what we can do is simply set our left hand side equal to -1, and on the right hand side we will replace x with 0. So we get 2 e to the power of 2 multiplied by 0. That's 0 plus c. So -1 equals 2 multiplied by 1, or basically 2 plus c, and therefore c is equal to -1 minus 2, which is -3. Well then, so now we can show that fx is equal to 2 e to the power of 2 x minus 3, which is our final answer for this problem. Thank you for watching.

Initial Value ProblemsFind the curve y = f(x) in the xy-plane that passes through the point (9,4) and whose slope at each point is 3√x.

Watch next

Initial Value Problems

Find the curve y = f(x) in the xy-plane that passes through the point (9,4) and whose slope at each point is 3√x.