Question

Find the function \(y=f(t)\) passing through the point (0,10) with the given first derivative. Use a graphing utility to graph the solution. $$ \frac{d y}{d t}=-\frac{3}{4} \sqrt{t} $$

Short answer

The function \(y=f(t)\) that passes through the point (0,10) with the given first derivative is \(y = -\frac{1}{2} t^{3/2} + 10\).

Step-by-step solution

Find the Antiderivative

The antiderivative of \(-\frac{3}{4} \sqrt{t}\) is found using the power rule of integration. The power rule states that the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration. Here, rewrite \(\sqrt{t}\) as \(t^{1/2}\). So, the antiderivative becomes \[ -\frac{3}{4} \int t^{1/2} dt = -\frac{3}{4} \cdot \frac{2}{3} t^{3/2} + C = -\frac{1}{2} t^{3/2} + C\]

Find the constant of integration

Substitute the given point (0,10) into the function to find the constant \(C\). \[10 = -\frac{1}{2} \cdot 0^{3/2} + C \implies C = 10\]

Write the Final Function

Substitute \(C\) into the function from step 1, yielding the final function \(y = f(t) = -\frac{1}{2} t^{3/2} + 10\).

Graph the Function

The function \(f(t) = -\frac{1}{2} t^{3/2} + 10\) is graphed using a graphing utility. Our domain is limited to \(t \geq 0\), because \(t\) is in a square root and must be nonnegative.