When we are presented with a mathematical equation containing an unknown, the process of identifying this mystery element is referred to as solving for a constant, provided an initial condition is available. This initial condition is typically a recognized value or group of values that the resolution is required to comply with. Frequently utilized in the fields of integral calculus and differential equations, this method proves to be indispensable.
| Topic | Problem | Solution |
|---|---|---|
| None | Four million $E$. coli bacteria are present in a … | The given function is in the form of an exponential decay function. We can convert it to the form \… |
| None | Painful bone metastases are common in advanced pr… | We are given that the initial dose of radioactivity, \(Q_{0}\), is 4 mCi. |
| None | A hot bowl of soup is served at a dinner party. I… | The initial temperature of the food is given by the function \(T(t)\) at \(t=0\). We can substitute… |
| None | Solve the following initial value problem. \[ \fr… | We are given the second order differential equation \(\frac{d^{2} s}{d t^{2}}=-4 \sin \left(2 t-\fr… |
| None | Find $f$ such that $f^{\prime}(x)=x^{2}-6$ and $f… | We are given the derivative of the function $f(x)$, which is $f'(x) = x^2 - 6$. We also know that $… |
| None | Find the particular solution determined by the in… | This is a first order ordinary differential equation. The general solution of this differential equ… |
| None | Find $s(t)$, where $s(t)$ represents the position… | Given the acceleration function \(a(t) = -6t + 2\). |
| None | Find $f$ such that $f^{\prime}(x)=\frac{8}{\sqrt{… | The problem is asking for a function \(f(x)\) such that its derivative \(f'(x)\) is equal to \(\fra… |
| None | Find $f$ such that $f^{\prime}(x)=10 x-7, f(9)=0$ | Given the derivative of the function \(f'(x) = 10x - 7\) and a point on the function \((9,0)\). |
| None | A chemical substance has a decay rate of $6.2 \%$… | The rate of change of the amount of the substance is given by the differential equation \(\frac{dN}… |
| None | Find $f$ such that $f^{\prime}(x)=10 x-3, f(2)=0$… | The problem is asking for a function \(f(x)\) such that its derivative \(f'(x)\) is equal to \(10x … |
| None | Find the unique function $f(x)$ satisfying the fo… | We are given the derivative of the function $f(x)$ as $f'(x) = 3^x$ and a condition $f(0) = 4$. |
| None | Consider the function $f(x)$ whose second derivat… | Given that the second derivative of the function $f(x)$ is $f''(x) = 10x + 3\sin(x)$, we can find t… |
| None | Let $x y=4$ and $\frac{d y}{d t}=3$ Find $\frac{… | Given the equation \(xy = 4\), we can differentiate both sides with respect to \(t\) to get \(x\fra… |
| None | A particle is moving with the given data. Find th… | \(a(t) = t^2 - 5t + 3\) |
| None | (b) A $10 k L$ tank of water provides water for a… | Let the amount of heavy metals in the tank at time t be A(t) in mg/kL. The tank is 3/4 full, so it … |
| None | Question 4: If \( \mathrm{f}^{\prime \prime}(\mat… | \int 8\sin(4x) dx = 8 \int \sin(4x) dx |