Some example types:

1. Minimizing potential energy of any realistic physical system. Examples:

• 0D: The point where a rolling ball/flowing water might* come to rest (*might not, if momentum carries it past the low point).
• 1D: The curve described by a hanging chain/flexible rope (in equilibrium).
• 2D: The surface of a soap film (in equilibrium).
• Generally, anything that doesn't move or change: look at the world and wonder. [For instance, how hard do you have to push or jump to move your classroom building out of its potential well?)

1. Maximizing a utility function subject to a budget constraint.
2. Fermat's principle: Light takes the path of least time.
3. Machine learning: minimize the loss function of a particular learning task. (It and the subfield, data mining, are hot fields right now.)
4. In a network of resistors connecting voltage $V_A$ to voltage $V_B$, the voltages at the nodes of the network minimize power dissipation.
5. Space trajectories that minimize fuel use (Mission Design).
6. When using the central difference formula $[f(x+h)-f(x-h)]/(2h)$ to approximate the derivative $f'(x)$ using floating-point numbers with a machine epsilon of $\epsilon$, what choice of $h$ minimizes the relative error of the approximation? [Note: This is a somewhat ill-defined problem, since the formula is exact (no truncation error, only round-off error) if $f$ is a polynomial of degree at most $2$. One could pick $f$ a certain function and value of $x$; even so, the exact solution is hard to obtain.]

Some example types:

1. Minimizing potential energy of any realistic physical system. Examples:

• 0D: The point where a rolling ball/flowing water might* come to rest (*might not, if momentum carries it past the low point).
• 1D: The curve described by a hanging chain/flexible rope (in equilibrium).
• 2D: The surface of a soap film (in equilibrium).
• Generally, anything that doesn't move or change: look at the world and wonder. [For instance, how hard do you have to push or jump to move your classroom building out of its potential well?)

1. Maximizing a utility function subject to a budget constraint.
2. Fermat's principle: Light takes the path of least time.
3. Machine learning: minimize the loss function of a particular learning task. (It and the subfield, data mining, are hot fields right now.)
4. In a network of resistors connecting voltage $V_A$ to voltage $V_B$, the voltages at the nodes of the network minimize power dissipation.
5. Space trajectories that minimize fuel use (Mission Design).
6. When using the central difference formula $[f(x+h)-f(x-h)]/(2h)$ to approximate the derivative $f'(x)$ using floating-point numbers with a machine epsilon of $\epsilon$, what choice of $h$ minimizes the relative error of the approximation? [Note: This is a somewhat ill-defined problem, since it may depend on $f$ and $x$.]

Some examples:

1. Minimizing potential energy of any realistic physical system. Examples:

• 0D: The point where a rolling ball/flowing water might* come to rest (*might not, if momentum carries it past the low point).
• 1D: The curve described by a hanging chain/flexible rope (in equilibrium).
• 2D: The surface of a soap film (in equilibrium).
• Generally, anything that doesn't move or change: look at the world and wonder. [For instance, how hard do you have to push or jump to move your classroom building out of its potential well?)

1. Maximizing a utility function subject to a budget constraint.
2. Fermat's principle: Light takes the path of least time.
3. Machine learning: minimize the loss function of a particular learning task. (It and the subfield, data mining, are hot fields right now.)
4. In a network of resistors connecting voltage $V_A$ to voltage $V_B$, the voltages at the nodes of the network minimize power dissipation.
5. Space trajectories that minimize fuel use (Mission Design).

Some example types:

1. Minimizing potential energy of any realistic physical system. Examples:

• 0D: The point where a rolling ball/flowing water might* come to rest (*might not, if momentum carries it past the low point).
• 1D: The curve described by a hanging chain/flexible rope (in equilibrium).
• 2D: The surface of a soap film (in equilibrium).
• Generally, anything that doesn't move or change: look at the world and wonder. [For instance, how hard do you have to push or jump to move your classroom building out of its potential well?)

1. Maximizing a utility function subject to a budget constraint.
2. Fermat's principle: Light takes the path of least time.
3. Machine learning: minimize the loss function of a particular learning task. (It and the subfield, data mining, are hot fields right now.)
4. In a network of resistors connecting voltage $V_A$ to voltage $V_B$, the voltages at the nodes of the network minimize power dissipation.
5. Space trajectories that minimize fuel use (Mission Design).
6. When using the central difference formula $[f(x+h)-f(x-h)]/(2h)$ to approximate the derivative $f'(x)$ using floating-point numbers with a machine epsilon of $\epsilon$, what choice of $h$ minimizes the relative error of the approximation? [Note: This is a somewhat ill-defined problem, since the formula is exact (no truncation error, only round-off error) if $f$ is a polynomial of degree at most $2$. One could pick $f$ a certain function and value of $x$; even so, the exact solution is hard to obtain.]