A construction firm is manufacturing precast concrete panels. The total cost function $C(x)$ describes the cost to produce $x$ panels. The firm notices that as production scales up, bulk material discounts reduce the average cost per panel, but eventually, overtime pay and machine maintenance cause costs to spike. To maximize profit, the firm determines the Marginal Revenue $R'(x)$ (the revenue from selling one additional panel) and sets it equal to the Marginal Cost $C'(x)$ (the cost to produce that specific panel). By finding the exact production level $x$ where $R'(x) = C'(x)$, the firm ensures they are not producing panels that cost more to make than they sell for, thus mathematically guaranteeing maximum profit.

When designing a horizontal curve for a highway, engineers must ensure drivers have adequate stopping sight distance. If a large retaining wall is built on the inside of the curve, it creates a visual obstruction. By modeling the highway curve and the line of sight as mathematical functions, engineers use Tangent Lines to determine exactly where a driver's line of sight intersects the obstruction. The slope of the tangent line (the derivative) dictates the required setback distance for the wall, ensuring the highway design complies with safety regulations.

Find the equations of the tangent and normal lines to the curve $y = x^2 - 4x + 3$ at $x = 3$.

Find the angle of intersection between the curves $y = x^2$ and $y = 2 - x^2$ in the first quadrant.

Find the equation of the tangent line to the curve $y = e^{2x}$ that is parallel to the line $y = 4x + 1$.

A particle moves along a line according to the position function $s(t) = t^3 - 6t^2 + 9t$. Find the velocity and acceleration at $t=2$.

For the particle in the previous example, when is the particle at rest, and when is it moving forward?

A company estimates that the cost (in dollars) of producing $x$ units of a product is $C(x) = 10000 + 5x + 0.01x^2$. Find the marginal cost of producing the 501st unit.

A manufacturer's revenue function is $R(x) = 50x - 0.5x^2$ and the cost function is $C(x) = 10x + 3$. Find the production level that maximizes profit.

A 10-meter ladder leans against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 m/s, how fast is the top of the ladder sliding down when the bottom is 6 meters from the wall?

Air is being pumped into a spherical balloon so that its volume increases at a rate of $100 \text{ cm}^3/\text{s}$. How fast is the radius increasing when the diameter is 50 cm?

A conical water tank with vertex down has a radius of 10 ft at the top and is 24 ft high. If water flows into the tank at a rate of $20 \text{ ft}^3/\text{min}$, how fast is the depth of the water increasing when the water is 16 ft deep?

A farmer wants to fence a rectangular area of $1000 \text{ m}^2$ using an existing wall as one side. Minimize the amount of fencing needed.

In open-channel hydraulics, the most efficient cross-section for a trapezoidal canal is a half-hexagon. Design a trapezoidal canal with side slopes of $60^\circ$ that must have a cross-sectional area of $50 \text{ m}^2$. Minimize the wetted perimeter $P$.

Find the intervals of concavity and the inflection points of $f(x) = x^3 - 6x^2 + 9x + 1$.

Sketch the curve of $f(x) = x^4 - 4x^3$ by finding critical points and intervals of increasing/decreasing.

Evaluate the limit: $\lim_{x \to 0} \frac{\sin x - x}{x^3}$

Evaluate the limit involving an exponential: $\lim_{x \to \infty} \frac{e^x}{x^2}$.

Evaluate the limit using L'Hopital's Rule for a product indeterminate form: $\lim_{x \to 0^+} x \ln x$.